104 CARPMAEL ON THE DETERMINATION IN TERMS OF A 



Let (^^!^^:il^=^, 2^-"=^^;theu 



(p{.r, 0) = i + J 2" ^hj^Jl J II 



I 



If / be iucroased iudefinitely, this reduces to 



cp (,;■ 0) = i + i r '"'-^ ^ do (vi) 



This result agrees with the value of / ^^'" •^' ^ </«, g-iveu in Todhuuter's Integral Cal- 

 culus (second edition, p. 246). 



Integrating (vi) with respect to x, in times between the limits and x, we obtain 



by (i)- 



,„ _ 1 ,„ - 1 ,„ — 3 _ .^ 



cos (x H - '±+A n ) + -:^ ^ - ;^ «'" "+ &c. 



9> (,,, m) = i ^ + _ y -. —^ rf« (^ n) 



aud integrating (vii) with respect to x, n times between the limits x — I and x + J, we 

 obtain by (iv) and (v) — 



« „ „i + II 



+ ^/ 



cos (x H - —J- -) (-^^) + |,,^^_i « - ^^c. 



H 



m + 1 



rf« 



(Viii) 



By differeutiatiug 2 m times with respect to x, we find — 



^ . cos (^x # + -2" '^J {.-fjr) af, (ix) 



The result (ix) agi-ees with those of Cauchy and Laplace, (vii) aud (viii) differ from 



"' J [x — i n) 

 Cauchv's in containing the terms J . ,— and i .,„_|_^ respectively. Equation 



■» " j jjj, I^AAt -j- 7i 



(ix), it will be noticed, is only a particular case of equation (viii). 

 Cauchy designates the integral 



J , » +1 



dx 







where P is any function of the variable x, aud 



the first terms of the development of P in ascending powers ofx-, A bciug the greatest 

 whole number not greater than a, an extraordinary integral, and represents it by 



t- P 



y,. «-rl 



djù 



( a; = 



I X' ^ Xj 



