196 Mr. A. Cayley on an Analytical Theorem connected with 

 We deduce hence 



or what is the same thing, 



-log(i + (l-^)(l+i)0)}, 



which may be converted into 



/i6 log(l + A) r' df 



J'^-l + b A l6 + /(l-a;) 



Miog(i+A) / r^ ^ r dt n 



or what is the same thing, 



hb log(l + A) 



p d^ 



Jo b + t-i 



l + b A J„ b + t—tx 



AMog(i+A) / r> ^^ f ^ ' 



"'■ A \j^(l + ft)(l + /0)-*(l + (H-^»)/0) J^b + {l + b)tO-a;{l+b)tQ. 



the obiect of the transformation being to express fx so that x 



1 log(l + A) 

 may only enter under the form . _t» • The factor ^ 



which multiplies the first of the three definite integrals, might 

 be reduced to unitj-^, but it is more convenient not to make this 

 change. -j^ 



Now if a fraction -r — ^ be operated upon by expanding in 



ascending powers of x, and multiplying the successive terms of 

 the development by Pq, Pj, Pg, &c., it is converted into 

 1 



(A2-2AB/ia7 + B2a?2)** 



Hence from the foregoing expression iov fx we pass at once to 

 the expression for ^(/i., x) ; that is, we have 



dt 



+ 



hb log(l + A) p 



<l>(fi,x)-^_^^ ^ j^ (A2- 2AB/z.r + B^a?^)* 



^Mog(l + A) ff dt r dl n 



A lJo(A'2-2A'BW + B'^.0* Jo(A"2-3A"BV + B"V)*J 



