588 Mr. J. J. Sylvester'8 Note on Burman's Law for 

 C(r : p, s la, t :t, &c.) 



which by Lemma 1 



+ &C. 



= C((p-l), {r-l):p,s:a, /:t, &c.) 

 H-{l4-rF((7-p)}C(r:/o, cr-1, (s-1) ic, /:t,&c.) 

 H-{l+rF(T-<7)}C(r:p, 5:cr, T-1, (/-1)t, &c.) 

 + &c. &c. 



r(cr— p), F(t— 0-), &c. meaning quantities which are respectively 

 zero when o- — l>p, r — l>o-, &c., and respectively units when 

 {(T—l)=p, T~l=(7, &c. ; foritwillbeobvious that if <r— l=p, 

 the quantity [r:p, a—i, {s — l) : a; t : t, &c.] becomes 



[(r + l):/3, (s-l):cr, ^:t, &c.], 

 and consequently when divided by G(r) . G(s) .G(/),&c., does not 

 give C((r + 1) : p, {s— 1) : o", / : t, &c.), but 



(1 +r) X C((r+1) :p, (5-I) : cr, ^ : r, &c. ), 



and so similarly for the cases of t — 1 = cr^ &c. 



Now let us suppose that we are considering any group ^pp ... to 

 r places, era... to s places, &c.), or more briefly {rip, sia-j tiT.,.), 

 the numerical coefficient of the term x^ . a;'^ . x^. . . . in the in- 



verse development of —^. 

 dvy- 



And first, suppose that p is not 2. 



The coefficient in question will evidently be made up exclusively 

 of the following parts (each, however, affected with the factor 



jf*— 1 „ 

 (— )''~^) derived from the expansion of 1-, for which the 



law to be established is supposed to hold, viz. 



C(p-1, (r-l):p, s'.a, tir.kc.') 

 + (l+rF(o— p))C(r:p, o— 1, {s-1):<t, /:t,&c.) |. ^^^ 

 + (l+rF(T-(7))C(r:p, s.a, r-l, (/- 1) : r, &c. ) 



+ &C. 



