326 Mr. Cankrien on the Problems of 



function of xfflp and V to contain x / y / x u y lft and our ex- 

 pression becomes 



U^|^+§^Hv^/rf*^ + »*/"*«.£ + 



Again, in order to obtain w / we must, as was remarked, 

 substitute x + 8 x for # in this expression (2), and then change 

 x and 8 x into x t and 8 ^ : as we are only in want of the terms 

 which are multiplied by the simple power of 8 x y and 8 x lft it 

 will be sufficient to make this substitution in U. Now we 

 found by (1) above 



V=/3 +/dxM + Vp + Qq-pQ f +&c. 

 and, .-. V = a+/dxY - a + fdx{(3 +fdxM} + 

 fdxp(?-Q')+fdxqQ + &c. 

 = « +fdx{(5 + fdxM] +j/(P-Q0- 

 fdxy (P-QJ f pQ-fdxpQ' + &c. 



Let 8 U denote that part of the new value of U, when x + $x 

 is substituted for x in this expression, which is multiplied by 

 the simple power of 8 x : then 



8U - **{£ +/rf*M} + 8j/(P-Q') + #(P-Q0'8*- 

 8^. < y(P-Q / ) / + 8^>.Q +pQ / .^-b.j)Q'+ &c. 

 = Ix [/3 +/^^r M} + 8# (P-QO + 8^. Q + &c. 

 but, VBjb = 8.r{/3 +/^Z^M} + _p&#(P-Q / ) +g8^.Q 

 + &c.by(l) 



.-. 8U = V8* + {ly-plx) (P - Q' + &c.) + 

 [{%p-q$x){Q-&c.) + &c. 

 If then we substitute in (2) and then change 8. r and ly 

 into 8 x, and dy /t we find for the value of u 



h/^* 4J + ^yk -S + &c - + u , + *>i + 



(&j/ / -p,8^)[P / -Q/ + &c.} + (8^ - ql X/ ) {Q,-&c.}+&c. 



• /"*' d V 



In this expression I dx - — denotes the value of 



J dx 777" when x is substituted for x in it. In like man- 

 ner we shall find 



