144 REV. T. P. KIRKMAN ON THE 



c?ii, &c. can be taken, which shall not have a positive sum, 

 (unless k=\) and that 180 times this sum, added to — 47556^, 

 will always give a number nearer zero than — 259200. 



When *6^_i=l, p is one of the ten numbers 0, 6, 12 . . . 

 55 ; and inspection readily shows that no k of the set do, c/q, 

 &c. can be taken consecutively, which shall not have a nega- 

 tive sum, (unless k=l) and shall not, when multiplied by 

 180, make with 50544A— 42120 a number less than 259200. 

 By continuing this kind of inspection, of (c?pH-c?p_6+ . . to A 

 terms) for any value *6^_c=l, the reader may convince him- 

 self that the number qP^ is really the integer nearest to the 

 sum of the terms in x; and the only proof that appears in any 

 case practicable of this point, is by inspection of the constant 

 in qPx' He will therefore excuse me, thus far, from the trou- 

 ble of writing down the 60 terms of that constant. 



Neither do I think it necessary to have them assigned in 

 order to the finding of 7?^?; for, in fact, there is a method of 

 obtaining the value of Dp above written, from the terms in x 

 of sPx, without any knowledge of the numbers dp in that ex- 

 pression ; and I am convinced, that all the terms in x of yV;^ 

 can by the same method be deduced from the terms in x of 

 gPar. But I shall reserve the proof of this, which is too long 

 to be here inserted, for a future communication. It is very 

 important that a method should be made out of finding nPx 

 from the terms in x of n-i^x, without the enormous labour of 

 specifying every term in the constant circulator of „_iPa;; for 

 the number of those terms increases very rapidly, as we ad- 

 vance to higher values of n. 



The preceding results, as far as the expression for qP^, are 

 identical with those given by Sir John Herschel in the Trans- 

 actions of the Royal Society for 1850; '* On the algebraical 

 expression of the number of partitions of which a given number 

 is susceptible." In that memoir the reader may see some 

 account of what has been before done on this subject by 

 Prof. De Morgan and Mr. Warburton ; and he will perhaps 



