714 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



Let X = i(l + tv) ; then substituting in equation (8) we get 



i£.l(i-'''OM+/3^ = /3 (29) 



Assume 



z= B^ + B, tv- + B.w* ... =25iW^ (30) 



then substituting in (29) we get 



S5,{2j(2J - i)w''-^- 2(2i + 2) (2t + l)w-'+ (2j + 4)(2i + 3)w'' + = + ifiw-'} = 4/3, 



2{2(2j -l)5,-2[i(2i- l)-^]fi;_,+ j(2j- l)5,_„}w'*-= = 2j3. 



am 



This equation leaves B^ arbitrary, and gives on dividing by i (2 i - 1), and putting in succession 

 8 = 1, j = 2, &c., 



B. -2 (l - j^) 5„=2^, (31). 



B,-2 (^1 - ^J£, + fi„ = 0,&c.; 



and generally when i > 1 , 



g. = g,,.+ Ag,_,- ^ ^^ g,_. (32). 



t (2j — I ) 



The constants 5,, B.,,... being thus determined, the series (30) will be an integral of equation 

 (29), containing one arbitrary constant. An integral of the equation derived from (29) by replacing 

 the second member by zero may be obtained in just the same way by assuming sr = C,, w + C, ?<'^ + ... 

 when Oi, C2... will be determined in terms of C,, which remains arbitrary. The series will both be 

 convergent between the limits w = - 1 and w = I, that is, between the limits x = and .r = 1. 

 The sum of the two series will be the complete integral of (29), and will be equal to (x — x'')~^f{x) 

 if the constants B„ Co be properly determined. Denoting the sums of the two series by F,, (««), 

 /"„ {w) respectively, and writing a {x) for (a; - x^)'^f{x), so that sr = o- (x), we get 



<7 {x) = F, (w) + F„ (tv), o- (1 - x) = F^ (w) - F„ (w) ; 



and since 2F^ (w) = a (x) - cr (l - x) = (x - x^)-^ (p (x) by (21), we get 



ff{x) =F,(w) +X(a!-x^)-^<{)(x), o-(l -x) = F,(w) - ^ix - x")-- (p (x) (33). 



To determine Bo we have 



5o=<7(i), (34). 



which may be calculated by the series (9). 



10. The series (9), (30) will ultimately be geometric series with ratios x, w", or x, (2,» — 1)-, 

 respectively. Equating these ratios, and taking the smaller root of the resulting quadratic, we 

 get « = 5^. Hence if we use the series (9) for the calculation of ct (.r) from .j? = to .r = ^, and 

 (30) for the calculation of a (x) from .r = 1 to ■» = -|, we shall have to calculate series whicli are 

 ultimately geometric series with ratios ranging from to t. 



Suppose that we wish to calculate tr (x) or z for values of x increasing by .02. The process 

 of calculation will be as follows. From the equation (2 + (i) J„= (3 and the general formula (12), 

 calculate the coefficients A^,, J,, A.,,., as far as may be necessary. From the series (9), or else from 

 the series (9) combined with the formula (15), calculate a (^) or B„, and then calculate 5,, B2... 

 from equations (31), (32). Next calculate ct (x) from the series (9) for the values .02, .04,. ...26 

 of X, and F^ (w) from (30) for the values .04, .08..., .44 of w, and lastly (x - aP)'^^ {x) for the 

 values .52, .54..., .98 oi x. Then we have o- (a') calculated directly from a; = to .r = .26 ; equa- 



