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



1 _ -2iu + e-9V-l) {l-^eOy—x) 

 l-d~ {l+2ucosd + u''){l-2l3cosd + l3^y ^ ' 

 of which the real part may be put under the form 



+ 1 « ^^c rt L «2 » • • • (19.) 



l+2«cos^ + «^"^ 1-2 13 cos d+ 13^* 

 if X and fi be so chosen as to satisfy the conditions 



\{l+^-) +fj,{l+u^)=2{l3-u), ...(20.) 



X/e-/xa = l-«/S, (21.) 



which give 



l-a^ /S^— 1 ,„^, 



^= ^T^' '^=m:F (""-^ 



The imaginary part of the expression (18.) changes sign 

 with 6, and disappears in the integral (14.); that integral 

 therefore reduces itself to the sum of the two following : 



^P' iSTtJo H-2aCOS^+«^ 



+ 



1^ r Jd 



S TtJ 1 — 



{^-i)de 



(23.) 



^STtJ 1— 2/3cos^ + ( 



32 



in which, by (16.), a + /3 has been changed to 4 5. But, in 

 general if a^ > h-, 



^^ " . . (24.) 



/: 



/: 



a + 6cos^ s/d^ — ly^ 

 the radical being a positive quantity if a be such ; therefore, 

 in the formula (23.), 



{\-o?)de . 



\ l+aacosdl + a^ - '"' ^ ' 



because, by (16.), a cannot exceed the limits and ^, 5 be- 

 ing supposed not to exceed the limits and 1, so that \—o^ 

 is positive. On the other hand, /3 varies from to 4-, while 

 s varies from to 1 ; and /S^ — 1 vvill be positive or negative, ac- 

 cording as 5 is greater or less than the positive root of the 

 equation 



5-- + 5 = i (26.) 



Hence, in (23.), we must make 



r^^^F^T^^ = 7r,or = -TT, . . (27.) 

 according as 



s> or K^^^f^l (28.) 



and thus we find, under the same alternative, 



X 



