BINOMINAL Si'MMATION 103 



Integrating by parts and separating the terms involving fe'^'^dt 

 from the terms containing c-'^ we obtain equations (3) and (4). 



To determine Rs = Ds^i/Di, note that equation (6) gives t = (log Y 

 — log yy- and set v{x) = {x — .Vo)/(log Y — log yY'- so that .v may 

 be written in the form 



.V = xo + iix)L 



This form for x gives the expansion (Lagrange's Theorem for the 

 simple case where f(x) = x; see "Modern Analysis" by Whittaker 

 and Watson) 



Zo 5 ! \ dx''-^ 



Comparing this expansion for .v with the previous expansion (9) for 

 dx/dt, we obtain 



Di = i'(.vo) 

 and 



Ds+i_ ^ ^ I 1 d^v'+' \ 



A ' \s\v{x)' dx^ ;.=.,* 



Up to this point no particular form has been attributed to the 

 function y{x). From now on we deal with the function which consti- 

 tutes the integrand of the integrals in equation (1). 



The function ^(.t) = x''~^{\ — .y)""'' gives the expansion (log Y 

 — log y) = {x — .ro)-[^o + Ai{x — xo) + .42(.v — .vo)- • • •]. where Xo 

 = {c — l)/{n — 1) is the value of x for which y(x) is a maximum and 



1 r ^^^+^dog F-logy) ] 



' (^v + 2)!L dx^^-^ J..., 



or 



We are now prepared to e^'aluate Rg- Set 



= ^0 + --li(.v - .Vo) + Ao{x - XoY • • ' 

 g, = d'g/dx". 



and 

 Then 



V = g-^'\ 



^ = (3/2)g-^/^[(5/2)gr - g.g-], 

 ^ = - 2g-^[gsg^ - 9g,g^g + 12gi3]. 



