MISCELLANEOUS TRACTS ON Some curious, and very interesting Subjects

Transcription

1 MISCELLANEOUS TRACTS ON Some urious and very interesting Sujets IN MECHANICS PHYSICAL-ASTRONOMY and SPECULA- TIVE MATHEMATICS; WHEREIN The Preession of the EQUINOX the Nutation of the EARTH S AXIS and the Motion of the MOON in her ORBIT are determined. By THOMAS SIMPSON F.R.S. And Memer of the ROYAL ACADEMY of SCIENCES at STOCKHOLM. LONDON Printed for J.NOURSE over-against Katherine-street in the Stand MDCCLVII.

2 [This extrat onsists of a paper on pages 64 to 5 of Misellaneous Trats... whih is a slightly expanded version of the paper pulished in Philosophial Transations of the Royal Soiety of London 49 (155) The original pagination is shown in square rakets. The notation is generally the same as in the original exept that for typographial simpliity a ar over an expression is replaed y raes... and a ar over an expression joined to a following vertial line is replaed y raes followed y a vertial line not joined to them as....] [64] An ATTEMPT to shew the Advantage arising y Taking the Mean of a Numer of Oservations in pratial Astronomy ALTHOUGH the method pratied y Astronomers. in order to diminish the errors arising from the imperfetions of instruments and of the organ of sense y taking the mean of several oservations is of very great utility and almost universally followed yet it has not that I know of een hitherto sujeted to any kind of demonstration. In this Essay some light is attempted to e thrown on the sujet from mathematial priniples: in order to the appliation of whih it seemed neessary to lay down the following suppositions. 1

3 P + P 1. That there is nothing in the onstrution or position of the instrument wherey the errors are onstantly made to tend the same way ut that the respetive hanes for their happening in exess and in defiit are either aurately or nearly the same. 2. That there are ertain assignale limits etween whih all these errors may e supposed to fall; whih limits depend on the goodness of the instrument and the skill of the oserver. These partiulars eing premised I shall deliver what I have to offer on the sujet in the following Proposition. PROPOSITION I. Supposing that the several hanes for the different errors that any single oservation an admit are expressed y the terms of the series where the exponents denote the quantities and qualities of the respetive errors and the terms themselves the respetive hanes for their happening; it is proposed to determine the proaility or [65] odds that the error y taking the mean of a given numer (n) of oservations exeeds not a given quantity. It is evident from the laws of hane that if the given series!" #$%!% &" expressing all the hanes in one oservation e raised to the th power the terms of the series therey arising will duly exhiit all the proposed ( ) oservations. But in order to raise this power with the greatest faility our given series may e redued to )( * +-/ (y the known rule for summing up the terms of a geometrial progression); whereof the th power (making 46589: ; ) will e #N <(>= ;@?A9&BDC #N #N (FEG;H?AI CJ ; whih expanded N eomes #N K? LM N Ö #N 1LM N #N P P 1LM ( into ; D Now to find from hene the sum of all the hanes wherey the exess of the positive errors aove the negative ones an amount to a given numer Q preisely it will e suffiient (instead of multiplying the former series y the whole of the latter) to multiply y suh terms of the latter only as are neessary to the prodution of the given exponent Q in question. Thus the first term ( ) of the former series is to e multiplied y that term of the seond whose exponent is 9R Q in order that the power #N #N of #N in the produt may e : ut this term (putting 9H Q 5TS ) will e E/S#I S eing the#n numer #N of N fators; and onsequently that the produt under onsideration will e E/S#IU(<. Again the seond term of the former series eing? LM the exponent of the orresponding term of #N the latter #N must therefore e?v4t 9W Q EX5YSZ?[4\I and the#n termn it-[66]self E/SH?]4^I^(_ #`3 L ; whih drawn into? LM gives E/S^?4\IV( for the seond term required. In the like #N manner #N the third term of the produt whose whole exponent is Q will e found EaS$?>4\I\(. And the sum of all the terms having the same given exponent will onsequently e [; > E/S#IU(< 2 #

