CHAP. Ill] 



PARTITIONS. 



121 



Hence 



Sl - 1 



s 2 Si 2 



Ss S 2 Si 





 

 



Sp Sp i Sp 2 s\ 



Set f(h/k) = or k, according as h is not or is divisible by k. 



Thus 



:)- 



m 



G. Battaglini 48 proved Sylvester's formula for the wave W q by means of 

 the special case (i = 1, , a r = 1) where the coefficient of x n in (1 x)~ r 

 equals the coefficient of I ft in e nt (l e~')~ r . To evaluate the waves, we 

 need the value of S: 



S = 2 ^zr", F a = Z^z*, ft = 2 A*/, 



^6 



where, in /S, the summation extends over all imaginary kth roots Xi of unity. 

 We can find c's such that 



F a fF b = C + dXi + 



+ Ck-l 



Since 2x, y = 1 for ,7 ^ k 1, we see that S is 70, 71, , jk-i, according 

 as n = 0, 1, , k 1 (mod k), where 7/ = A;cy Co Ci c k -i, 

 and 27,- = 0. Hence we obtain Cayley's 44 prime circulator [with k q for a q ] 



F. Brioschi 49 proved Sylvester's 43 theorem by use of Cauchy's theory of 

 residues. He noted that, if ai, , a r are all primes, 



w m = - i: 



in s =i 





1 or o, 



i=\ 



according as m is or is not one of the a's. Here y\, - - -,yi are the primitive 

 wth roots of unity, and /3i, j8 2 , are the a's not divisible by m. Applica- 

 tion is made to 2x\ + 3x 2 + 5xs = n. 



A. Cayley 50 wrote (pi 1 - -p n r r ) for the partition of n into n\ parts pi, 

 n 2 parts p z , etc., where pi > p 2 > It is conjugate to the partition 



of n. For example, (6 3 2 2 2 ) and (5 2 3 I 3 ) are conjugate partitions. Given 



48 Memorie della R. Accad. Sc. Napoli, 2, 1855-7 (1857), 353-363. 



49 Annali di sc. mat. e fis., 8, 1857, 5-12. 



"Phil. Trans. Roy. Soc. London, 147, 1857, 489-499; Coll. Math. Papers, II, 417-439. 

 Reviewed by E. Betti, Annali di mat., 1, 1858, 323-6. 



