IV. STATICS AND DYNAMICS OF DEFORMABLE MEDIA.

Transcription

1 V. TT ND DYNM OF DEFOMLE MED. 48. Defoabe eu. Naua sae a efoe sae. The heoes of he efoabe e a he efoabe suface ha we scusse ea a ve aua ae o evsog a oe geea efoabe eu ha he oe ha s habua cosee he heo of easc a sees o us o acheve he goa ha was usue b LOD ELVN a HELMHOLTZ he heoes of gh a ages. ose a sace M ha s escbe b a o M whose cooaes wh esec o hee fe ecagua aes O O O. e a ega hese cooaes as fucos of he hee aaees whch ae chose a aba ae; howeve o sf we suose ha hese cooaes ae ake o be eee vaabes. ff a -ecagua a o each o M of he sace M whose aes M M M have eco coses ; ; wh esec o he aes O O O a whch ae fucos of he eee vaabes. The couous hee-esoa se of a such as M w be wha we ca a efoabe eu. Gve a sacee M M o a o M ; e be he cooaes of he o M wh esec o he fe a O. ao eow he a M wh a oao ha w uae bg s aes o ageee wh hose of a a M ha we aff o he o M. e efe ha oao b gvg he eco coses ; ; of he aes M M M wh esec o he fe aes. The couous hee-esoa se of a such as M w be wha we ca he efoe sae of he efoabe eu ue coseao whch w be cae he aua sae s oga sae. 49. eaca eees ha eae o he saes of he efoabe eu. Fo ease of oao we soees ouce he ees sea of he seue as eesse b he fouas: so w suffce o kee he. Deoe he cooes of he veoc of he og M of he aes M M M wh esec o hese aes b whe aoe vaes a as he oe of e. Lkewse e be he oecos o hese aes of he saaeous oao of he a M eave o he aaee. e eoe he aaogous uaes fo he a M b ζ a whe he ke he a M ae efee o he fe a O. The eees ha we ouce befoe ae cacuae he usua fasho; acua oe has:

2 THE DEFOMLE MEDUM The ea eee of he efoe eu M whe efee o he eee vaabes s efe b he foua: s ε ε ε whch ε ε ε ae cacuae b he foowg oube fouas: 45. ε ε ε Deoe he oecos of he sege OM oo he aes M M M b such a wa ha he cooaes of he fe o O wh esec o hese aes becoe. e have he foowg we-kow fouas: 46 whch gves ew eessos fo ζ.

3 4 THEOY OF DEFOMLE MED 5. Eessos fo he vaaos of he veoces of asao a oao of he a eave o he efoe sae. uose ha oe eows each of he as of he efoe sae wh a fe sa sacee ha a va a couous fasho wh hese as. Deoe he vaaos of ; ; b ; ; esecve. The vaaos ae eesse b fouas such as he foowg: 47 b eas of he hee aua fucos whch ae he cooes of wekow saaeous oao ha s aache o he fe sa sacee ueso wh esec o M M M. The vaaos ae he oecos of he fe sa sacee fe b he o M oo O O O. The oecos of hs sacee oo M M M ae euce eae a have he vaues: 48. e oose o eee he vaaos ζ fe b ζ esecve. Fo he fouas 44 we have:. eace b s vaue a wh he aaogous vaues; we oba: 49. a fouas 46 gve us hee fouas he fs of whch s:. eace wh he vaues as gve b fouas 49; we oba:

4 THE DEFOMLE MEDUM 4 5 whch we have ouce he hee sbos efe b fouas Euca aco of efoao o a efoabe eu. e eseve he oaos of sec. 49 a ouce he kow ua whch s efe b he foua: D D a whose suae whch s foe b he ue fo ucao of eeas s eesse as a fuco of ε ε ε b he foua:. ε ε ε ose a fuco of wo fe cose osos of he a M.e. a fuco fo o a he fs evaves wh esec o. e oose o eee he fo ha us ake oe fo he ega: whe ake ove a aba oo of he sace M o have u vaao whe oe subecs he se of a as of he efoabe eu ake s efoe sae o he sae aba fesa asfoao of he gou of Euca sacees. efo hs aous o eeg such a wa ha oe has:

5 4 THEOY OF DEFOMLE MED whe o he oe ha he og M of he a M s subece o a fe sa sacee whose oecos o he aes O O O ae: 5 a a a ω ω ω ω ω ω whee a a a ω ω ω ae s aba cosas a s a fe sa ua ha s eee of a whe o he ohe ha he a M s subece o a fe sa oao whose cooes aog he aes O O O ae: ω ω ω. Obseve ha he ese case he vaaos ζ ; of he eghee eessos ζ ; ae u sce hs esus fo he we-kow heo of ovg faes a as we a oeove vef eae b eas of fouas 49 a 5 b eacg ; b he acua vaues. esus fo hs ha we oba a souo o he ueso b akg o be a aba fuco of a he eghee eessos ζ ;. e sha ow show ha we hus oba he geea souo of a obe ha we ow ose. To ha effec we eak ha he eaos 44 e us o eess he fs evaves of he e coses wh esec o b eas of hese coses a usg we-kow fouas. O he ohe ha fouas 4 e us o hk of eessg he e coses b eas of ζ a he fs evaves of wh esec o o b eas of ζ a he fs evaves of wh esec o o fa b eas of ζ a he fs evaves of wh esec o. Fuheoe s useess hs case fo us o ake a hohess o he oe of souo because s cea ha we w o oba a oe geea fo ha he oe ha we sae wh b suosg ha he fuco ha we seek s a aba fuco of a a he fs evaves wh esec o a of ζ ; whch we cae b usg he oaos b wg:. ce he vaaos ζ ; ae o-u he acua case oe eaks ha hee s a sa whch we sha uae escbe fo whch we have b vue of fouas 5 he ew fo fo fo a a a a ω ω ω : a of wha foows we suose ha he eu s suscebe o a ossbe efoaos so ha as a esu he efoe sae a be ake absoue aba.

6 THE DEFOMLE MEDUM 4. e eace wh he vaues 5 a wh he vaues ha oe euces b ffeeao. e se he coeffces of a a a ω ω ω ; we oba he foowg s coos: whch ae ees f we assue ha he eessos ha fgue have bee euce o he saes ube. The fs hee show us as oe a eas foesee ha s eee of. The as hee eess ha ees o he fs evaves of wh esec o o b he eea of he uaes ε ε ε ha wee efe b he fouas 45. Fa we see ha he ese fuco has he eakabe fo: ζ ; whch s aaogous o he oe ha we ecouee befoe fo he efoabe e a he efoabe suface. f we u b he voue eee of he sace M he he ouc so obae s a vaa he gou of Euca sacees ha s aaogous o he voue eee of he eu M. us as he coo vaue of he egas: ake ove he eo of a suface of he eu M a he eo of he coesog suface of he eu M esecve eees he voue of he

7 44 THEOY OF DEFOMLE MED oa boue b he suface. Lkewse f we assocae he sae s he oo of he aco fo he assage fo he aua sae M o he efoe sae M he we a he fuco o he eees he efo of a efoabe eu a we sa ha he ega: s he aco of efoao fo he eo of he suface he efoe eu. O he ohe ha we sa ha s he es of he aco of efoao a a o of he efoe eu whe efee o he u of voue of he uefoe eu a ha s he es of ha aco a a o whe efee o he u of voue of he efoe eu. 5. The eea foce a oe. The eea oe a effo. The effo a oe of efoao a a o of he efoe eu. ose a aba vaao of he aco of efoao of he eo of a suface he eu M ae:. vue of fouas 49 a 5 of sec. 5 we a we:. e a he GEEN foua o he es ha ec efe o he evave wh esec o oe of he vaabes. f we e eoe he eco coses wh esec o O O O of he eeo oa o he suface ha bous he eu befoe efoao a he aea eee of ha suface b he hs gves:

8 THE DEFOMLE MEDUM 45. e: F G H X Y

9 46 THEOY OF DEFOMLE MED Z L M N we have: X Y Z L M N F G H. f we fs ec ou aeo o he e ega ha fgues he eesso fo he we ca he seges ha have he og a M a whose oecos oo he aes M M M ae X Y Z a L M N esecve he eea foce a eea oe a he o M efee o he u of voue of he uefoe eu. Ne ecg ou aeo o he suface ega ha fgues : we ca he seges ha ssue fo he o M a have oecos F G H a o he aes M M M esecve he eea effo a eea oe of efoao a he o M of he suface ha bous he eu efee o he u of aea of he suface. a efe o M of hese as s uaes ee o o he eco of he eeo oa o he suface. The ea vaa f he ego ueso s vae a he eco of he eeo oa oes o chage bu he chage sg f hs eco s eace b he oose eco. uose ha oe aces a suface Σ he eo of he efoe eu ha s boue b he suface such a wa ha Σ ogehe wh a oo of suface uue ccuscbes a subse of he eu a e eoe he es of he eu ouse of he subse. Le Σ be he suface of M ha coesos o he suface of M a e a be he egos of M ha coeso o he egos a of M. Mea seaae he wo subses a. Oe a ega he wo seges F G H a ha ae eee b he o M a he eco of he oa o Σ ha os owas he eeo of as he eea effo a oe of efoao a he o M of he foe Σ of he

10 THE DEFOMLE MEDUM 47 ego. a oe a ega he wo seges F G H a as he eea effo a oe of efoao a he o M of he foe Σ of he ego. easo of ha eak we sa ha F G H a ae he cooes wh esec o he aes M M M of he effo a oe of efoao ha ae eee a M o he oo of he eu M a ha F G H a ae he cooes wh esec o he aes M M Mof he effo a oe of efoao ha ae eee a M o he oo of he eu M. The obsevao ae a he e of secs. 9 a 4 o he subec of eacg he a M b a a ha s vaa eae o a be eeae hee whou ofcao. 5. Vaous was of secfg he effo a oe of efoao. e:. a eese he oecos oo M M M of he effo a oe of efoao esecve ha ae eee a he o M o a suface ha has a eo oa a he o M ha s aae o he cooae as O O O ha coesos o he e befoe efoao. ee suffces o eca ha oe has aea agee o eace he ees whch coeso b hs oao o he ces esecve wh. f ou eca ha effo a oe of efoao ae efee o he u of aea of he uefoe suface. The ew effos a oes of efoao ha we efe ae eae o he eees ouce he eceg seco b he foowg eaos: F G H X Y Z

11 48 THEOY OF DEFOMLE MED L M. N e oose o asfo hese eaos o oes ha ae eee of he vaues of he uaes ha we cacuae b eas of ha fgue he. ee hese eaos ea o he seges ha ae aache o he o M o whch we gave he aes. sea of efg hese seges b he oecos o M M M we a efe he b he oecos o he ohe aes; he ae oecos w be coue b eaos ha ae asfos of he eceg oes. Moeove he asfoe eaos ae obae eae f oe eaks ha he oga fouas have se a eae eeaos b he auco o hese ovg aes of aes ha ae aae o he a he o O.. e cofe ouseves o he coseao of fe aes O O O. Deoe he oecos of he eea foce a eea oe a a aba o M of he efoe eu oo hese aes b X Y Z a L M N esecve a he oecos of effo a oe of efoao o a suface whose eo oa has he eco coses befoe efoao b F G H a esecve. The oecos of he effo a he oe of efoao ae eoe b a esecve. The asfos of he eceg eaos ae obvous: F G H X Y Z L eesg eeao o oe s he aaog wh he oe gve b. NT-GULHEM he coe of he acs of as.

12 THE DEFOMLE MEDUM 49 M N eaos ha ae he hee-esoa geeaaos of he wo-esoa euaos of LOD ELVN a TT.. Now obseve ha we a eess he e coses b eas of hee aua fucos; e be hee such aua fucos. e: ϖ ϖ ϖ. The fucos ϖ of so efe sasf he eaos: ϖ ϖ ϖ ϖ. ϖ ϖ a oe has: ϖ ϖ ϖ o. Le ϖ eoe he oecos oo he fe aes O O O of he sege whose oecos oo he aes M M M ae ; ϖ we have: ϖ ϖ ϖ

13 5 THEOY OF DEFOMLE MED b vue of whch he ew fucos ϖ of sasf he eaos: ϖ ϖ ϖ ϖ ϖ ϖ.. ga we ake he eak whch w be of use ae o ha f oe es eoe he vaaos of ha coeso o he vaaos of he oe w have: ϖ ϖ ϖ ϖ ϖ ϖ whch ae he oecos oo he fe aes of he sege whose oecos oo M M Mae. Now se: ϖ ϖ ϖ ϖ ϖ ϖ L ϖ L M N ϖl M N M ϖ L M N ϖl M N N ϖ L M N ϖ L M N. ao we ouce he foowg oaos: Π ϖ ϖ These fouas a seve o efe he fucos ϖ ec a he subsuo s efe b: ϖ ϖ ϖ ϖ.

14 THE DEFOMLE MEDUM 5 Χ ϖ ϖ Σ ϖ ϖ he sea of he ae sse whch ehe o fgue we have he foowg: Π ϖ L ϖ ϖ ϖ ϖ wh wo aaogous euaos. f oe eaks ha he fucos ζ of gve se o he fouas: ϖ ϖ ϖ ϖ ϖ ha esu fo he efg eaos of he fucos ϖ a he e ees ha he vef he oe a gve he eceg sse he ew fo: Π L wh wo aaogous euaos.. The eceg euaos ha we ouce aso cosue he geeaao of he oes we eveoe a eae wok. e a asfo he such a wa as o oba he geeaao of he we-kow euaos of he heo of easc ha eae o effo. To ha effec w suffce o eouce he eho we aea eoe he wok ha we eoe. To abbevae he wg e X Y Z al M N eoe fo he oe he ef-ha ses of he asfoao eaos whch efe o X Y Z L M N esecve a obseve ha oe a suae he weve eaos ha we esabshe b he foowg: E. a F. OET. ee éoe su a héoe e éascé; aes e a Facué es sceces e Tououse

15 5 THEOY OF DEFOMLE MED µ µ µ X Y Z L M N { G F µ H } µ µ whch µ µ µ ae aba fucos a he egas ae ake ove he suface of he eu M a he oa boue b. f we a GEEN foua he he eao ha we woe becoes he foowg oe: N M L Z Y X µ µ µ µ µ µ H G F µ µ µ µ µ µ µ µ µ µ µ. µ e seek he asfo of hs ae eao whe oe akes he fucos of fo he ew vaabes. f oe es ϕ eoe a aba fuco of ha becoes a fuco of he he eeea fouas fo he chage of vaabes ae: ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ.. ϕ ϕ ϕ ϕ

16 THE DEFOMLE MEDUM 5 hese fouas o he fucos µ µ µ. h awas eog he suface of he eu M ha coesos o he suface of M we fuhe eoe he oecos oo O O O of he eea foce a eea oe ae o he o M b X Y Z L M N whch ae efee o he u of voue of he efoe eu M a he oeco oo O O O of he effo a he oe of efoao ha ae eee a he o M of b F G H efee o he u of aea o. Fa ouce he eghee ew aua fucos b he fouas: a he aaogous oes ha ae obae b eacg: wh: a he b: esecve. e oba he asfoe eao: N M L Z Y X µ µ µ µ µ µ H G F µ µ µ µ µ µ µ µ µ { } µ µ µ whch he egas ae ake ove he suface of he eu M a he oa boue b wh esgag he aea eee of.