4 ? [; > E/S^?4\IV( [; > E/S^?d#4\IU(?>;? [; > E/S^? 4\IU(?>;?e From whih general expression y expounding Q y suessively the sum of the several hanes wherey the differene of the positive and negative errors an fall within the proposed limits Eh Qi? Q I will e found: whih divided y the total of all the hanes or (e= ;\?>9KBjC (e= ;^?] BjC will e the true measure of the proaility sought. From whene the advantage y taking the mean of several oservations might e made to appear: ut this will e shewn more properly in the next Proposition; whih is etter adapted and to whih this is premised as a Lemma. REMARK If e taken!; or the hanes for the positive and the negative errors e supposed aurately the same; then our expression y expunging the powers of. will e the very same with that shewing the hanes for throwing [S points preisely with n die eah die having as many faes ( 4 ) as the result of any single oservation an ome out different ways. Whih may e made to appear independent of any [6] kind of alulation from the are onsideration that the hanes of throwing preisely the numer Q with die whereof the faes of eah are numered?v9??g?;? \9 must e the very same as the hanes y whih the positive errors an exeed the negative ones y that preise numer: ut the former are evidently the same as the numer =9k;B$( Q (or AS ) with the same n die when they are numered in the ommon way with the terms of the natural progression and so on; eause the numer on eah fae eing here inreased y 9H[; the whole inrease upon all the ( ) faes will e expressed y = 9[;B$( ; so Q as there was efore for the numer Q ; sine the hanes for throwing any faes assigned will ontinue the same however the faes are numered. that there will e now the very same hanes for the numer =9!";BR( PROPOSITION II. Supposing the respetive hanes for the different errors whih any single oservation an admit of to e expressed y the terms of the series l_3 l m G = 9n<;B j:# o R (whereof the oeffiients from the middle one Ep9n<;I derease oth ways aording to the terms of an arithmetial progression); it is proposed to find the proaility or odds that the error y taking the mean of a given numer EpqGI of oservations exeeds not a given quantity Following the method laid down in the preeding proposition the sum or value of the series here proposed will appear to e +r.s&t33 +1.G0D2Gu v w 3 +1x (eing the same with the square of the geometrial progression 2 y(>= ;gzz\[ { { { [ BI. And 3

5 the power thereof whose exponent is q (y making 5}q and 4}5~9Z; ) will therefore e & p g(y= ;M?WLBDC (Z= ;?Z BjCJ 5z L:? LM & p M 1LM p? into ;U #N #N #N \.[68] V Ü Whih two series eing the same with those in the preeding Prolem (exepting only that the exponents in the former of them are expressed in terms of q instead of ) it is plain that if S e here put 5 qƒ9 Q (instead of 9 Q ) the onlusion here rought out will answer equally here; and onsequently that the sum of all the hanes wherey the exess of positive errors aove the negative errors an amount to the numer Q preisely will here also e truly defined y [; > E/S#IU(<? [; > E/S^?4\IV( [; > E/S^?d#4\IU(?>;? [; > E/S^? 4\IU(?>;?e But this general expression as several of the fators in the numerators and denominators mutually destroy eah#n other #N may e transformed to another more ommodious. Thus the quantity E/S#I in the first line y reaking the numerator and denominator will eome E z;i E ] I E I S EaSz;I EaS>I EaS I EaS?A;I E [;I E >I E I S whih y equal division is redued to =SV?A;#B =S?eDB =S? B { { SUz; ; 5 supposing 5 SV 5zqƒ9@ Q. #N #N appears that EaSO?W4\I [69] 5 Ž o Ž Ž whole given expression (making Š Š5?Œ#4?>;?e? [ E?>;IU(< Šj?A; Š?d Šj? [ E?>;IU( Š Šj?A; Š Šj?e Š Š??>;?e? E?>;I*ˆ In the very same manner y making SŠm5 SH?Œ4 and Šm5 SŠK EX5?>4\I it E? ;I Consequently our Š Š Š? 4 ) will e transformed to [ Š Š Šj?A; Š Š Š?e E?A;I(?A; Š Š ŠK? E?>;I(?>;?d 4

