496 tRAi^SACTIOJJS OF SECTION A* 



Now, from the diil'erential eijuation it follows that the left-haud side of (7) 



Hence, equating separate!}' the coefficients of cos (j) and sin (j), we find 



and 



^^^^-HS'ft = (9) 



ax dx (Lv ' 



From (9) it follows that S ,® = constant = c (say). 



dx 



Inserting this value in (8), we find 



{l-»y}B'.«S-<g)W. . .(,0, 



Now, since S = R^ = P'^ + Q'^, and is therefore of the form 1 + terms containing 

 inverse (even) powers of .t, it follows that the value of c'^is unity. 



and 



.-. S'l'^ = l, is. R<1+^°) = 1 . . . (11) 



dx \ dx/ 



(l-'in_i)s...sS-j(g)'.l . . . <12) 



This last equation will enable us to expand S. 



Diflerentiating, and writing n" — \^2k for brevity, we have, after reduction — ■ 



x'^^V':^)dx-'^d;?=^' 



L.r-* X- dx J dx dx'' 



Now let S = 1 + "« + "^ + . . . Then (13) becomes 



Ix^ a-' a' J x^ x" 



wlience n.^ = 1:, 



a, = la,{2k-2), 



and generally, 



2m„. = - i2r(2r - l)(2r - 2)a.„-^ + 2(2/' - l)^-a„.-,. 



2r-l 



2r 



{»'-(^-'')}».,.. 



Therefore, finally, 



S-R'-l + i 4««-l 1^3_(4««-l)(4«'-3') ,^^. 



