1872.] Anniversary Meeting. 21 



As (n) increases without limit, this equation may be written 



or 



Let 3l converge to a certain quantity p, then p will be given by the 



u h-l 



equation 



Let p x be the greatest root of this equation ; then the series will be con- 

 vergent if is less than unity, or if x be greater than p v 

 u n _ x x 



It appears from my last paper that in certain cases a quantity of the 

 form 



\/ a-\-bx-\- V c + ex 



may be regarded as a solution of a differential equation of the form here 

 considered. Hence we are able, by means of the principles given in 

 this paper, to represent, under these circumstances, 



fdoGS/ a+bce+ V c+eco 



by a converging series when (x) does not exceed an ascertainable value. 

 This method evidently admits of a great variety of applications. 



November 30, 1872. 

 ANNIVERSARY MEETING. 

 Sir GEORGE BIDDELL AIRY, K.C.B., President, in the Chair. 



Mr. Abel, for the Auditors of the Treasurer's Accounts on the part 

 of the Society, reported that the total receipts during the past year, 

 including a balance of .£28 2s . 2d. carried from the preceding year, 

 amount to £4308 14s. 8c7., and that the total expenditure in the same 

 period amounts to £3860 17s. 10c?., leaviug a balance at the Bankers 

 of .£424 0s. 3c?., and <£23 16s. 7d. in the hands of the Treasurer. 



The thanks of the Society were voted to the Treasurer and Auditors. 



