110 The Rev. B. Bronwin on the Integration of some 



z=x--P+^{f{iJ-^x) + '\){ij-otx)]-\-x-^P+- {af'(i/ -Sx) 



+ i\I/'(.?/-aa;)} +a;-2'^+3{a,<p"(?/ — €a') +^4'" {y-^'^)} + 



to the term containing x~p inclusive. The constant coeffi- 

 cients a, a I, &c., b, ij, &c. are easily determined, and could 

 not be conveniently put down. 



Make k=hT>', and this becomes 



If IS ='D''- hi, by a paper of mine in the Mathematician for 

 November last, this reduces to 



x'^p +{k-2px)tc=0, 



or rather 



x~^ +{k-2px)u=D-P0 = A + AiX + Ap-ix"-'. 



ax 



Integrating 



ft /» k 



r^ x-^Pn=t dxx-^P-^ e~l- (A + Ai^; +Ap-ixP-'). 



Restoring the value of k, and changing A, A„ &c. into 

 /(«/)» /i(i/)j &c., we have 



u=x'PeJ'"Jdxx-'"-'J^f (y-^) +xf,{y- ^) . . . . 



=<P(.y)+^<Pi(3/) .... +'•*'■•''' 'P2p(i/)+«''' 4/ (2/+^). 



We must sometimes change the order of the terms, as in 

 this example ; for if in substituting in the given equation we 

 begin at the wrong end of the series, we shall have <^,J (^dy, 

 fffdy"^, &c. instead of <p, <p', f", i?cc. 



Thus 



z = \yp-Ui=xP^'(^{y)-^xP <p,(2/) . . . + <P;,+ i(?/) +^'""4'(i/ + ^) 



