440 f Mr. J. W. L. Glaisher on the Verification 



uneven powers of c vanish through the differentiation ; so that 

 although the right-hand side of (4) (omitting the factor c, which 

 is unaltered by differentiation), viz. 



— { 1 4- 2e- 2c ^ /w cos 2<a? + 2e" 4c ^cos 4.2? + ...}, 



involves both even and uneven powers of c, the expression de- 

 rived from it by differentiation, viz. 



—j- {1 + 2e-* 2 cos 2x + e" 4c2 cos 4# + . . . V, 



V 7T * 



is only a function of c 2 . 



The identity (1) could be just as well deduced from the well- 

 known equation 



jj. e a(*-x) + e -a(TT-x) ^ I CQS<r COS 2^7 . . 



Ya e a "-e-™ ~ 2^ + l 2 "^ 2 + 2 2 + a 2 + " ' * '' 



as from (3) ; but the method adopted has the advantage that it 

 is much easier to prove (3) in an elementary way than (5). In 

 fact, although (5) is demonstrated at once by Fourier's theorem, 

 it does not admit of being established very simply by ordinary 

 algebra (see De Morgan's 'Diff. and Int. Calc' p. 668). It 

 will be observed that in the proof given above, with the ex- 

 ception of the series for the cotangent, nothing higher than 

 algebra is required, as the continued differentiation may, as 

 usual, be replaced by a comparison of coefficients. 



The principle of the process employed consists in operating 



(d \ n ~ l 

 — r- 1 and using the two lemmas 



(/7v«-l e -bVn fn ft 2 



-i) V = V"--^" T * v; « 



/ A n - 1 J_ = A /2^. e -n e - a (7) 



V dti) n + a V7 ' * ' * {7) 



and 



n being infinite. The first lemma may also be written 



( 



— T-)e- b ^ n =— 7 =.e- n .e * 



In the 'Messenger of Mathematics' for September 1872 I 

 applied the formulae (6) and (7) to deduce the integral 



»°° */—. i 2 



e-^cosbxdx^-^- e~~*& , . . . (8) 



i 



2a 



