Mr. A. Cayley on a System of Algebraic Equations. 341 



But, as appears from what precedes, a denumerant may be equal 

 to zero, or may denote a number of partitions taken negatively; 

 and it is not allowable, in the place, e.g., of the first denumerant, 

 to write simpliciter, number of partitions of ctm—a/jb in respect of 

 ab—a(3, ac—ay, &c. The notion of a Denumerant is, in fact, an 

 important generalization of the notion of a number of partitions. 



2 Stone Buildings, W.C., 

 October 4, 1860. 



XLIV. On a System of Algebraic Equations. 

 By A. Cayley, Esq.* 



npHE determination of a, b, c from the system of equations 



b^ + ca=zfjbj 



tf + ab — v, 



in the case where \, fi, v have the values 16, 17, and 18 respect- 

 ively, is the problem known as Colonel Titus's Arithmetical 

 Problem. See Maseres' ' Tracts on the Resolution of affected 

 Algebraic Equations,' Lond. 1800. If for shortness we put 



(J — V — C 1 , 



then the third equation gives b—- ; and substituting this value 



of b in the two other equations, we have 



2 , ac ^ 

 a 2 + — =\, 



a 



<r 2 



■^ + ca=fi-, 



or what is the same thing, 



a 3 —\a +crc=0. 

 c« 3 -^ 2 + o- 2 =0, 

 And from these equations, eliminating «, we have 



<7 4 -3cV+(3c 4 -2X^)o- 2 -c 2 (c 4 -X ) a)c7 + cV-^(^ + ^ 3 ) 

 + \V=0, 



where a — v— c 2 . The equation in c 2 is thus of the fourth order ; 

 and in like manner, if instead of c 2 we take <t as the unknown 

 quantity, and substitute therefore for c 2 its value v— -a, the equa- 

 tion in <r will be also of the fourth order. And effecting the 



* Communicated by the Author. 