6 2 whih expression is to e ontinued until some of the fators eome nothing or negative; and whih with H5 ; is the very same with that exhiiting the numer of hanes for points preisely on die eah having 4 faes. And in this ase where the hanes for the errors in exess and in defet are the same the solution is the most simple it an e; sine from the hanes determined answering to the numer preisely the sum of the hanes for all the inferior numers to may e readily otained eing given (from the method of inrements equal to?a;?e? Š?d z Š Š?A; Š Š?e z Š Š ŠK?>; Š ŠJŠ?d?> Š?A; E I?A Š Šj? E IV(?A; Š Š ŠK? E IU( The differene etween whih and half ( 4?>; Š??e E?>;I( E IU( ) the sum of all the hanes (whih differene I shall denote y ) will onsequently e the true numer of the hanes wherey the errors in exess (or in defet) an fall within the given limit (Q ); so that L will e the true measure of the required proaility that the error y taking the mean of q oservations exeeds not the quantity proposed. [0] But now to illustrate this y example from whene the utility of the method in pratie may learly appear it will e neessary in the first plae to assign some numer for 9 expressing the limits of the errors to whih any oservation is sujet. These limits indeed (as has een oserved) depend on the goodness of the instrument and the skill of the oserver: ut I shall here suppose that every oservation may e relied on to five seonds; and that the hanes for the several errors?g #Š Š? Š Š? Š Š?g Š Š?; Š Š f Š Š!; Š Š Š Š Š Š Š Š inluded within the limits thus assigned are respetively proportional to the terms of the series Whih series is muh etter adapted than if all the terms were to e equal; sine it is highly reasonale to suppose that the hanes for the respetive errors derease in proportion as the errors themselves inrease. These partiulars eing premised let it e now required to find what proaility or hane for an error of or 5 seonds will e when instead of relying on one the mean of the oservations is taken. Here 9 eing 5 and q@5 we shall have EX5 qgi5 ; 4Eh5 9! ;I5 and Eh5 qƒ9r Q IR5 Q ; ut the value of Q if we first seek the hanes wherey the error exeed not one seond will e found from the equation 5 š!; ; where either sign may e used (the hanes eing the same) ut the negative one is ommodious: from whene we have Q EX5?VqGIO5? ; and therefore 5 Š5 f Š Š&5k Whih values eing sustituted in the general expression aove determined it will eome 1œ - E ;I( ;V - EG;I( ^? Ÿ E ;IO( P - EG;IM? 1ž 1Ÿ - * f 5k# : and this sutrated from ( 5 for the value of Gœ (U G ) leaves orresponding: therefore the required proaility that the error y taking the mean of the six oservations exeeds not a single seond will e truly measured y the fration Ÿ-ž1ž-ž3 P žg 3ž1ž-1 *- -ž ; and onsequently the odds [1] will e as to 5

7 = x 2 Š x ( x 9? to 1. But the odds or proportion when one single oservation is taken is only as 16 to 20 or as ƒ ž to 1. To determine now the proaility that the result omes within two seonds of the 5 Š or near as truth let e made 5?g ; so shall Q Eh5?gqGIg5?; ; therefore f Š Š 5 ; and our general expression will here ome out = ; whene 5 ; f ;; K;. Consequently -œ-1*- Ÿ- 3 -ž-ž1- *1G 1ž will e the true measure of the proaility sought: and the odds or proportion of the hanes will therefore e that of to or as 29 to 1 nearly. But the proportion or odds when a single oservation is taken is only as 2 to 1: so that the hane for an error exeeding two seonds is not th part so great from the mean of six as from one single oservation. And it will e found in the same manner that the hane for an error exeeding three seonds is not here 1- th part so great as it will e from one oservation only. Upon the whole of whih it appears that the taking of the mean of a numer of oservations greatly diminishes the hanes for all the smaller errors and uts off almost all possiility of any large ones: whih last onsideration alone is suffiient to reommend the use of the method not only to Astronomers ut to all Others onerned in making experiments or oservations of any kind whih will allow of eing repeated under the same irumstanes. In the preeding alulations the different errors to whih any oservation is supposed sujet are restrained to whole quantities or a ertain preise numer of seonds; it eing impossile from the most exat instruments to take off the quantity of an angle to a geometrial exatness. But I shall now show how the hanes may e omputed when the error admits of any value whatever whole or roken within the proposed limits or when the result of eah oservation is supposed to e exatly known. [2][ Fig. 20 in the margin at this point] Let then the line AB represent the whole extent of the given interval within whih all the oservations are supposed to fall; and oneive the same to e divided into an exeeding great numer of very small equal partiles y perpendiulars terminating in the sides AD BD of an isoeles triangle ABD formed y the ase AB: and let the proaility or hane wherey the result of any oservation tends to fall within any of these very small intervals N e proportional to the orresponding area NMQ< or to the perpendiular NM; then sine these hanes (or areas) rekoning from the extremes A and B. inrease aording to the terms of the arithmetial progression it is evident that the ase is here the same with that in the latter part of Prop. II; only as the numer 9 (expressing the partiles in AC or BC) is indefinitely great all (finite) quantities joined to v or its multiples with the signs of addition or sutration will here vanish as eing nothing in omparison to 9. By whih means the general expression E I? Ž Ž Ž E I( % Ž Ž o Ž Ž Ž Ž E I( o i- ª there determined will here eome E Im? Ž Ž Š E Ig( 5 *«««P w? o Š Š? o Š Š Š 5zqƒ9M (wherein Q Š&5? Q Š Š&5?$#9 Š Š Š5? 9 ) and therefore the value of in the present ase eing *««w ()? =?9&BDC =?e9&bdc i it is evident that the proaility ª of the error s not exeeding the quantity (in taking the mean of q oservations) will truly e deformed y ;? *««w (?A; n? ( (?A; C 6

