1878.] Mr. W. H. L. Russell on certain Definite Integrals, 129 



February 28, 1878. 



Sir JOSEPH HOOKER, K.C.S.I., President, in the Chair. 



The Right Hon. Sir William Henry Gregory was admitted into the 

 Society. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " On certain Definite Integrals." By W. H. L. Russell, 

 F.R.S. Received January 10, 1878. 



The integrals in the three preceding papers may nearly all be in- 

 cluded under the following general theorem : — 



Let <p(x)=u Q J r uiX-\-Uo,x r + . . . u r x y "+ • • • which series is, of course, 

 supposed to be convergent, and let II be a general functional symbol, 

 such that 



n0(fl»)=« DX (O> J +mix(1) »+«*x(2) • • • 



and let x (0)=/TO0, x (l)=/UYd0 . . . x (r)=jW r d0 ... the inte- 

 grals being supposed to be taken within certain limits : then 



(62.) fU(p(Vx)de=Ucp(x). 



The following integrals require for the most part other methods to 

 determine them : 



(63.) T — =*JL.(!±vJ™_V 



K J ) (l + x 2 )Xl-2acosx + a 2 ) 4 l-a 2 \ e-a (e-af f 

 More generally we may obtain 



(64) T - • (65.) f* dx.x am ra 



v ) (l + x 2 y(l-2acosx + a 2 ) J (l + xy(l-2acosx + a 2 ) 



(66) f°° C ^ X sinaa? (67) f°° ^ X cosaa; 



Jo(l + aj 2 ) r sin&# Jo(l + a? 2 ) r cos bx 



/nn\ f 00 dx sinaa? {fsQ } 1°° cos ax 



(70.) 



cos bx 

 (x — a 2 x)dx 



o ((1 + a + cr) sin ax — a sin Sax) (l + x 2 )' 



( 7 1 ) f°° (l — a^d x 



Jo ((1 + a + a 2 ) cos ax + a 



cos Sax)(l + x 2 ) r 



VOL. XXVII. 



