On a New Formula in Definite Integrals. 53 



superimposed strata of igneous rock in the latter region may- 

 represent not only the direct effect in loco of more or less par- 

 oxysmal thrusts, but also the reflex action of the simultaneously 

 progressing anticlinals in the high Sierras. 



VII. A New Formula in Definite Integrals. 

 By J. W. L. Glaisher, M.A*. 



1. INTEGRATE the identity 



"o-V 2 +«2* 4 -- • '= ffts -^o (i^ip + A%(~ ^3~, . . (I) 



(where Aa n =a n+l — a n ) between the limits zero and infinity, and 

 the right-hand side becomes 



j (yt^ 5 



IT 



= r 2 (fl o cos 2 0-Afl o tan 2 0cos 4 + .. .)sec 2 &/0 



Jo 



IT 



= r 2 (tf -A« sin 2 6 + A 2 tf o sin 4 0- . . .)dd 

 Jo 



tt/, 1 A , 3 1 A2 5 3 1... \ 



so that 



i 



00 77" 



(tf — a l x 2 + a 2 x 4 - — . . .)d%= - a_i. (2) 



The definition of the symbol E is contained in E« n = « n+1 ; and 

 of course, a n being only defined for n a positive integer, #-.£ is 

 without meaning. But in cases where a n involves factorials, 

 there is a strong presumption, derived from experience in similar 

 questions, that the formula will give correct results if the conti- 

 nuity of the terms is preserved by the substitution of gamma 

 functions for the factorials. This I have found to be true in 

 every case to which I have applied (2). 



* Communicated by the Author. 



