FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES

Transcription

1 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria FINITE DIFFERENCE CONCEPT Give a fuctio f() the first differece of the fuctio is defied Af() = f( + )-f(). (NOTE: There is a more geeralized fiite differece ivolvig a step of size h but this ca be reduced to the above by a liear trasformatio.) EXAMPLES f() = +3, Af() = ( + V +3 - ( +3) = f() = \Af() = 3( + V + 7( + )+- (3 +7+) = Fidig the first differece of a polyomial fuctio of higher degree ivolves a cosiderable amout of arithmetic. This ca be reduced by itroducig a special type of fuctio kow as a geeralized factorial. A geeralized factorial GENERALIZED FACTORIAL W W = x(x - )(x - ) - (x- + ), where there are factors each oe less tha the precedig. To tie this i with the ordiary factorial ote that The first differece of x' ' /?W =! EXAMPLE x< 4 > = x(x-)(x-)(x-3). is foud as follows: Ax () = (x + )x(x - V - (x-+3)(x-+j- x(x - )(x - ) - (x-+)(x- + ) = x(x - )(x - ) - ( x - +3)(x - +)[x+ - (x - + )] = x ( - }. Note the ice parallel with takig the derivative of x i calculus. To use the factorial effectively, i workig with polyomials we itroduce Stirlig umbers of the first ad secod kid,. Stirlig umbers of the first kid are the coefficiets whe we express factorials i terms of powers of x. Thus x () = x, x () = x(x~ ) = x -x t x (3) = x(x- )(x-)(x-3) = x 3-3x +x x (4) = x(x - )(x - )(x - 3) = x 4-6x 3 + llx 6-6x. Stirlig umbers of the first kid merely record these coefficiets i a table. Stirlig umbers of the secod kid are coefficiets whe we express the powers of x i terms of factorials. x = J * x = x -x+x = x^) +x^ x 3 = x 3-3x +x + (3x - 3x) +x = x (3) +3x( ) +x () As oe example of the use of these umbers let us fid the differece of the polyomial fuctio 4x - 7x 4 + 9x 3 - x +3x-. Usig the Stirlig umbers of the secod kid we first traslate ito factorials, 3

2 4 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES Givig power of * TABLE OF STIRLING NUMBERS OF THE FIRST KIND TABLE OF STIRLING NUMBERS OF THE SECOND KIND Coefficiets otx( k ) x> -7x 4 9x 3 -x 3 3x TABLE OF FACTORIALS,() yw y(v y() M) x( ) + 33x (4) + 67x (3) + 33x () +4x (l) -. Usig the formula for fidig the differece of a factorial the first differece is give by 0xW + 3x (3) +0x () +66x (l) Now we traslate back to a polyomial fuctio by usig Stirlig umbers of the first kid. The resultig polyomial fuctio is x x x x 0x( 4) x (3) x^ x () x 4 + x 3 +x +9x

3 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES 6 A POLYNOMIAL FUNCTION FROM TABULAR VALUES From the above it is evidet that the first differece of a polyomial of degree is a polyomial of degree/? - ; the secod differece is a polyomial of degree - ; etc., so that the th differece is a costat. The ( + ) st differece is zero. As a matter of fact sice at each step we multiply the coefficiet of the first term by the power of x, the/7 differece of a 0 x + a x ~ + a x ~ a _ x + a \sao! Coversely if we have a table of values ad fid that the r th differece is a costat we may coclude that these values fit a polyomial fuctio of degree r. For example for fix) = x 3-7x + 3x-8 we have a.table of values ad fiite differeces as follows. X f(x) Af(x) A*f(x) A*f(x) i The problem is how to arrive at the origial formula from this table. Suppose that the polyomial is expressed i terms of factorials with udetermied coefficiets bg, bi, b, The problem will be solved if we fid these coefficiets fix) = b 0 +b x () +b x () + b 3 x (3) + b 4 x (4) + b x () + - Af(x) = b t +b x () " +3b 3 x m +4b 4 x (3) +b x (4) + - A f(x) =!b +3*b 3 x () +4*3b 4 x () + *4h x( 3 ) + - A 3 f(x) = 3!b 3 +4*3*h 4 x () +*4*3b x () A 4 f(x) = 4!b 4 + *4*3*b x () + -. Set* = O. Sice ay factorial is zero forx = O we have from the above: Hece f(0) = b 0 or b 0 = f(0) Af(0)=bi A H0) =!b A 3 f(0)= 3!b 3 A 4 f(0)= 4!b 4 or or or or bi ' = Af(O) bi- -- A f(0)/! bs -- A 3 f(0)/3! fm = o) + LHO)xW + L f^-x^ + L 3 f - x^ + A 4 f x W + b 4 - ' A 4 f(0)/4!.