17 54 THEOY OF DEFOMLE MED Oce oe a GEEN foua o he es ha efe o he evaves of µ µ µ wh esec o a e eoe he eco coses of he eeo oa o he suface wh esec o he fe aes. ce µ µ µ ae aba he becoe: F G H X Y Z L M N. The sgfcace of he eghee ew aua fucos esus eae fo he eaos ha we us fou. ee s cea ha he coeffces of he eessos fo F G H eese he oecos oo O O O of he effo ha s eee a he o M o he suface whose eeo oa s aae o O a ha he coeffces of he eessos fo ae he oecos oo O O O of he oe of efoao a M eave o he sae suface. The coeffces of a of gve se o a aaogous eeao ega o sufaces whose eo oas ae aae o O a O. The aua fucos ha we us ouce a he euaos ha eae he o o aea o have bee evsoe a fo ha was ha geea u ow; o ou kowege he have bee cosee o he acua case whch he e uaes ae u a he fs wok o ea ha ueso sees o be ha of VOGT. LDEM VOGT. Theoesche ue übe e Eascäsvehässe e sae bhauge e kögche Geseschaf e sseschafe u Göge The fs seco ee: beug e Gugechuge aus e ahe oaä begabe Moeküe has 49 ages -5 he seco oe ee: Uesuchug es eassche Vehaes ees es aus ksasche ubsa auf esse Maefäche kee äfe wke we see e wkee auge ägs e eae cosa s s 48 ages 5-. Oe a kewse cosu he wok of VOGT: L Éa acue e os coassaces su éascé es csau eo esee a he eaoa ogess of hscs covee as 9 T whch he aues o

18 THE DEFOMLE MEDUM 55 cocuso we obseve ha f oe efos a chage of vaabes he s euaos ha vove X Y Z F G H such a fasho as o ouce he oga vaabes he oe eae fs euaos whose fs hee cosue he geeaao of he euaos ha wee esabshe b OUNE. 54. Eea vua wok. Theoe aaogous o hose of Vago a a- Guhe. eaks o he aua fucos ha wee ouce he eceg seco. e gve he ae of eea vua wok o he efoe eu M fo a aba vua efoao o he eesso: T F G H e X Y Z L M N. e efe o he oaos of sec. 5 a e eoe he oecos oo he fe aes of he sege whose oecos oo M M M ae such a wa ha oe has fo eae: uo awas suosg ha he aes ueso have he sae oeao. Ths beg he case suose as sec. 5 ha oe gves he aba fucos µ µ µ he sgfcace efe fo he fouas: µ µ µ. e he see ha he evous-obae eaos bewee he aua fucos ha we ouce seves o o eess he foowg coo: he a of he vua sacees sec. 5 ae gve o he efoe eu he eea vua wok T e s gve ehe b he eao: T e OON Mé. e ca. T. XV. 84 see. 89. so cosu LMO O he oagao of a subace a gosaca oae eu oc. Lo. Mah. oc. Nov. 89; LOVE Tease o he Maheaca Theo of Easc ab. Uves ess s e. 89 e. 96; OME u es éuaos gééaes e éascé u. De a oc. Mah. De Face T. XXX. 8- a

19 56 THEOY OF DEFOMLE MED } { whee he egas ae ake ove he efoe eu o b he eao: e T whch he egas ae ake ove he uefoe eu because he foua we gave above: e F G H T. N M L Z Y X o seve as he efo of eea vua wok a aso be we: e F G H T N M L Z Y X b vue of he sgfcace of X Y N F G a kewse: e F G H T

20 THE DEFOMLE MEDUM 57 X Y Z L M N b vue of he sgfcace of X Y N F G. a wh he foua: T e whch s ae o a aba oo of a eu ha s boue b a suface. ce us be eca u b vue of he vaace of ue he gou of Eucea sacees wh he vaaos gve b fouas 5 ae: a b: a ω ω a ω ω a ω ω ω ω ω a fo hs a he eessos fo T e o whch we us ss we cocue ha oe has: F X H G L Z Y a fou aaogous euaos. These s fouas ae eas euce fo he oes ha oe oa wes b eas of he ce of sofcao. Oe a age ha he foe s vaabe hese fouas. The aua fucos ha wee ouce he eceg aagahs ae o he o oes ha a be evsoe; f we cofe ouseves o he coseao he we s a a few obvous eaks. efo we have ouce wo sses of effos a oes of efoao eave o a o M of he efoe eu. The fs ae he oes ha ae eee o sufaces ha have he oa aae o oe of he fe aes O O O befoe efoao. The seco ae he oes ha ae eee o sufaces ha have he oa aae o oe of he sae fe aes O O O. The fouas ha we have cae gve he ae eees b eas of he foe; howeve b a eae souo whch we sha o so o efo oe obas covese he foe eees es of he ae. Now suose ha we have ouce he fuco. The foe effos a oes of efoao have he eessos we aea gave a oe eae euces he eessos es of he ae fo hs. Neveheess hese cacuaos oe a secf he fucos ha oe us ouce accog o he The assage fo eees efee o he u of voue of he uefoe eu a aea of he foe o he eees efee o u of voue fo he efoe eu a he aea of he foe suffce eae ha suffces o cofe ouseves o he foe as we have oe fo eae.

21 58 THEOY OF DEFOMLE MED aue of he obe a whch w be fo eae o a hee aaees b eas of whch oe eesses. f oe ouces a f oe coues o e eoe he fuco ha ees o he fs evaves of wh esec o o a he fs evaves wh esec o a s obae b eacg he ffee uaes ζ he fuco ζ wh he vaues as gve b fouas 4 a 44 he oe w have: Π Χ Σ. 55. Noo of eeg of efoao. Theoe ha eas o ha of aeo as a acua case. Evso he wo saes M a M of he efoabe eu boue b he sufaces a a cose a aba seuece of saes ha sa wh M a e wh M. To ha e suffces o cose fucos of a oe vaabe h ha euce o esecve whe h s eo a euce o he vaues esecve fo o-eo h eave o M. f we ake he aaee h va a couous fasho fo o h he we oba a couous efoao ha es us o ass fo he sae M o he sae M. Fo hs couous efoao cose he oa wok efoe b he foces a eea oes ha ae ae o he ffee voue eees of he eu a b he effos a oes of efoao ha ae ae o he suface eees of he foe. To oba hs oa wok suffces o egae he ffeea so obae fo o h sag wh oe of he eessos fo T e he eceg seco a subsug he aa ffeeas ha coeso o he cease h h fo he vaaos of ; he foua: Fo such aua fucos oe a ake fo eae he cooes of he oao ha akes he aes O O O aae o M M M esecve.

22 THE DEFOMLE MEDUM 59 T e gves he eesso h fo he vaue of T e a we oba: h h h h fo he oa wok. The wok ueso s eee of he eea saes a ees o o he eee saes M a M. Ths eas us o ouce he oo of eeg of efoao whch us be sgushe fo ha of he aco of efoao ha we evous evsoe. e sa ha s he es of he eeg of efoao efee o he u of voue of he uefoe eu. The ooso ha we us ecoue whch eees he oa wok ha s efoe b he eea foces a oes as we as he effos a oes of efoao ha ae ae o he foe gves LEYON heoe whe we cose a fe sa efoao a secf he eu. ee fs ouce s he hohess a we efe o sec. 58 fo he oe geea fo ha s a se fuco of ε ε ε. e a he evso he fouas: Ω ε Ω ε Ω ε Ξ Ξ Ξ as efg a chage of vaabes ha eaces he ees ε ε ε wh he ees Ω Ω Ω Ξ Ξ Ξ. vue of hs chage of vaabes becoes a fuco of Ω Ω Ω Ξ Ξ Ξ. Havg sa hs we ass o fe sa efoaos a u ouseves o he suao evsoe sec of ou ee éoe su a héoe e éascé; a becoe uaac fos of e e e g g g a of N N N T T T ; he ae s u o a faco of ¼ wha oe cas he ao fo o. he hs s of ssue a he case of fe sa efoaos oe obas he foowg eesso fo he oa wok:. LMÉ sees o have bee cee wh akg LEYON heoe kow hs Noe o he oes eus T. XXXV he hs Leços su a héoe ahéaue e éascé es cos soes s e. 85 e. 866; ee was o he s of Febua 858 ha he foowg oe aeae: LEYON Méoe su e ava es foces éasues as u cos soe éfoé a aco e foces eeeues oes eus T. XLV so cosu TODHUNTE a EON Hso of he Theo of Easc ec. secs. 4 a 67-7.

