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

 which easily gives, by (24.), 



^(«)=^yo 3:::cos^2^ = Wo 3:^3^^=^^ ^- (^2.) 



And whenp = 1, we have 



1—5= - i(cos^ + \/^Tsin^), (43.) 



1 <?^ ^ 



2^;^ ^ZI^ = - TT-i (cos 6-\/ —Isind) dOy (44.) 



of which the integral, taken from ^ = ~ tt to ^ = tt, is F(l ) = 0. 



6. Let us consider now this other integral, 



^ , , \ r^ ^"d0 , , 



^(^)=W-.^^ ^''-^ 



The expression (13.) gives 



^"n)lPn = F{p) + G{p); (46.) 



therefore, by (34.), we shall have 



G ip) = s;;^; p„, if i^>^3-i. . . . (47.) 



For instance, let p = 1 ; then multiplying the expression (44.) 

 ^^ -5« = -(l+i,^V=T)«^ (^g) 



the only term which does not vanish when integrated is 

 1 mr~^dd, and this term gives the result 



G (1) = «, (49.) 



which evidently agrees with the formula (47.), because it is 

 well known that 



P„ = 1 when;; =1, (50.) 



the series (2.) becoming then the development of (1 —x)~^. 



7. On the other hand, let phe <. 'V/S — 1 ; then, obsei'ving 

 that, by (33.), 



F(p)= i2-2p)-i=X^n^o^„, (51.) 



we find, by the relation (46.) between the functions F and G, 



= -(P„+P.+, + P.+ .+ -)-J'""' 

 For instance, let p = o ; then, by (40.) and (45.), 



that is. 



G(0) = :l_^JriJY" ^^(sin^)" 



^ 27r J-ttI-^-I sin^' i^^.) 



G(0)=^-^y^ iTihTF-' (^*-> 



if n be either = 2i—l, or = 2i. Now, when j5 r= 0, P„ is 



