PROFESSOR .1. J. SYLVESTER AND MR. J. HAMMOND 



D (q - r) + $ E (q - r)* + &c., ad inf. 

 + * ................. (1) 



The law of the coefficients will then be established by proving that 



If there were any terms, of an order superior to that of (q r)*, whose indices 

 did not obey the assumed law, any such term would make its presence felt in the 

 course of the work ; for, in the process we shall employ, the coefficient of each term 

 has to be determined before that of any subsequent term can be found. It was in 

 this way that the existence of terms between (q r)* and (q r)* was made manifest 

 in the unsuccessful attempt to calculate the coefficient of (q r)*. It thus appears 

 that the assumed law of the indices is the true one. 



It will be remembered that p, q, r, . . . , are the halves of the sharpened 

 Hamiltonian Numbers E + 1 , E, 'Ei H _ v . . . , and that consequently the relation 



.. .^.., -._!- 2) 



1.2 1.2.3 



may be written in the form 



- 1 _L 1 ) _ r(2r-l)(2r-2) s(2s - 1)(2 - 2) (2s - 3) 



1 2 2.3 2.3.4 



t (2t - 1) (2* - 2) (2< - 3) (2< - 4) (2 - 1) (2 - 2) (2 - 3) (2 - 4) (2tt - 5) 

 2.3.4.5 2.3.4.5.6 



- .................. ....... (2) 



The comparison of this value of p with that given by (l) furnishes an equation 

 which, after several reductions have been made, in which special attention must be 

 paid to the order of the quantities under consideration, ultimately leads to the deter- 

 mination of the values of A, B, C, . . . , in succession. 



Taking unity to represent the order of q, the orders of 



p, q, r, s, t, u, v, w, . . . 

 will be 



2 li i i i t^, 3*2, 3* 



Hence, after expanding each of the binomials on the right-hand side of (!) and 

 arranging the terms in descending order, retaining only terms for which the order is 

 superior to , we shall find 



* In the text above Q represents some unknown function, the asymptotic value of whose ratio to 

 (? ~ r )* > 8 n t infinite. 