23 6 THEOY OF DEFOMLE MED To be oe secfc f we suose ha we have : µ e g µ e e e g g g 4ee 4ee 4ee he we have: N N N N N N T T T N T µ µ µ µ o: µ N T N N N NN NN NN T T T µ µ Oe sees ha oe has ecovee he esu of LMÉ ecse f oe eaks ha he oa wok of he eea foces a effos o he foe obvous euces o he cae eesso he case of fe sa efoaos. 56. Naua sae of he efoabe eu. he eceg we sae wh a aua sae of a efoabe eu a he we wee gve a sae we cae efoe. e cae he fouas ha e us o cacuae eea foce a he aaogous eees ha ae aoe o he fuco fo he efoabe eu a eese he aco of efoao a a o. s befoe e us so fo a oe o hs oo of aua sae. U ow he ae s a sae ha has o bee subece o a efoao. age ha he fucos ha efe he efoe sae ee o oe aaee a ha oe ecoves he aua sae fo a acua vaue of hs aaee. The ae he sees o us o be a seca case of a efoe sae a we ae e o ae o a he oos eag o he ae o. hou chagg he vaues of he eees ha ae efe b he fouas of sec. 5 oe a eace he fuco wh hs fuco augee b a aba efe fuco of a f oe s se b he ea of aco ha we assocae o he assage fo he aua sae M o he efoe sae M he oe a f oe efes suose ha he fuco of ha s efe b he eesso: s eca u; howeve he vaues obae fo he eea foce a he aaogous eees wh ega o he aua sae w o ecessa be u. e sa ha he efe he eea foce a he aaogous eees eave o he aua sae. E. a F. OET. ee éoe su a héoe e éascé. 77.

24 THE DEFOMLE MEDUM 6 ou wa of seakg he aua sae eses sef as he a sae of a seuece of efoe saes a sae ha we sa wh oe o su he efoao. s a esu oe s e o ea ha s o ossbe o ake oe of he efoe saes a he oe ha we have he aua sae a a ha hs us be ue such a wa ha he eees ha we efe sec. 5 eea foce a oe eea effo a oe of efoao whch wee cacuae fo he ohe efoe saes have he sae vaues f oe efes he fs of hese eees o he u of voue of he efoe eu a he seco of hese o he u of aea of he efoe suface. Ths ueso a eceve a esose o f oe ouces a secfes he oo of he aco ha coesos o he assage fo oe efoe sae o aohe sae. The ses hohess cosss of assug ha hs ae aco s obae b subacg he aco ha coesos o he assage fo he aua sae M o he fs efoe sae M fo he aco ha coesos o he assage fo he aua sae o he seco efoe sae M. h ega o M f we eoe he uaes ha ae aaogous o ζ eave o M b he we ae e o ao he foowg eesso fo he aco of he efoao eag o he assage fo he sae M o he sae M: 5 { } whch oe a we f s he vaue of fo M : 5 whch we have e eoe he suface of M ha coesos o fo M a eoes he eesso: { }. Fuheoe fo he eak ae a he begg of hs aagah oe a f oe efes subsue he foowg eessos fo : 5 e a he seak of he foce effo ec. sce we ega he aua sae as he of a seuece of saes fo whch we kow he foce effo ec. U ow he foce effo ec. wee efe fo us o whe hee was a efoao caabe of afesg a easug he. Oe us eak ha ae o aaogous o because he ae o foe b eas of he cooaes of M he sae wa ha ae foe b eas of.

25 6 THEOY OF DEFOMLE MED whch eoes he eesso: f oe eaks ha oe has fo eae:. he s cea ha ag fouas ha ae aaogous o hose of sec. 5 o eessos 5 o 5 a sag wh M as he aua sae bu whe suosg ha M s efee o he sse of cooaes a assug ha he fouas of sec. 5 ae ofe as a coseuece w gve he sae vaues fo he eeo foce a oe eave o he sae M efee o he u of voue of M as we as he sae vaues fo he effo a he oe of efoao efee o he u of aea fo. Theefoe we a cose M o be a efoe sae fo whch M s a aua sae ove ha he fuco assocae wh he sae M s acua o. ofog o hese caos suose o f eas ha he eea foce a oe ae gve b eas of se fucos of a eees ha f he oso of he a M. uose oeove ha he aua sae s gve. e a cose he euaos of sec. 5 eag o he eea foce a oe o be aa ffeea euaos he ukows a he hee aaees b eas of whch oe a eess. The eessos ζ ae he fucos of awas seg ha oe cacuaes b eas of fouas 4 a 44. uose ha X Y Z L M o wha aous o he sae hg X Y Z N L M N ae gve fucos of. The eesso s afe subsug fo he vaues of ζ b eas of fouas 4 a 44 a efe fuco of whch we coue o eoe b a he euaos of he obe a be we: s we sa a he begg of hs seco hs es us o geeae he oo of aua sae ha we fs ouce. sea of akg hs wo coeso o he ea of a acua sae we a a oe geea fasho ake coeso o he ea of a aba sae sag fo whch we a su he efoao. The fac ha we ouce a he begg of he heo sees o ake M a a acua oe; howeve oe us o cose as ahg bu he cooaes ha seve o efe he ffee ea a o o M. Oe has chose hese cooaes a acua fasho a eao o a acua eu oe ha oe us as a esu a aeo o M he coe of fe sa efoaos.

26 THE DEFOMLE MEDUM 6 X Y Z L M N whch L M N ae fucos of ha esu fo he efos of sec. 5. esus ec fo he fouas of he eceg aagahs ha a oe eae wa of efg X Y Z L M N a be suae he eao:.e. : T e F G H X Y Z L M N 57. Noos of he a a he. he su of efoabe ea as he su of efoabe es a sufaces s aua o a acua aeo o he oke ea ha ae escbe b he efoabe ea. Ths aous o evsog seaae a coseg as s aua fucos. Ths s wha we kewse eess b agg ha oe goes he esece of he as ha eee he efoabe eu a ha oe kows o he veces of hose as. f we ao ha vewo oe o evso he aa ffeea euaos ha oe s e o hs case he we a ouce he oo of he a a we ae e o a esug cassfcao of he vese ccusaces ha a be ouce b he eao he.

27 64 THEOY OF DEFOMLE MED Theefoe a a su ha eses sef s ha of he eucos ha eae o he eao of he. Lkewse he coesog acua cases whch he aeo s ece aos ecusve o he oke ea ha ae escbe b he efoe eu M oe a soees absac fo M a as a esu fo he efoao ha es us o ass fo M o M. s we aea sa fo he efoabe e a suface he a a be eoe aohe fasho. e a ake acua hoheses o a he eu M; a of hs aous o evsog acua efoaos of he fee efoabe e. f he eaos ha we ose ae se eaos bewee ζ as w be he case he acaos ha we sha su we a accou fo hese eaos he cacuao of a euce oe acua fucos fo. The eesg ueso ha hs oses s ha of oucg hese acua fos s a o cose he geea ha seves as he o of eaue as beg he soe sese. e hus have a heo ha w be secfc o he acua efoaos bough o gh b he gve eaos bewee ζ. e cof ha b eas of he heo of fee efoabe ea oe a heefoe cobe he acua cases a ove a coo og o he euaos ha ae he esu of seca heoes ha oe ecoues hscs. he acua cases oe soees fs oesef he oe ccusaces o avo he coseao of hese efoaos; ea he us soees be coee. Ths s wha oe a o acca acaos whe oe evsos fe sa efoaos. Take he case whch he eea foce a oe efe o o he fs evaves of he ukows a ; he seco evaves of hese ukows w be ouce o hese aa ffeea euaos o fo ; howeve he evaves of fgue o ζ a hose of show u o. Oe heefoe sees ha f ees o o ζ o o o he hee w be a euco he oe of he evaves ha ee o he aa ffeea euaos. Hee we eae he fs of hese wo cases whch coesos o he oa heo of easc fo aea ea a o he heo of he vaous eheea ea ha ae evsoe he oce of uous waves. 58. ase whch ees o o ζ a s eee of. How oe ecoves he euaos ha eae o he efoabe bo of he cassca heo a o he ea of hosacs. uose ha ees o o he uaes ζ a o o. The euaos of sec. 56 whch euce o he foowg: of ou coseaos heeofoe a be ae us he sae o aea ea as o vaous eheea ea. e have ecae he wo ae o be va a wha we eose s as we sa o beg wh a heo of aco fo eeso a ovee. To have a oe coee ea of he oo of ae we sha ea ae o how oe us aoach he ae fo he coce of eo accog o he ofou vewo ha LMNN ouce o eecc.

