(2.) 



14 The Rev. B. Bronwin on the Lwerse Calculus 



Operate with (— j on both members; then 



{-iyT{n)T{p)^ ^ = {£)J^ ^v-d,^a + a^) ; 

 or 



^ 

 We may put ip [a) under a different form by making 



a+d?= -. The forms of f [a) obtained in (1.) and (2.) differ 



from those given by Mr. Boole in the Cambridge Mathema- 

 tical Journal, No. 20 ; but by varying the process a little, we 

 might obtain his results. We may observe that the least 

 valueof 7win(l.) must be greater than (—1), and in (2.) greater 

 than n+p or /. 



In (p(^) = e^^(p(0), which is Taylor's theorem (D standing for 



-r-), change (^[x) into <p (s-^), and then x into log x\ we have 



(p(a:) + .r»<p(eO) (a.) 



Tlierefore, also, 



<^{a—x) = {a — x)^<^{s% 

 and 



r''x'^-^dx<^[a — x)=i< J x'^-'^dx{a—x)^ \<^{^^) 

 Consequently 



and 



rwimi)(„_^)D.„^(,o)=^(,_^), 



or 



= / xP-'^dx'\>{a—x)', 



nn)np) Y{S^rU) "''^'■^^^°) ^fo""'" ^-^^(^-^^ 



Operating with ( y-) on both members, we find 



