176 Mr. W. H. L. Russell on certain Integrals. [June 15, 



Especially in cases of valvular disease of the heart is it likely to be of 

 great service ; and we think it advisable to bring again before the notice 

 of physiologists and physicians this power of the veins, which, although 

 so long known, appears of late years to have been overlooked. 



XXIII. On certain Integrals.^' By W. H. L. Ecjssell, F.R.S. 

 Received June \o, 1876. 



The follo^^^.ng are certain definite integrals which will, I hope, be 

 found interesting : — 



r c?2e-(^+l)^+^-~^=-(l-r+r(r-l)l- ....). 

 Jo 



C"" d sine cid 77 , l + 



1 ~, = — log • 



J 1 — cos-d 2x ^ 1 — X 



sin dQ _ TT tan-\^' 

 Jo l+c^''cos'0 X 



TT 



dde ^ 3^ cos (cos^ e sin 30) = ^ v^^ 



( dddsmd ^ 1 + x' cos' = — { 07 Vl + x' + log, (x-\-Vl-\- x^) } . 

 Jo 



r*lT 4. ^ 



\ 0sin0{ VirH^c^0 + l^l-x cos 6}dd = -^{(1 + xf - (1 -xf}. 



Jo ^''^^ 



p 0sin0(Z0 7r_^^ l-^xs/2 + x'' jr_^^^_^W2 



l + a;^cos^0""2tr + 1-^'"^'* 



TT 



log, cos { e"" '''' cos (x cos 20 - 20) + e""" '''' cos (x cos 20 + 20) }dd ■■ 

 I* + a cos' + V 1 + (2a + a') cos^ = ^ V a + 2. 



2a? 



2 



.COS2 0COS (a sin COS 0) 



Pcze ^ 



Jo 1 + 2 ^^0^2 COS (a sin cos 0) + e^^ cos^ 2(eH 1) 

 I r?0 Vl-|-e««o^'^ cos (usin0cos0)+ '^l + 2e«'=o^'^cos(asin0 cos0)-f e ""^o^'^ 



Jo ''^^ 



V2 l + e'^ 

 ^ COS COS (0 — a sin cos 0) — cos^ 



2eacos2 (.os(a sin0co3 0) + e2»°o«'^ 4(e?_l)* 