28 THE DEFOMLE MEDUM 65 X L Y M Z N whch ees o o we show ha f oe akes he se case whch X Y Z L M N ae gve fucos of he he hee euaos a be sove fo a oe fa obas hee aa ffeea euaos ha fo ou hoheses efe o o he a o a he fs a seco evaves. Fs evso he acua case whch he gve fucos L M N ae u; he sae w be ue fo he coesog vaues of he fucos of oe of he sses N M L L M N L M N. esus fo hs ha he euaos: aou o:.e. whose eeao s eae. Hag sa hs obseve ha f oe of he wo osos M a M s assue o be gve a ha f oe euces he fucos L M N fo hs as sec. 5 he he case whch hese hee fucos ae u oe a ave a hs esu accea oe o sf he eoso a o cae oe eas wha we ae aug o we suose ha X Y Z L M N o o efe o he evaves of.

29 66 THEOY OF DEFOMLE MED.e. fo a cea se of acua efoaos; howeve oe a ave a hs esu fo a efoao M sce s a coseuece of he aue of he eu M.e. of he fo of. ose hs ae case whch s acua eesg; s he a se fuco of a he s eessos ε ε ε whch ae efe b he fouas 45. The euaos euce fo sec. 5 a 5 euce o ehe: X F Y G Z H whch oe has: k k k k k k ε ε ε k. o o : X F Y G Z H whch oe has: Ξ Ξ Ω Ξ Ω Ξ Ω Ξ Ξ The a s coee he; we a aso coceve ha we have a se oke eu. oae E. a F. OET. ee Méoe su a héoe e éascé

30 THE DEFOMLE MEDUM 67 Ξ Ξ Ω Ξ Ω Ξ Ω Ξ Ξ Ξ Ξ Ω Ξ Ω Ξ Ω Ξ Ξ whch we se ε Ξ Ω o abbevae oao o we ge : X F Y G Z H whch oe has: Ξ Ξ Ξ Ω Ω Ω a aaogous fouas fo has he sgfcace ha we gave sec. 5 whch we sha eca a oe. s oe sees we ecove he couous efoabe eu as s eae he oa heo of easc. acua eesg case s obae b ookg fo a fo fo ha gves he ees: fo a Oe fs ha us be a se fuco of a he eesso whch s efe b he fouas : oae E. a F. OET. ee Méoe su a héoe e éascé

31 68 THEOY OF DEFOMLE MED ε ε ε fo whch oe a see uo eakg ha f oe efes o he evous fouas ha gave us as a fuco of he oe has: a wo aaogous sses; sce s assue o be a se fuco of a oe has as a esu:. f we cose he acua case whch ees o o a f we assue ha we ae gve X Y Z eesse as fucos of he he euaos ueso whch ae: X Y Z F G H uo seg becoe hose whch seve as he bass fo hosacs. The a eu M aeas o b wa of a oe a eace he ukow wh he ukow ha s eae o b he eao. f he fuco whch s o gve s he he oe has he eceg euaos whch s a aua fuco whose sgfcace s we kow. w suffce fo us o cae ha he case whch he fucos L M N ae o-u coses he heo of a he eheea ea ha have bee cosee fo he su of uous waves fo MULLGH o LOD ELVN bu hee he heo of hese ea s coee echaca. e kewse eo ha he os geea oae E. a F. OET. ee Méoe su a héoe e éascé. 4. These fouas ae acua he oes o age 47 of ou ee Méoe su a héoe e éascé. oae DUHEM. Hoaue Eascé cousue.

32 THE DEFOMLE MEDUM 69 case whch he ace of he evaves of he aco wh esec o he oaos eas he eesso fo he eea oe eas he os aua ae o he oo of agec uco ha was ouce b MXELL. 59. The g bo. e have cosee he acua case whch oes o ee o a ffee seca cases of hs case. Oe a ave a he ohe ea ha wee cosee a eas a b he auhos ehe b he su of acua efoaos o b he su of ew ea ha ae efe b a heo of cosas ha ofs fo he esus ha we aea acue. Fo eae sa wh he se case whch he a s he.e. b efo s a oke eu whch s a fuco of ε ε ε.. e a age ha oe as aeo o o he efoaos of he eu fo whch oe has: ε ε ε. he efos of foces ec. suffces o ouce hese hoheses a f he foces ae gve o ouce hese s coos. he ae case he habua obes whch coeso o he gve of he fuco a o he geea case whch he ε ae o-u a be ose o fo acua gves. f we suose o ha he fuco ha s obae b akg ε ε ε ε s gve ha oe oes o kow he vaues of he evaves of wh esec o ε ε... fo ε ε... so ha s he he we see ha fo eae becoe s aua fucos ha oe us ao o such a wa ha fo he case whch he foces ha ac o he voue eees ae gve we have e aa ffeea euaos e ukows he case o whch oe us ao accesso coos. Now we eak ha oe kows how o egae he sse: ε ε ε. ce he efoao s suose couous he ega coesos o a sacee of he se of he eu; hus eas fo us o eee he s cosas of egao a he aua fucos f he foces a effos ha ac o he eu ae gve a we suose ha X ae kow as fucos of he he s euaos of sec. 54 wh he sfcaos e fo he fo of whe ae o he ee bo eee he s egao cosas. To coee he ocess wha eas s fo us o uae eee f we eave ase he obe of hs uae eeao he oe sees ha we ecove he habua obes of he echacs of g boes whch oe gh oa suose ha he he fuco ees o o.. e a age ha we seek o efe a eu whose efo aea akes he coos:

33 7 THEOY OF DEFOMLE MED ε ε ε o accou su gees. oe o efe he ew eu whe hkg aog he sae es as befoe we fuhe efe F N b he e: F X N. Howeve hs e us o oge ho b vue of he fac ha ε. ohe wos we evso a eu whch he heo us esu fo he a oseo ao of he coos ε o he kowege of a fuco ε ε a s aua fucos µ µ 6 of b eas of he e: µ ε µ ε µ F 6 X whch aous o seg ε he geea heo ha ecee whch oe has eace wh µ ε µ 6 ε. s oe sees we coe ow o he heo of easc ea ha coeso o he fuco of ε ε whe oe escs oesef o he su of efoaos ha coeso o ε. Theefoe f we cose he case of a he he f we suose ha we kow s he vaue ha a ake suaeous whe ε he we ecove he habua heo of he g bo. Obseve ha f we accou fo he coos ε a o b a chage of aua fucos he we ae e o eace wh µ ε µ 6 ε he cacuaos ha eae o he geea eu a we kewse f fouas ha coe ow o he su of a easc eu whch we ae cofe o sug efoaos ha coeso o ε. Uo suosg ha µ µ 6 ae ukow we oce oe coe ow o heo ha coses he habua heo of he g bo. Fo hs ae vewo we eu o he eoso ha oe a ake abou he eas of LGNGE. acua we a obseve ha he case whch X Y Z ae gve as he aa evaves wh esec o of a fuco ϕ of he euaos whch X Y Z fgue ae oe ohe ha he euaos ha oe s e o whe oe seeks o eee he eeu of he ega: gve he coos: ϕ ε ε ε.

