Sii* W. R. Hamilton on certain discontinuous Integrals. 291 



that is, 



But, by (17.), 



F(;') = ^(l±l), (29.) 



F(P)=-^, or = (30.) 



2. (36.) 



the function F(p), oi- the definite integral (14.), receives 

 therefore a sudden change of form when p, in varying from 

 — 1 to 1, passes through the critical value 



p = V3-1; (32.) 



in such a manner that we have 



F {p) = (2 - 2p)-i, if ;j < -/ 3 - 1 ; .... (33.) 

 and, on the other hand, 



Y{p) = 0, ifp> ^/3-l (34.) 



For the critical value (32.) itself, we have 



5 = ^^^^' '=' = 2 V^-% yS = 1, . . . (35.) 



and the real part of (18.) becomes 

 l — a. 

 1+2 a cos + d 

 multiplying therefore by d 6, integrating from ^ = to ^ = tt, 

 and dividing by tt, we find, by (25.) and (14.), this formula 

 instead of (29.), 



F(''' = TTr = TJ (3'-) 



that is, 



F (i?) = i (2-2;))-*, if p = \/S -I. . . . (38.) 



The value of the discontinuous function F is therefore, in 

 this case, equal to the semisum of the two different values 

 which that function receives, immediately before and after the 

 variable j) attains its critical value, as usually happens in other 

 similar cases of discontinuity. 



5. As verifications of the results (33.), (34.), we may con- 

 sider the particular values p = 0, p = 1, which ought to give 



F(0) = 2-4, F(l) = (39.) 



Accordingly, when/; = 0, the definitions (9.) and (14.) give 



■& = V — 1 sin ^, (40.) 



1 /^ dd _\ p^ de 