4 6 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES [FEB. This is kow as Newto's forward differece formula. We ca fid the quatities f(o), Af(O), A f(0), A 3 f(0), A 4 f(0), - from the top edge of our umerical table of values provided the first value i our table is 0. f(x) = -8+x + 6x^/ + 30x (3) /3! = -8 + x + 8x - 8x + x 3 - x + lox = x 3-7x +3x- 8. Stirlig umbers of the first kid ca be used i this evaluatio. SUMMATIONS INVOLVING POLYNOMIAL FUNCTIONS Sice a polyomial fuctio ca be expressed i terms of factorials it is sufficiet to fid a formula for summig ay factorial. More simply by dividig the k th factorial by k! we have a biomial coefficiet ad the summatio of these coefficiets leads to a beautifully simple sequece of relatios. To evaluate k >,et E k = * () meaig that the value is a fuctio of. The +l A<p() = Yl k ~ L k = + - Now A/? = 7 ad A ( >/ =. Hece <p() = k = () /++C = ( + l)/ + C, where the C is ecessary i takig the ati-differece jsice the differece of a costat is zero. This correspods to the costat of itegratio i the idefiite itegral. To fid the value of C let/7 = /. The Hece / = U/ + C so that C = 0. ) k = ( + )/= («/') a well-kow formula. Next, let p r ) - - ^, A*M~z {>+>)-< ( f e r)-(«r) The differece Hece k = l = 7 shows that C = 0. The sequece of formulas ca be cotiued: ad i geeral l k+ 3 ) = l 4 3 ) = E (*;;) = (\ + ;r) Oe could derive the formula for the summatio of a factorial from the above but proceedig directly:

5 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES 7 u Hece, Takig = r, so that C = 0. kw = y(), Aip() = ( + ) (r). k= y M = y km = < + D (r+) +C. LJ r+ rl = (r+v (r+l) /(r+) h r+ Agai there is a oteworthy parallel with the itegral calculus i this formula. For examples we take some formulas from L. B. W. Jolley Summatio of Series, EXAMPLE.(4) p. 8, ] (3k - )(3k+) = *+ *8 + 8* +- This equals f; m +3k-)= ; m< ) + m< i )-) = 9 (^i^'* (JL±M^-. ( + i)+c. Takig =, * = 6* - * + c so that = J (3k- )(3k+) = = ( ). EXAMPLE. (0) p. 0 ]T k(k + 3)(k + 6) = *4*7 + 4**k + 3*6*9 + -= ]T (k 3 + 9k + 8k) = J^ + C (k(3) + m W + 8k() > = (+ )( 4 > + u ( + )< 3 ) +8(]l+j l + c = (ju-^[( -)(-)+6(-)+6]+C = (+)( + 6)( + 7)/4 + C. Settig/? =,*4*7 = **7*8/4 + c so that C = 0 ] P k(k + 3)(k + 6) = (+)(+6)( + 7)/4. EXAMPLE 3. (49) p. 0 ] T (3k-)(3k + )(3k + 4) = *4*7 + 4*7*0* This ca be chaged directly ito a factorial: 7 (k-/3)(k+/3)(k + 4/3) = 7 (k+4/3p> givig 7( + 7/3)^/4 + C = (3 + 7)(3 + 4)(3 + )(3 -)/ + C. Settig=, 8 = {0*7*4*) * so that C = 6/