8 x? ( o?e i whih may whih may e represented y the urvilinear area CNFE or-[3]responding to the given value or asissa CN ( 5 ). Now though the are all of them here supposed to e indefinitely great yet they may numer 9 and Q e exterminated and the value of the expression determined from their known relation to one another. For if the given ratio of to 9 of CN to CA e expressed as that of to 1 or whih is the same if the error in question e supposed the part of the greatest error; then Q eing 5zqƒ9 ( 5Fqƒ9K Q ) will e 5[qƒ9K ^qƒ9 and therefore 5Fq#(V= ; B ; whih let e denoted y ± : then y sustitution our last expression will eome ;? 3««w () ±? = ±?>;BDC = ±?edbjc o = ±? BjC i whih series is to e ontinued till the quantities ± ±?e; ±?_ eome negative. As an example of what is aove delivered let it e now required to find the proaility or odds that the error y taking the mean of six oservations exeeds not a single seond; supposing (as in the former example) that the greatest error that any oservation an admit of is limited to five seonds. Here q eing 5[ EX5kqGI5 ; and 5 œ we have ± Eh5zqD(^= ;K? BI5 i ; and therefore the proaility will e equal to ;O? *««w ()= = G-x i jbjc G?e= ;( i BDC = $( i jbdc Gm?]=$( ; i BDC $( fki jbjc GB^5 f i # nearly; so that the odds that the error exeeds not a single seond will e as 068 to 02332; whih is more than three to one. But the proportion when one single oservation is relied on is only as 36 to 64 or as 9 to 19. In the same manner taking 5 œ it will e found that [4] the odds of the error s not exeeding two seonds when the mean of six oservations is taken will e as 0985 to 0015 nearly or as 65 to 1; whereas the odds on one single oservation is only as 6 4 to 36 or as Ÿ to 1: so that the hane for an error of two seonds is not G th part so great from the mean of six as from one single oservation. And it will farther appear y making 5 that the proaility of an error of three seonds here is not P - th part so great as from one single oservation: so that in this as well as in the former hypothesis almost all proailities of any large error is ut off. And the ase will e found the same whatever the hypothesis is assumed to express the hanes for the errors to whih any single oservation is sujet. From the same general expression y whih the foregoing properties are derived it will e easy to determine the odds that the mean of a given numer of oservations is nearer to the truth than one single oservation taken indifferently. For if ² e put EX5 ;V? I_5 ³ and 5 then y eing 5Fq ² the quantity (>µ E I ±? = ±?>;BDC?A; = ±?edbjc i (expressing the proaility that the result falls within the distane ² of the greatest limit) will here y sustitution eome q E I ( µ ²? = ²? BjC?>; = ²?d BjC i whih in the ase of one single oservation (when qm5 ; and 5k ) is arely ² and its fluxion ²V ² : therefore if we now multiply y ²U ² the produt #q #N (>µ E I ² ²? = ²? BDC ²U ²?>; = ²?d BjC ²U ²ji

9 will give the fluxion of the proaility that the result of q oservations is farther from the truth or nearer to the limits than one single oservation taken indifferently. And on- P sequently the fluent thereof whih is #N 3««w x into 0 [5]? (\¹Uº «t #N º u v 0D2 t N º u v 0» ( ¹ º «t #N º u v 0 2 t Nº u v 0» will when ² 5 ; e the true measure of the proaility itself whih in the ase aove proposed where q5% and 5 ; will e found 5 fki and onsequently the odds that the mean of six is nearer to the truth than one single oservation as 55 to 245 or as 151 to 49. 8