34 THE DEFOMLE MEDUM 7. e scuss a h oceue fo cosug a eu fo whch he heo awas eas o he sae euaos a whch w be a g case of he oga heo. Ths oceue agees wh he fs oe a a aso be ae o he cases of he efoabe e a suface. age ha he ha seves o efe he oga eu s vaabe a o f eas suose ha he vaues of ε ae eveoabe a MLUN sees a eghbohoo of eo b he foua: whch eeses he se of es of he h egee. ssue ha he coeffces of whch a ee o cease efe he vaao. f we wa o coseve a fe vaue he we us suose ha ε e owas eo. ohe wos we a he cose o efoaos ha sasf ε. ohe wos he bo ha we aoach he a ake o sacees of he se. e a suose ha oe akes he evaves whch aoach s ε whe vaes a ae we sha escbe kewse va as a coseuece of a sue efoao fo hs eu. To ea hs a oe ecse fasho age ha he coeffces of ee o oe aaee h such a wa ha whe h es owas eo he coeffces of cease efe. To f eas suose ha he ae coeffces ae ea wh esec o. Lkewse age ha whch efe he efoao h ueso va wh h such a wa ha ε e o eo. ao we suose ha ε ae fe sa of fs oe wh esec o h; fo eae ε gh be eveoe owes of h a he fs es of ha eveoe ae he oes h. h hese coos es o eo a e o cea s whch a be ε fucos of. Theefoe f we cose he euaos of sec. 5 ha seve o efe eea foce a oe he we ae fa e o fouas ha e us o efe he a whch ae oe ohe ha euaos of ou o of eaue whch he oo of he fuco has saeae a whch s aua fucos F G H fgue. 6. Defoabe ea oo. The heo of oo fo he efoabe e a ha of he oo of he efoabe suface ese heseves ve aua as seca cases of he heo of he efoabe suface a ha of he efoabe eu. To see hs suffces o gve oe of he aaees of he suface o eu he sgfcace of e. s we w o evso he sacs of ea of eso geae ha hee hee we us eose he heo of oo of a efoabe eu ec oae THOMON a TT. Tease vo.. a.. 7 sag wh he h e ow.

35 7 THEOY OF DEFOMLE MED wha foows; howeve we eveheess gve a fo ha s ee aaogous o he oe ha we cae fo he acs of efoabe e a he efoabe suface. ose a sace M ha s escbe b a o M whose cooaes ae wh esec o he hee fe ecagua aes O O O a ao a ecagua a o each o M of he sace M whose aes M M M have he eco coses ; ; wh esec o he aes O O O esecve a whch ae fucos of he eee vaabes. The couous hee-esoa se of such as M a be cosee as he oso a a efe sa of a efoabe eu ha s efe he foowg fasho: Gve he o M a sacee M M whch s a fuco of e a he oso of he o M a s u fo. Le be he cooaes of he o M whch we cose o be fucos of. ao eow he a M wh a oao ha akes s aes fa agee wh hose of a a M ha we ao o he o M. e efe ha oao b gvg he eco coses ; ; of he aes M M Mwh esec o he fe aes O O O. Lke hese coses w be fucos of. The couous hee-esoa se of as M fo a gve vaue of e w be wha we ca he efoe sae of he efoabe eu cosee a he sa. The couous fou-esoa se of as M ha s obae b akg va w be he aeco of he efoe sae of he efoabe eu. Fo ease of wg a oao he seue we soees ouce as we aea he ees sea of. e coue o eoe he cooes of he veoc of he og M of he aes M M M aog hese aes b whe aoe vaes a he oecos of he saaeous oao eave o he aaee of he a M o hese sae aes b. e eoe he aaogous eessos fo he a M b ζ a whe oe efes he ke he a M o he fe aes O. he e vaes a he oo of he a M s efee o he fe a O he he og M has a veoc whose cooes aog he aes M M M w be esgae b ζ a he saaeous oao of he a M w be efe b he cooes. The eees ha us ouce ae cacuae as sec. 49; fs oe has he fouas: 54 55

36 THE DEFOMLE MEDUM 7 o whch we ao he foowg: f oe ow ouces he sco bewee he oaos fo he evaves wh esec o e eeg o whehe oe akes o fo he eee vaabes. uose ha oe eows each of he as of he aeco of he efoe sae wh a fe sa sacee ha vaes a couous fasho wh hese as. h he sae oaos as sec. 5 we have: Eucea aco of efoao a oo fo a efoabe eu oo. ose a fuco of wo fe cose osos of he a M.e. a fuco of a of a he fs evaves wh esec o. e oose o eee he fo ha us ake oe fo he uaue ega:

37 74 THEOY OF DEFOMLE MED whe ake ove a aba oo of sace M a he e eva bewee wo sas a o have a u vaao whe oe subecs he se of a as aog wha we ae cag he aeco of he efoabe eu ake s efoe sae o he sae aba fesa asfoao of he gou of eucea sacees. efo hs aous o eeg such a fasho ha oe has: whe o he oe ha he og M of he a M s subece o a fe sa sacee whose oecos o he aes O O O ae: 6 a a a ω ω ω ω ω ω whch a a a ω ω ω ae s aba cosas a s a fe sa ua ha s eee of a whe o he ohe ha hs a M s subece o a fe sa oao whose cooes aog he O O O aes ae: ω ω ω. suffces fo us o eea he easog ha we ae befoe wh sevea eses oe o see ha he ese fuco has he eakabe fo: ζ ζ whch s aaogous o he oe we ecouee fo he efoabe e suface a eu a es. e sa ha he ega: s he aco of efoao a oo he eo of he suface of he efoe eu oo a he eva of e bewee he sas a. O he ohe ha we sa ha s he es of he aco of efoao a oo a a o of he efoe eu whe ake a a gve sa a efee o he u of voue of he uefoe eu a he u of e. f we gve he sae sgfcace as sec. 5 he s he es of ha aco a a o a a gve sa whe efee o he u of voue of he efoe eu a he u of e.

38 THE DEFOMLE MEDUM The eea foce a oes; he eea effo a oe of efoao; he effo oe of efoao ua of oo a he oe of he ua of oo of a efoabe eu oo a a gve o a sa. ose a aba vaao of he aco of efoao a ovee he eo of a suface of he eu M a he e eva bewee he sas a ae:. vue of fouas we a we:. e a GEEN s foua o he es ha ec vove a evave wh esec o a of he vaabes a efo a egao b as ove he es ha ec vove a evave wh esec o e. f we e esgae he eco coses wh esec o he fe aes O O O of he eeo oa o he suface ha bous he eu befoe efoao a he sa a esgae he aea eee of ha suface b he we oba:

39 76 THEOY OF DEFOMLE MED. s sec. 5 se: F G

40 THE DEFOMLE MEDUM 77 H a ao:. O he ohe ha se: X Y Z L M N. Ths akes: H G F { }. N M L Z Y X

41 78 THEOY OF DEFOMLE MED f we fs cose he uaue ega ha fgues he eesso fo he we ca he seges ha have he og a M a whose oecos o he aes M M M ae X Y Z a L M N he eea foce a eea oe a he o M a he sa efee o he u of voue of he oso of he eu a he sa esecve. f we he cose he e ega ha s ake ove e a he suface he we ca he seges ha ssue fo he o M whose oecos o he aes M M M ae F G H a he eea effo a eea oe of efoao a he o M of he suface ha bous he efoe eu a he sa. a efe o M of hese as s uaes ee o o he eco of he eea oa o he suface. The ea vaa f he ego we ca M vaes bu he eco of he oa oes o chage a he chage sg f hs eco s eace b he oose eco. uose ha oe aces a suface Σ he eo of he efoe eu ha s boue b he suface whch ehe aoe o wh a oo of he suface ccuscbes a subse of he eu a e eoe he es of he eu ouse of. Le Σ be he suface of M ha coesos o he suface of M a e a be he egos of M ha coeso o he egos a of M. Mea seaae he wo subses a ; oe a ega he wo seges F G H a ha ae eee fo he o M a he eco of he oa o Σ ha os o he eeo of as he eea effo a oe of efoao a he o M of he foe Σ of he ego. a oe a ega he wo seges F G H a o be he eea effo a e of efoao a he o M of he foe Σ of he ego. easo of hs eak we sa ha F G H a ae he cooes of he effo a oe of efoao ha s eee o he oo of he eu M a M aog he aes M M M a ha F G H a ae he cooes of he effo a oe of efoao ha ae eee o he oo of he eu M a M aog he aes M M M. Fa f we cose he e ega ove he voue of M a he sa whose vaues ae ake a he eee sas a he we ca he seges ha have he ogs a M a whose cooes aog he aes M M Mae a he ua of oo a he oe of he ua of oo a he o M of he efoe eu M a he sa esecve. 6. Dvese secfcaos fo he effo a oe of efoao he ua of oo a he oe of he ua of oo. s sec. 5 se:

42 THE DEFOMLE MEDUM 79 ; whch a eese he oecos o M M M esecve of he effo a oe of efoao ha ae eee a he o M of a suface ha has a oa ha s aae he as O O O ha we escbe b he e befoe efoao. ee suffces o eca ha we aea agee o eace he ees ha coeso o he ces b hs coveo wh. eca ha hs effo a oe of efoao ae efee o he u of aea of he uefoe suface a he sa. The ew effos a oes of efoao ha we us efe ae eae he eees ha he ouce he eceg seco b he foowg eaos: F G H X Y Z L M. N Oe a oose o asfo he eaos we us woe eee of he vaues of he uaes ha fgue he ha ae cacuae b eas of. ee hese eaos eae o he seges ha ae aache o he o M o whch we gave he aes. sea of efg hese seges b he oecos o M M M we a us as we efe he b he oecos o ohe aes; he ae oecos w be coue b eaos ha ae asfos of he eceg oes. Moeove he asfoe eaos ae obae eae f oe eaks ha he oga fouas

43 8 THEOY OF DEFOMLE MED have se eeaos b he auco of aes ha ae aae o he ovg aes a he o O.. s sacs we cofe ouseves o he coseao of he fe aes O O O. Le X Y Z a L M N eoe he oecos of he eea foce a he eea oe a a aba o M of he efoe eu a a sa oo hese aes a e F G H a be he oecos of he effo a he oe of efoao o a suface whose eeo oa has he eco coses befoe efoao a he sa. Le a be he oecos of he effo a he oe of efoao a e a be he oecos of he ua of oo a he oe of he ua of oo. The asfos of he eceg eaos ae obvous: F G H X Y Z L M. N eesg eeao o oe s he aaogue of he oe gve b. NT-GULHEM he coe of he acs of as.

44 THE DEFOMLE MEDUM 8. Now obseve ha we a eess he e coses b eas of he hee aua fucos. e: ϖ ϖ ϖ. The fucos ϖ of so efe sasf eaos ha we have we sevea es aea: ϖ ϖ ϖ ϖ ϖ ϖ a oe has: ϖ ϖ ϖ ϖ ϖ ϖ whch. f we e ϖ eoe he oecos oo he fe aes O O O of he sege whose oecos oo he aes M M M ae ϖ he we w have: ϖ ϖ ϖ

45 8 THEOY OF DEFOMLE MED b vue of whch he ew fucos ϖ of sasf he eaos: ϖ ϖ ϖ ϖ ϖ ϖ. Oce oe we ake he eak whch w seve us ae o ha f oe es eoe he vaaos of ha coeso o he vaaos of he oe w have: ϖ ϖ ϖ ϖ ϖ ϖ whch ae he oecos oo he fe aes of he sege whose oecos oo M M M ae. Now se: ϖ ϖ ϖ ϖ ϖ ϖ L ϖ L M N ϖl M N M ϖ L M N ϖ L M N N ϖ L M N ϖ L M N. ao ouce he foowg oaos: These fouas a seve o efe he fucos ϖ ec a a be subsue fo: ϖ ϖ ϖ ϖ.

46 THE DEFOMLE MEDUM 8 ϖ ϖ Π ϖ ϖ Χ ϖ ϖ Σ ϖ ϖ Π ϖ ϖ Χ ϖ ϖ Σ a sea of he ae sse whch ehe o fgue we have he foowg: ϖ ϖ ϖ Π L ϖ ϖ Π ϖ ϖ ϖ ϖ ϖ wh wo aaogous euaos. f oe eaks ha he fucos ζ a ζ a gve se o he fouas: ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ha esu fo efg eaos fo he fucos ϖ a he e ees he vef he oe a gve he eceg sse he ew fo:

47 84 THEOY OF DEFOMLE MED Π L Π wh wo aaogous euaos.. Fa we sha subec he eceg wo euaos ha we ouce o a asfoao ha s aaogous o he oe ha e us sec. 5 o he geeaao of he euaos of he heo of easc ha eae o effo. To abbevae he oao e X Y Z L M N eoe fo he oe he ef-ha ses of he asfoao eao ha efes o X Y Z L M N esecve a obseve ha oe a suae he weve euaos we have esabshe b he foowg: µ µ µ X Y Z L M N { G F µ H } µ µ whch µ µ µ ae aba fucos a he egas ae ake ove o he oe ha he e eva bewee he sas a a o he ohe ha he suface of he eu M a he oa bous. f we a GEEN heoe a egae b as he he eao ha we us woe becoes he foowg oe: N M L Z Y X µ µ µ H G F µ µ µ { } µ µ µ µ µ µ µ µ µ µ µ

48 THE DEFOMLE MEDUM 85 µ µ µ µ µ µ. µ e seek o asfo hs as eao whe oe akes he fucos fo ohe ew vaabes whe esevg. e a he eeea fouas fo he chage of vaabes ha we ecae sec. 5 o he fucos µ µ µ. h awas cag he suface of he eu M a he sa ha coesos o he suface of M. Moeove e X Y Z L M N be he oecos o O O O of he eea foce a eea oe ha ae ae o he o M a he sa a efee o he u of voue of he efoe eu M a e F G H L eoe he oecos o O O O of he effo a oe of efoao ha ae eee a he o M o efee o he u of aea of. Fa ouce as sec. 5 eghee ew aua fucos b he fouas: a he aaogous oe ha s obae b eacg: b a he b

49 86 THEOY OF DEFOMLE MED esecve wh he ua havg he sae eesso as sec. 5. e oba he asfoe eao: X Y Z Lµ Mµ Nµ F G H µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ whch he egas ae ake ove o he oe ha he e eva bewee he sas a a o he ohe ha he suface of he eu M a he sa a he oa bous wh esgag he aea eee of. Oce aga we a he GEEN foua o he es ha efe o he evaves of µ µ µ wh esec o a a egao b as of he es ha vove he evaves of µ µ µ wh esec a e eoe he eco coses of he eeo oa o he suface a he sa wh esec o he fe aes. ce µ µ µ ae aba he becoe: F G H X Y ce he fe of vaao acua vaes wh we efo ha egao b as b he eea of assg o he vaabes. e suose ha s osve a eua o.

50 THE DEFOMLE MEDUM 87 Z L M N. The sgfcace of he eghee ew aua fucos esu eae fo he eaos ha we us woe. ee s cea ha he coeffces of he eessos of F G H eese he oecos oo O O O of he effo ha s eee a he o M o a suface whose eeo oa s aae o O a ha he coeffces of he eessos fo ae he oecos oo O O O of he oe of efoao a M eave o he sae suface. 64. Eeo vua wok; heoes aaogous o hose of Vago a a- Guhe. eaks o he aua fucos ha wee ouce he eceg aagahs. O a efoe eu M bewee he sas a a aba sae of vua efoao we gve he ae of eea vua wok o he eesso: e { } T F G H F G H. e efe o he oaos of sec. 6 a oeove e be eoe he oecos oo he fe aes of he sege whose oecos oo M M M ae such a wa ha oe has fo eae: whch we ae awas suosg ha he aes ueso have he sae soso. Ths beg he case suose as sec. 6 ha oe has gve he aba fucos µ µ µ he sgfcace ha s efe b he fouas: µ µ µ.