6 8 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES [FEB. (3k - )(3k + )(3k + 4) = (3 + 7)(3 + 4)(3 + )(3 - )/ + 6/. SUMMATIONS THROUGH NEGATIVE FACTORIALS Startig with the relatio x( m h( x - m )W = x( +") set m = -. x(~ >*(x+)( ) = x( > =. T\\extfoxex ( - )=/(x+)( ). Possibly this bit of mathematical formalism seems ucovicig. Suppose the we defie the egative factorial i this fashio. Ax^ = /[(x+ + )(x+)(x+- ) ~.(x + ) - /[(x +)(x + - )(x + -) -(x + )(x + )] = /Ux + Hx + - ) -. fx +)][/(x+ + ) - /(x + )] = -/[(x + + )(x + )(x + -)-(x + ) = -x^~^ showig that the differece relatio that applies to positive factorials holds as well for egative factorials defied i this fashio. Cosequetly the ati-differece which is used i fidig the value of summatios ca be employed with egative factorials apart from the case of -. EXAMPLE. /[k(k + )(k+)] = J < k -) ( ~ 3) = ( - )/(-) + C = -/[( + )( + )] + C. Settig =, /6 = -/(*3*) + C, so that C = /4 J f/fkfk + )(k + )] = /4 - /[( + )( + )J. EXAMPLE. Jollev, No. 0, p. 40 /[(3k - )(3k + )(3k + 4)] = (/7) /fk - /3)(k + /3)(k + 4/3) = (/7) fk - /3) (~ 3 > ~ = (/7)( - /3) (' )/(-) + C = - /[6(3 + 4)(3 + ) + C. Settig =, /(*4*7) = -/(6*7*4) + C; C= /4 /[(3k - )(3k + )(3k + 4) = /4 - /[6(3 + 4)(3 + ) EXAMPLE 3. Jolley, No.3, p. 40 * (k - )/[k(k + )(k +) = /[(k + )(k +) - f/fkfk + )(k + ). h=i The secod summatio was evaluated i Example. The first gives k<- ) = ( + )(- l )/(-) + C. Altogether, the result is -/( +) - /4 + /[( + )( + ) + C. Settig =, /6 = -/3 - /4 + / + C so that C = (k - )/[k(k + )(k + ) = 3/4 - /( +) + /[< +)( + ).

7 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES 9 Let there be two fuctios f() ad g(). The DIFFERENCE RELATION FOR A PRODUCT A f()g() = f( + )g( + ) - f()g() = f( + )g( + ) - f( + )g() + f( + )g() - = f( + )Ag()+g()Af(). This will be foud useful i a variety of istaces. SUMMATIONS INVOLVING GEOMETRIC PROGRESSIONS A geometric progressio with terms ar ca be summed as follows: f()g() ]T ar^ But Ar = r + - r = r (r - Hece = y(), Ay() = ar y() = Y, Settig =, a = ar/(r - ) + C so that C = -a/(r - ). Hece, ar = A (arh) = arh /( r - V + c - The summatio ar *" = a(r - )/(r- ). kr k = <p(), A#() = ( + )r + \ A(r + ) = ( + )r +l (r- ) + +l r usig the product formula o page 8 with M the first fuctio as ad the secod as r +. HeCe ( + )r +l = A [r + /(r - )] - r + /(r - ). A~ ( + )r +l = r + /(r - ) - r + /(r - ) + C. Settig =, r = r /(r - ) - r /(r - ) + C; C = r/(r - V. Accordigly kr k = r + /(r - ) - r + /(r- ) + r/(r - ). EXAMPLE. ]>? k*3 k = *3+*9+3*7 + 4*8+*43 = 64. By formula *3 6 / /4 + 3/4 = 64. FIBONACCI SUMMATIONS A Fiboacci sequece is defied by two iitial terms Ti ad T accompaied by the recursio relatio Accordigly T+ ~ T + T - SUM OF THE TERMS OF THE SEQUENCE T k = <p(), Ay() = T +, AT = T + - T = T _ t. Settig =., Ti = T3 + C or C = Ti - T 3 = -T

