364 Professor Sir William Rowan Hamilton on 



A == 2 1 " 2 ^"y c0 ^(giLf) 2/, co S (2^^^) 

 2 P 2 2 *-W<> V * ) 3o7^ ' ' V 9 ') 



if, according to a known form of development, which the fore- 

 going reasonings would suffice to justify, we write 



- x L_ + _i = l +A 2 A 2 + A 4 /* 4 + A 6 /* G +&c. . (20.) 

 e — 1 



If p be a large number, the rapid and repeated changes of 



COS ( 2 79 T i v i 



sign of the numerator of the fraction i-J—. 1 produce 



n COS X l 



nearly a mutual destruction of the successive elements of the 

 integral (19.), except in the neighbourhood of those values of 

 x which cause the denominator of the same fraction to va- 

 nish; namely those values which are odd positive multiples 



of — (the integral itself being not extended so as to include 



SB 



any negative values of %). Making therefore 



jra(S»-.l)-I + C0) . . . . (21.) 



in which i is a whole number > 0, and w is positive or nega- 

 tive, but nearly equal to ; we shall have 



cos (2p x — x) = (— 1)^ T * ~ sin (2p co — co), 



exactly, and cos x = (— I)*co, nearly ; changing also 

 2 P 



to the value which it has when w = 0, namely 



/2\ 2p ~2v 



I — j (2 i — 1) ; and observing that 



/<>> 7 sin (2 p co — co) , 



d co v-£ '- = 7r, nearly, . . (22.) 



even though the extreme values of co may be small, if p be very 

 large ; we find that the part of A 2 , corresponding to any one 



value of the number z, is, at least nearly, represented by the 

 expression 



{-\f- l 2{2i-\)- 2p 



(2^-1)^ 5 • • • (23.) 



which is now to be summed, with reference to z, from i — 1 

 to i = oo. But this summation gives rigorously the relation 



s (0 ><-ir , '=(i-8-*')S (0 7 r2 '' . (24.) 



we are conducted, therefore, to the expression 



e-^y 



