120 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



is the "prime circulator to the period a," if a q = 1 or according as q is 

 divisible by a or not, and 



A t + A l+i + + 4 (x _Dn.i = (t = 0, 1, -, I - 1; X = all}. 



He showed how to evaluate the A's and then the coefficients A, , L, M 

 of the non-circulating part. Next, he evaluated the number P(0, 1, , k) m q 

 of partitions of q into m terms 0,1, , k, with repetitions allowed, known 

 to be the coefficient of x q z m in (1 2) -1 (l xz)~ l - (! x k z)~ l . 



Finally, Cayley proved that the non-circulating part of the fraction 

 <f>(x)/f(x) is the coefficient of 1/t in 



l-xe 1 /(<T*)' 



Cayley 45 later considered his last formula for tj>(x) = 1, obtaining a 

 formula equivalent to Sylvester's theorem, and applied it to find 



, 2, ,6)< ? . 



Cayley 46 noted that P(0, 1, , m) e q - P(0, , m)\q - 1) is the 

 number of asyzygetic covariants of degree 6 and order q of a binary quantic 

 of order m. Thus it is the coefficient of x e in the expansion of a given 

 function. He calculated the literal parts of covariants by Arbogast's 

 method of derivatives. 850 



F. Brioschi 47 started with Euler's remark that the number C s of parti- 

 tions of s into r parts ^ n is the coefficient of x 3 z r in the expansion of 



Z = (1 - g)-i(l - xz)- 1 - --(I - x n z}~\ 

 Now Z = 2\l/(x)z r , where 



f(x) 



*(*) = 



-x 1 ") 



Since \j/(x) is unaltered by the interchange of n and r, C s equals the number 

 of partitions of s into n parts ^= r. Let i, 2 , be the roots oif(x) = 0; 

 0i, 02, the roots of <f>(x) =0 and set 



1 1 



Then 



-T7IT = 81 + S 2 X + 







(7) Ci = si, 2C 2 = CiSi + s 2 , , pC p = C p _iSi + - + Cis p -i + s 



p . 



Phil. Trans. R. Soc. London, 148, I, 1858, 47-52; Coll. Math. Papers, II, 506-512. 



48 Phil. Trans. R. Soc. London, 146, 1856, 101-126; Coll. Math. Papers, II, 250-281. Cf. 



F. Brioschi, Annali di Mat., 2, 1859, 265-277. 

 47 Annali di sc. mat. e fis., 7, 1856, 303-312. Reproduced by Faa di Bruno. 92 