8 60 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES [FEB. T h - T + - T SUM OF THE SQUARES OF THE TERMS T h = *(), A<p() = T +. The ati-differece bears a strog resemblace to itegratio i the differetial calculus. Just as we kow itegrals o the basis of differetiatio so likewise we fid ati-differeces o the basis of differeces. Thus we try various expressios to see whether we ca fid oe whose differece is the square of T +i. Hece A T T +i = T +t T + - T T +t = T +i (T + - T ) = T +. Settig/? = a, r f = T a T a+ +C k=oi Hece T = k+a = V("h Aip() = T ( +l)+a', ^+a T ++a~ T +a = ^+l+a- J ] k=m c - T a (T a - T a +i) T a T a _i, Yl T k ~ T T +l ~ T a T a -i. k=a SUMMATION OF ALTERNATE TERMS Settig k = m, A T ( +l)+a = T+l+a + C* ] T T k+a = ^H+a + ^- k=m T~m+a ~ Tm+l+a + C ' Y ^k+a ~ ^+l+a~ ^m-l+a k=m SUM OF EVERY FOURTH TERM T 4k+a = $(), A<pM = T4 +4+a A 7~4+a = T4+4+a ~ 7~4+a = ^4+3+a + "^4++a ~ ^4++a + T~4+l+a = ^4+3+a + ^4+l+a To meet this situatio we itroduce a quatity M ^ = T- + T +. Now V- + V+l = T _ + T + T + T + =-T i + T +T + T + T +i = T. To obtai a differece which gives T we start with V. By a process similar to that for T Cosequetly, A " 4 " +fl = V4+3+a + V4++a = U++a ' A " ' T4+4+a = (V4++a>/ + C = T 4k+a Settig =, 0 = T 4+a - V 6+a /, Yl T 4k+a = (U+l+a + l~4+3+j/ ~ < T +a + ^7+J/^ + T 4+a

9 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES 6 EXAMPLE. We use the terms of the sequece begiig,4., 4,,9,4, 3, 37, 60,97,:7, 4,4, 66,076,74, 87,4M37,933,9308, 34, 049,8790,3339, 49, ,6097, 90706,46766, Leta=Z T 4k+ = T 6 + T 0 + T 4 + T 8 + T = = 980. By formula we have (T 3 + T )/-(T 7 +T 9 )/ + T 6 = ( )7- ( )/ + 3 = 980. SEQUENCE WITH ALTERNATING SIGNS <-> kt k+a = *pm, &<P(") = (-l) + T + +a, V +a = T +l+a +T - +a k=m A(-D V +a = (-D +l V ++a -(-) V +a = (-) +l [V ++a +V + J = (-) + T ++a. Hece Let = m. <-> T k+a = (-D (V ++ J/ + C = (-D [T +a + T +a+ ]/ + C. k=m (-) m Tm+ a = (-D m [Tm+a +T m++j / + C E (' V h Tk + a = (- V [T +a + T +a+ ]/ + (- ) m+l [T m+a + T m+a+ ]/ + (- ) m T m+a. k=m Usig the,4 sequece oce more 7 Jl (- D h T k+3 = -T 9 + T -Ti 3 + T -T 7 = = -33. By formula we have -(T 7 + Ti 9)/ + (T 9 + Ti t )/ -T 9 = -( )/ + (97 + 4)/ -97 = -33. POWER of. GEOMETRIC-FIBONACCI SUMS y k T k = <p(); Atp() = + T + A T = + T + T = (T. + T ) = V, where we have used the product relatio o page 8 ad itroduced the sequece defied by V = T _i + T + i. Sice L V = * T (followig the same steps as for T ) ip() = A- ( + T + ) = + V + / + C. Settig/?^ 7,T ±4V / + C. Hece k T k = + (T + T + )/ + (6T -4T 3 )/. EXAMPLE. kl ~k = * + 4*4 + 8*+6*9+3*4 = 60 (,4 sequece).

