24 REPORT 1871. 



of n any integer greater than zero, it may sometimes be useful to be provided 

 with an easy test to secure ourselves against the omission of any of them. Such 

 a test is furnished by the following theorem :— 



2(l—x+xy—xy2. . . .)=0. 



thus, ex. g>:,i{ x-\-2y+3z+4it+ .... =4, the solutions are five in number, viz. 



(1) y=2, 



(2) t = l, 



(3) X=l 2=1, 



(4) x=2y = l, 



(5) x = 4, 



the values of the omitted variables in each solution being zero. The five corre- 

 sponding values oil~x+xy. . . . are 



1, 1, 0, 1, -3, 

 whose sum is zero. 



The theorem may be proved immediately by expressing the denumerant (which 

 is zero) of the simultaneous equations 



fX+2y+3z+...=n,. 



\x+y+z+ =0,1 



in terms of simple denumerants according to the author's general method, or by 

 virtue of the known theorem, 



(i-t)a-i'){i-t').... 



= 1-_1_+ *' «' . <" -L 



This gives at once the equation 



1 ^ +-^ 1. 



Hence the coefficient of t" in the above wi-itten series for all values of w other than 

 zero is zero. But it will easily be seen that the coefficient of t" in the first term is 21, 



in the second term 2a^, in the third Sa'y, &c. ; so that 2(1— aj+a'y ) = 0,aswasto 



be shown. Thus we have obtained for the problem of indefinite partition a new 

 algebraical imsymmetrical test supplementing the well-known pair of transcen- 

 dental symmetrical tests expressible by the equations 



y n(a+y+g...) _g„_i 

 n.rnyn2... ' 



(2-y+y+z.. n(.r+y+2...) _Q, 

 nxnijnz. .. 



* Subject of course to the conditions that n is greater than 1. If x, y, z,.. .,(o repre- 

 sents any solution in positive integers of the equation 



*+2y+32 . . . +ru, = r, 

 it IS easy to see that 



2(-)«+y+. . . . +n(£±H:iiid:^)= 1, _i, or 0, 

 nx n^ iiuj 



according as n, in regard to the modulus r+l, is congruent to 0, 1, or neither to nor 1, 

 for the left-hand side of the equation is obviously the coefficient of x» in the development of 



1 . 1-.T 



I.e. 



On making r=cc , this theorem becomes the one in the text. It obviously affords a 

 remarkable pair of independent arithmetical quantitative criteria for determining whether 

 or not one number is divisible by another. 



