132 
MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
for all values of q from q— — \ to^= — (D— 1); and q being 0 or positive, 
The preceding investigations show the general form of the function P(a, c, ...)^, 
viz. that 
P(a, h, c, + 2 /( Ao, A,, .. A;_,) per/,, 
a formula in which k denotes the number of the elements a, h, c, . . &c., and / is any 
divisor (unity excluded) of one or more of these elements ; the summation in the case 
of each such divisor extends from r=0 to r=/i:— 1, where A" is the number of the 
elements a, b, c, . . . &c. of which / is a divisor; and the investigations indicate how 
the values of the coefficients A of the prime circulators are to be obtained. It has 
been moreover in effect shown, that if D=ad-6 + c+.., then, writing for shortness 
P( 9 ) instead of P(tt, b, c, ..)q, we have 
P(^) = 0 
for all values of q from q= — 1 to — (D— 1), and that q being 0 or positive, 
P(-D-9) = (-)>^->P(9); 
these last theorems are however uninterpretable in the theory of partitions, and they 
apply only to the analytical expression for P(y). 
I have calculated the following particular results : — 
P(l,2), =i{2</+3 
+ (1,-1) pci- 2, 1 
P(l, 2, 3)? =^|6j‘+36?+47 
+ 9(1, —1) per 2, 
+ 8(2,— i,— 1) pcr3,j 
P(l,2, 3. 4)q =^|25r='+309^+135^+175 
+ (9^ + 45)(l, — 1) pcr2, 
+ 32 (1,0, -1) per 3, 
+ 36 (1,0, -1,0) per +1 
P(l, 2, 3, 4, 5)9=g^|309"+9009='+9300^^+38250^+50651 
+ (1350^+10125) (1,-1) pcr2, 
+ 3200 (2, -1, — 1) pcr3, 
+ 5400 (I, 1, — 1, — 1) pcr4, 
+3456(4, - 1, - 1 , - 1 , - 1 ) per 5, 1 