10 6 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES [FEB. By formula [ 6 (4 +37) +6-4*]/ = 60. THE SUMMATION E ' k Tk. The direct approach leads to a apparet Impasse. We wish to fid the iverse differece oir + T +i. it is of the form A[r k T +i +r ]. Assume that This approach parallels what is doe i the solutio of differetial equatios. k,j, ad A are udetermied costats. Takig the differece ad settig it equal to r + T +i we have A[r k+ T +r) +i T - t +r k (r- l)t + +rhr- )TJ = r + T +. Replacig 7" w _^ o the left-had side by T +i - T ad equatig coefficiets of T + i ad T gives: From the secod/ = k +. The the first gives A [r k (r -) + ri + ] = r + \ r k+ + rhr - D - r> + = 0. A[r k+ -r k +r k+ J = r +. Lettig k = + ad A = /(r +r- ) establishes equality. Hece E rkj k = (r + T +i +r + T )/(r +r- ) + C, C = (-r T 0 -rt t )/(r +r- ) EXAMPLE (,4 sequece) By formula, THE SUMMATION r k T k = [r +l T +l +r + T -r To-rT l ]/{r +r-). 3 k Tk = 3*7+3 *4+3 3 * *9 + 3 s *74 = 430. (3 6 *3+3 7 *4-7-3)/ = 430. FIBONACCI-FACTORIAL SUMMATIONS ktk = *M Ay() = ( + )T +l AT = ( + )T ~l + T AT + = ( + )T +l + T + A~U + )T +l = T + - T +3 + T 3 +C = i which we have used the formula A~ T + = T +3 - T 3 = gives kt k Tt = T 3 -T 4 + T 3 +C; C = 0

11 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES 83 so that V kt k = T + -T +3 + T 3. Note that this is also A~ ( + )T +l, a fact that is used i the ext derivatio. EXAMPLE (,4 sequece) J ktk = *+*4+3* + 4*9 + *4 = 30. By formula * = 30. THE SUMMATION i k( h h = spi) A^W = (+0T + &( h + = ( + lph + +T + k () Tk = () T + - ( - )T T +4 -T 4 + C i which the formula for the previous case was used. tor =, T = T 4 - T +T 6 - T 4 + C; C = -T 3 VERIFICATION (,4sequece) k( h k = ( h + - ( - )T +3 +T +4 - T 4 - T 3 By formula J k () Tk = *0*+**4 + 3** + 4*3*9 + *4*4 = 46. *4*37-*4*60+*97-*9-* = 46. For/7 = J, E THE SUMMATION k^tu = <p{) A$() = { + )( 3 >T+i ^(3) T + = ( + D (3) T + +3( k + k(3)r k = (3) T + -3(-) () T (-)T +4-6T + +6T 6 + C. 6T 3 = 6T -6T 6 +6T 7-6Tg + 7T 6 + C; C = 6T k< 3 h k = ( 3 h + - 3( - Ph ( - )T +4-6T + + 6T 7.

