186 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(63) |,S'4 = l-2«2+i.s^ ^, = s_|s3 + ,Jjs^ 

 [^g = 1 — 3 s- + s* — tV s^ etc. 



VIII. The Form of/(s). The E's. 

 To simplify the integration of equation (29), we put 

 f{s) = ^ (s) • u, 

 where u is an integral of the equation 



-^ + s • u = 0, 



ds 



say 



' _£: 



u = e 2 ' 



where e represents the base of natural logarithms, so that 



(64) f(s)=6{s) -e-S. 

 Writing as an abbreviation for 6 (s), we have 



9/ 



so that (29) becomes 



[ 



57^-^-^J^ 2=2-r^-e 2' 



_S2 



because e 2 is independent of the p's (and P's) ; therefore, 



which is a differential equation for (s).' 



The assumption II that <^ (x) is developable by Taylor's theorem 

 throughout the range of possible values of x implies that if/ ($), f(s), 

 and (s), are similarly developable ; we may therefore put 



(66) ^(^)=XtI''' 







where A, (for each value of ^) is a function of the P's alone. This de- 

 velopment implies that we shall approximate more and more nearly to 



