Sir W. R. Hamilton on certain discontinuous Integrals. 289 



hence, by the property (3.)> P» is the coefficient of q" in the 

 development of 



((^|)!^)% p., 



it is therefore also the coefficient of <7° in the development of 



If then we make, for abridgment, 



^ =p + Jocose + i/^=T(l - |^)sin d,... (9.) 



we shall have the following expression, which perhaps is new, 

 for P„ : 



P„ = -^ f ^"cie; (10.) 



and hence, immediately, the required sum (5.) may be ex- 

 pressed as follows : 



^in)0P„:>f=—J_je-^-^; ... (11.) 



in which it is to be observed that x may be any quantity, real 

 or imaginary. 



3. We have therefore, rigorously, for the sum of the n first 

 terms of the series 



Po + P. + P^ + ..., (12.) 



the expression 



n-l 1 /^ TT ^1— S" 



of which we propose to consider now the part independent of 

 7i, namely, 



T., X I r^T (Id , , 



F(^) = W-.T3^' (1^^-) 



and to examine the form of this function F of p, at least be- 

 tween the limits/? = — 1, j» = 1. 



t. A little attention shows that the denominator 1— d may 

 be decomposed into factors, as follows : 



1-3 = - J^-(« + e9'/f) (l-;Sf-V-i); . . . (15.) 

 in which, 



a = 2s(l-s), /9=2s(l+s), .... (16.) 



and p = 1 -2s2; (17.) 



so that s may be supposed not to exceed the limits and 1, 

 since p is supposed not to exceed the limits — 1 and 1. Hence 