12 64 FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES [FEB. VERIFICATION (,4 sequece) By formula for = 6, 6 ]T k( 3) T h = 6* + 4*9 + 60*4+70*3 = *60-60*97 + 4* 7-6*4 +6*37 = The formulas for the ext two cases are writte dow ad the patter that is emergig is oted. K (4) T h = ( 4 h + - 4( - ) (3) T +3 + ( - ph +4-4( - 3)T + +4T +6-4T 9 ]T k&t k = ( h + -(- )( 4 h +3 +Q( -)( 3 h +4-60( -30T +s + 0( - 4)T +6-0T T t t. The patter may be described as follows: For the r th differece:. The first term \s( r )T +.. For the portio, both ad r go dow by at each step. 3. For the T portio the subscript goes up by at each step for /* + steps. 4. The sigs alterate.. The coefficiets are the product, respectively, of the biomial coefficiets for r by 0!, \,l,,r!, respectively. 6. The last term is rlt r +i with sig determied by the alteratio metioed i 4. With the aid of these factorial formulas it is ow possible to fid polyomial formulas. For example. The first few formulas for the powers are give herewith. S ^ = E [k (4) +6k( 3 ) + 7k( ) + k( )]T k. X k T k = ( +)T + -(-3)T +3 -T 6 k 3 T k = :( )T + - ( JT T 6 + T 3 Y, k 4 Tk = ( 4 + ~ )T + - ( )T +3-3T 8 - T 7 ]T k T k = ( )T + - ( WO l6)t T 9 +30T 6 + T 3. I these formulas cosiderable algebra has bee doe to reduce the umber of terms dow to two mai terms by usig Fiboacci shift formulas. GENERAL SECOND-ORDER RECURSION SEQUENCES Give a secod-order recursio sequece govered by the recursio relatio T + = P lt +PT -l

13 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES m to fid Y, T k = <p() A^p() = T + ^[T+PlT-ll = T + +P T -T -P T = (Pi+P -DT. Provided P± + P - is ot zero, Tk = (T +i + P T )/(P +Pi-) + C. For/7=, T t = (T +P T )/(P +Pi- C = D + C [(P l -)T -T ]/(P +Pi-D Tk = [T + +PT + (Pl ~ DTl - T ]/(P +P t - ). EXAMPLE: T + = T -3T _ t 3,7,6,09,467,008; J T k = = 6. By formula (008-3* *3-7)/( ) = 6. If SUM OF TERMS OF A THIRD-ORDER SEQUENCE Such a sequece is boud by a recursio relatio of the form T + = PlT +PT-l+P3T -. ]T T k = ip(), A0/W = T +i A (T + IP3 + P)T -l + P3 T -) = T + + &3 + ^T + P3 - Hece \\Pi+P +P 3 -\ = T + + (P 3 +P -)T - P T _ t - P 3 T _ = (Pi +P +P3-DT. E is ot zero, f fe = {T»+ + (P 3 + P )T " + P 3 T-ilMPl +P+P3-D Ti+T = [T 3 +(P 3 +P )T +P 3 T ]/(P +P +P 3 -) + C C = [(Pt +P - )Ti + (Pi - )T - T 3 ]/(Pi +P +P 3 -) E T k = [T +l +(P 3 +P )T +P3T -i + (Pi+P -)T l +(P l -)T -T 3 ]/(P l +P +P 3 -) EXAMPLE. T +i = 3T + T _ t - T =. Next term is 64. By formula ( * + * - 4)/3 =. + C

