570 Mr. stokes, on THE CRITICAL VALUES OF 



and the next odd multiple. The equation (86) has also an infinite number of real positive roots 



lying between each odd multiple of - and the next even multiple. The negative roots of (85) and 



(86) need not be considered, since the several negative roots have their numerical values equal to 

 those of the positive roots ; and it may be proved that the equations do not admit of imaginary 

 roots. The values of X in (84) must now be restricted to be those given by (85) for the first line, 

 and those given by (86) for the second. It remains to satisfy (82) and (83). Now let 



/(2/) + /(-2/) = 2/i(2/), f(y)-f{-y) -2f.{y), 



F{y) + Fi-y) = 2 F, (y), F(y) -F(-y) = 2F, (?/) : 

 then we must have for all values of y from to /3, and therefore for all values from - /3 to 0, 



2^ Z- cos Xy =/,(?/), '2BLcos\y = F^(y) (87), 



•^C M sin ,xy=f,{y), ^DM sin i^y = F,(y) (88), 



where Z, = e"" - e'*', M = c"" - e""', 



,11 denoting one of the roots of the equation 



ix fi. cot IX (i= - hfi (89), 



and the two signs 2 extending to all the positive roots of the equations (85), (89), respectively. 

 To determine A and B, multiply both sides of each of the equations (87) by cos \'ydy, X' being 

 any root of (85), and integrate from y = to y = ^. The integral at the first side will vanish, by 



virtue of (85), except when X' = X, in which case it will become — (2X/3 + sin 2X/3), whence A 



4X 



and B will be known. C and D may be determined in a similar manner by multiplying both sides 



of each of the equations (88) by sin fx'ydy, fx' being any root of (8.9), integrating from y = to 



y = fi, and employing (89). AVe shall thus have finally 



M = 42X(2X/3 + sin 2X^)-'(6*' -£-*")-' |(e''C-'> - e-^i— ") ^ f,(y) cosXydy 



Jo 



+ (e" - £-") £ F^ {y) cos \y dy} cos \y, 



+ 42^t(2,xi3 - sin2^/3)-'(e^'- e-n" {(e"'"'" - e"""-"*) £\f,{y) sin ^y dy 



+ (e-" - e-"-^) £^ F,(y) sin ,xydy} sin,xy (90). 



46. Such is the solution obtained by a method similar to that employed by Fourier. A 

 solution very different in appearance may be obtained by expanding u in a series SFsinm^r, and 

 employing the formula (fi). We thus get from the equation (79) 



f]-V 2m 



n'Y+— {/(y) -(-!)» F(y) ] = 0, 

 dy' IT 



which gives 



1 r'-' 

 ■jr 



Y= Ae"y + Be-'" -- f {f(y') - (- l)"F(y')} {e"^^-'-''^ - g-"^^-"'') dy'; 