51 88 THEOY OF DEFOMLE MED e he see ha he eceg eaos we obae bewee he ew aua fucos eess o he foowg coo: f a aeco of he efoe eu s gve a of he vua sacees of sec. 6 he he eea vua wok T e s gve b ehe he eao: T e whch he egas ae ake ove he e eva bewee he sas a a he efoe eu o b he eao: T e

52 THE DEFOMLE MEDUM 89 whch he egas ae ake ove he e eva bewee he sas a a he uefoe eu a he sa because he foua ha we gave above: e { } T F G H F G H whch seves o efe he eea vua wok a aso be we: e { } T F G H X Y Z L M N b vue of he sgfcace of X Y Z L M N F G H a kewse: T e F G H X Y Z L M N b vue of he sgfcace of X Y N F G. a wh he foua: T e ae o a aba a of he eu ha s boue b a suface a he e eva bewee he sas a. ce us be eca u whe he vaaos ae gve b he fouas 6 of sec. 6 ae:

53 9 THEOY OF DEFOMLE MED a ω ω a ω ω a ω ω b vue of he vaace of ue he gou of Eucea sacees a ae gve b: ω ω ω a ha hs s ue fo a vaues of he cosas a a a ω ω ω we cocue fo he eessos fo T e ha us sse o ha oe has: { } F { } X H G L Z Y a fou aaogous euaos. hese fouas oe a age ha he foe s vaabe. The aua fucos ha wee ouce he eceg aagahs ae o he o oes ha oe a age. Uo cofg ouseves o he coseao we a he sae se eaks as sec. 54. efo we have ouce wo sses of effos a oes of efoao eave o a o M of he efoe eu a he sa. The fs of he ae he oes ha ae eee o sufaces ha have he oa aae o oe of he fe aes O O O befoe efoao. The seco ae he oes ha ae eee o sufaces ha have he oa aae o oe of he sae fe aes O O O afe efoao. The fouas ha we cae gve he ae eees es of he foe; howeve b a eae souo whch we w o eaboae uo oe vese obas he foe eees es of he ae. Now suose ha oe ouces he fuco. The fs effos a oes of efoao have he eessos we aea cae a oe eae euces he eessos fo he seco oes. Howeve hese cacuaos oe a secf he fucos ha oe us ouce accog o he aue of he obe a whch ae fo eae a hee aaees b eas of whch oe eesses. The assage fo he eees ha ae efee o he u of voue of he uefoe eu a he aea of he foe o he eees ha efe o he u of voue of he efoe eu a he aea of he foe a he sa s suffce eae ha suffces o cofe oesef as we have oe o he fs fo eae. Fo such aua fucos oe a ake fo eae he cooes of he oao whch akes he aes O O O aae o M M M esecve.

54 THE DEFOMLE MEDUM 9 f oe ouces a f oe coues o e eoe he fuco ha ees o he fs evaves of wh esec o o a he fs evaves wh esec o ha ae obae b eacg he vaous uaes ζ ζ he fuco ζ ζ b he vaues he ae gve b fouas a 55 he oe w have: Π Χ Σ Π Χ Σ Π Χ. Σ 65. Noo of eeg of efoao a oo. e us eak ha ou ese eoso coas he sacs of efoabe ea as a seca case. ee suffces o cose a evesbe vua ofcao he sese of DUHEM sea of evsog a eaabe vua efoao as we have oe. Ths obsevao eas us o cose he oo of he eeg of efoao a oo. e oose o eee he wok oe b eea foces a oes as we as eea effos a oes of efoao ha ee o a aba e eva fo a ea ofcao. Fo hs suffces o cacuae he eeea wok eave o e. The ae s: { }. G F Y X

55 9 THEOY OF DEFOMLE MED f oe eaces X Y F G b he eesso as a fuco of he aco a f oe efos a vese cacuao o he oe ha e us o he efo he oe eae obas b vue of he ODZZ euaos: whch we have se: E E. acua f oe coses he case whch oes o coa ec such a wa ha s u he he eceg vaue becoes he ffeea wh esec o e of he eesso: E whch a be cae he eeg of efoao a ovee a he sa. hs o he scusso we ee o ake sevea oa geea eaks ha w f fuhe acao wha foows he heo of Eucea aco. The o oo of Eucea aco of efoao a oo ha suffces fo us fushes a ve eee case a cosucve efo of he ua of oo a he oe of he ua of oo he effo a oe of efoao a he foce a eea oe. Oe a sgush a aca a a a sac a he foce a he eea oe b goug o he oe ha he es ha coa o he aca acceeao a o he ohe ha he es ha coa o wha oe a ca he keaca acceeao; hs sco obvous eesses a eeso of LEMET s ce. a suose ha eea wok s u a ha he eeg of efoao a oo eas vaa e. e hus oba he oo of cosevao of eeg whch s asaes o he hohess ha he eu s soae fo he eea wo. u we ecove a of he fuaea eas of cassca echacs a s afes ha he acua fo ha he ake he ae coe us be wha oe evsos fo he sae of oo a efoao a fesa eghbohoo of he aua sae whch oe suoses ha a s evaves ae u. 66. a sae a aua saes. Geea caos o he obe ha e us o he coseao of efoabe ea. he foegog we cosee he aeco of he efoe sae a afe escbg he a oso M of ha efoe sae a a efe sa we efee o he oso M a a aba sa. oseaos ha ae aaogous o he oes we eveoe sec. 56 a whch he aaee ha was hus ouce s ow eace b e a be eeae

56 THE DEFOMLE MEDUM 9 hee f we ake oe of he efoe saes a he oe ha we abue o he a sae M. Howeve oe a aso age ha he fucos ha eee he aeco of he efoe sae ee o oe aaee a ha oe sgushes a acua vaue of hs aaee. Oe hus efes a seuece of saes ha oe a ca aua saes a he aeco a be cae he aeco of aua saes. Oe a use he ew aaee as we ou Noe su a aue u o e u cos vaabe a su acua he aeco of he efoe saes ha fe cose o he aeco of he aua saes. ofog o he evous caos suose o f eas ha he eea foce a oe ae gve b eas of se fucos of he eees ha f he oso of he a M. e a cose he euaos of sec. 6 ha eae o he eea foce a oe as aa ffeea euaos ha eae o a hee aaees b eas of whch oe eesses. Ths vewo s he oe ha eses sef os aua. The eessos ζ ζ w be fucos of seg as awas ha we a cacuae b eas of fouas a 55. uose ha X Y Z L M o wha aous o he sae hg X Y Z N L M N ae gve fucos of. fe subsug he vaues of ha oe euces fo fouas a 55 he eesso s a efe fuco of: ha we coue o eoe b a he euaos of he obe a be we: X Y Z L

57 94 THEOY OF DEFOMLE MED M N whch L M N ae fucos of ha esu fo he efos of sec. 6. Ths eas o he fouas of he eceg aagahs ec a wa ha s oe eae ha he efo of he X Y Z L M N a be suae he eao:.e. : T e { } F G H X Y Z L M N. 67. Noos of he a a he. ase whch ees o o ζ ζ a s eee of. Eeso of he cassca acs of efoabe boes. The gosac eu a kec asoo. The coseaos ha we eose evous ega o he he a a he ae aso acabe o he efoabe eu oo. suffces o s a ha a he w coeso o a he oo. acua we sha eae he case whch ees o o he uaes ζ ζ bu o o he. The euaos of sec. 66 he euce o he foowg: X Y L M

58 THE DEFOMLE MEDUM 95 Z N whch ees o a he show us ha f we ake he se case whch X Y Z L M N ae gve fucos of he he hee euaos o he gh a be sove fo. Oe heeb fa obas hee aa ffeea euaos ha b ou hoheses efe o o a o a he fs a seco evaves. age he acua case whch he gve fucos L M N ae u; he sae w be ue fo he coesog vaues of he fucos a of he sses: N M L L M N L M N. Fo hs esus ha he euaos: aous o:.e. o: whch oe a ee as sag ha he oo of he efoabe bo ueso whch cosues he cassca heo of easc as a seca case gves se o a oe whose hee cooes ae: To sf he eoso a o cae oe eas wha we ae aug o we suose ha X Y Z L M N o o efe o he evaves of.

59 96 THEOY OF DEFOMLE MED a hus has he effec of esog he euaes:. Havg sa hs we obseve ha f oe sas wh a aeco ha s suose o be gve a euces he fucos L M N as sec. 6 he he case whch hese hee fucos ae u oe a ave a he esu ha accea eses sef.e. fo a cea se of acua aecoes; howeve oe a ave a hs fo a aeco M as a coseuece of he aue of he eu M a s oos.e. fo he fo of. age he ae case whch s acua eesg; s he a se fuco of a e eessos ε ε ε ϕ ϕ ϕ v ha s efe b he foowg fouas: ε ε ε ϕ ϕ ϕ. v The a s coee he; hus we a aso age ha we have a s oke eu.

60 THE DEFOMLE MEDUM 97 The euaos euce sec. 6 a 6 euce o ehe: X F Y G Z H whch oe has: k k ϕ ε k k ϕ ε k k k ϕ ε v v ϕ v v ϕ v v ϕ o o: X F Y G Z H whch oe has: Φ Ξ Ξ Ω Φ Ξ Ξ Ω Φ Ξ Ξ Ω