14 66 FORMULA DEVELOPMENT THROUGH FINITE DSFFERENCES [FEB. The recursio relatio is FOURTH-ORDER SEQUENCES T +i = PlT+PT-l+P3T-+P4T -3. A etirely similar aalysis as was made for third-order sequeces leads to the formula where Tk = [T +l+(p + P3+P4)T+(P3+P4)T-l+P4T -]/(Pl+P+P3 + P4-l) + C, C = [(Pi +P+P3- VTi +(P t + P -)T + (P l -)T 3 -T 4 ]/(XP i - ). EXAMPLE. T + = 3T + T. t - 4T _ + 3T = 909. Next term is 03. By formula ( *6 + 4*3 + *4-6 ) / 3 = 909. FIBONACCI-COMBINATORIAL FORMULAS These are closely related to the Fiboacci-factorial formulas discussed o pp. 3-. However the added simplicity of these formulas merits a listig of the first few to show the patter. ()7* = [ AT + - T +3 + T 3, (I T k = () T + -["J \T T +4 - T ' [)\Tk= \ 3)T + -["]%+! +[ J )T +4-T + + T 7 S (4 ) T k = ( 4 ) ^ + ~ ( W 7 ) T +3- h \^ \ T +4-\~~ i ) T + + T +6 ~ T 9 Sequeces govered by FIBONACCI EXTENSION: SUMMING MORE TERMS T +l - T + T -t + T _, where three rather tha two precedig terms are added at each step have a summatio formula E T k = (T+l +T + T rl + Ti- T 3 )/. For sequeces govered by T + = T + T _i + T + T _ 3, where the four previous terms are added Tk = (T+l +3T +T _ t + T _ +T t + T - T 4 )/3. Where five previous terms are added at each step: Y, T k = (T+l +4T +3T t + T _ + T _ 3 +3T t +T + T 3 - T )/4. Where six previous terms have bee added at each step: T k = (T+l +T +4T _ +3T _ +T _ 3 + T _ 4 +4T t +3T +T 3 + T 4 - T 6 )/.

15 978] FORMULA DEVELOPMENT THROUGH FINITE DIFFERENCES i7 EXAMPLE. By formula = 4. (403 + *04 + 4*04 + 3*3 + * / = 4. CONCLUSION Fiite differeces have wide applicatio i formula developmet. There are, of course, may situatios i which the use of this method leads to difficulties which other procedures ca obviate. But where applicable the results are ofte obtaied with such facility that other procedures seem laborious by compariso. A GOLDEN DOUBLE CROSTiC MARJORIE BICKNELL-JOHNSON Wilcox High School, Sata Clara, Califoria 90 Use the defiitios i the clue story which follows to write the words to which they refer; the eter the appropriate letters i the diagram to complete a quotatio from a mathematicia whose ame appears i the last lie of the diagram. The ame of the book i which this quotatio appeared ad the author's last ame appear as the first letters of the clue words. The ed of each word is idicated by a shaded square followig it. CLUE STORY The mystic Golde Sectio Ratio, ( + >/ )/, called (A-,A-) (the latter most commoly), occurs i several propositios i (A-3, A-4) o Sie segmets ad (A-) This Golde Cut fasciated the aciet Greeks, particularly the (D-) who foud this value i the ratio of legths of segmets i the (D-) ad (D : 3) ad who also made studies i (D-4). The Greeks foud the proportios of the Golde Rectagle most pleasig to the eye as evideced by the ubiquitous occurrece of this form i art ad architecture, such as (C-) or i sculpture as i the proportios of the famous (C-) ; however, they may fiave bee copyig (C-3) for the Golde Proportio occurs frequetly i the forms of livig thigs ad is closely related to the growth patters of plats, as (C-4, C-, C-6) i which occur ratios of Fiboacci umbers. The Golde Sectio is the limitig value of the ratio of two successive Fiboacci umbers (amed for (G-) ), beig closely approximated by the (G-, G-3) By some mathematicias, the beauty of the (N) relatig to the Golde Sectio is compared to the theorem of the (D-) ad to such results from projective geometry as those see i Pascal's "Mystic (B) " or eve i the applicatios of mathematics i the Pricipia Mathematica of (I) while the costat ( + V&)/ itself is rivalled by (E-) ad (E-) Ufortuately, ot all persos fid mathematics beautiful. (H-) was oe of the four braches of arithmetic give by the Mock Turtle i Alice i Woderlad, ad the card player's descriptio of the sequece,,3,4, 7,,8, 9, 47, would be (H-) while some have to have all mathematics of practical use, such as i readig a (M) [The solutio appears o page 83 of the Quarterly.)