384 PROCEEDINGS OF THE AMERICAN ACADE:MY § 7 



hence 



dT" + ' ^ ^ -^ V3 ^3 J 

 + i?(^e" + ^+ 0^ + 2) €«+■ (^ e^T-\-± e^T^)\-\ 



The ratio of the term containing A in the 7i -\- i^' deriv- 

 ative to the corresponding term in the ;^^'' derivative is 



(;2 _|_ i) g (1_ -^ ± er) ; that of the B term is 



JL _[- (;2 -f- 2) e(-^ -)- 4_ er) ; that of the last term is 



^ _|_ (2;/ -f- i) e/^-I .+ A. ^t\ so that the succes- 

 sive derivatives ultimately form a divergent series. 

 Referring, however, to equation IX. we find 



/ _ ^ . 7/ _ (^^o . / _ I ^'g° etc 

 (;/ + i)' = - ^ "^"'^ 



(;^ + 2)' = 



n ! dr 



(;z + I)/ «^/" + ' 

 so that the ratio of the (;/-|-2)''to the (^/ + i)'' term in 

 the series, 



e = a' ^ /// + rV^ 4- dt^ + eV^ H [- ;//"- 



+ (;/ + i)7" + (;/ + 2)'r + ' + etc., 

 will be 



-(;2 _|_ ly n-^1 d^e. 



The last factor can be represented as the quotient of two 

 sums, each of a number of terms; the ratio of no term in the 

 numerator to the corresponding term in the denominator 

 can by any possibility exceed 



