180 Prof. J. J. Sylvester's Pr oof of the undemonstrated 



.-. a.o.u-oa.u=((ft*o)-(0*n))u 



- ib i-^-^4o—-^-^i- ii i 



If now /oa p .b q ,c r . . .1*, where /o is a number, be any term in 

 . U, we have 



P + q + r+...+t=j | b thesi 



... &.O.U-O.O.U, 



i. e. 



Y — 4- A — 4- — 4-Z— ^ 

 \ c?a d& dc " ' d// 



= 2/o(i; — 2w) (a p . &*. c r . . . F) 



= (ij— 2iv)TJ, as was to be proved. 



If now for U we write D a differential in a?, we have 

 OD = ; and therefore 



O.O.D = SD, 

 where 8= if— 2w. 



Again, 



O . 0(0 . D) - . 0(0 . D) = (ij-2(w + 1))0 . D ; 



for . D is of the weight w + 1 ; 



.-. n 2 .0 2 .D=n.OSD4-(S-2)O.O.D 

 = (2S-2)O,0.D 

 =S(2S-2)D. 



Similarly it will be seen that 



O 3 . 3 .D = S(2S-2)(3S-6)D, 



and in general 



n 2 . 2 .D = S(2S-2)(3S-6) . . . (q8-(q 2 + q)*)B 



= (1.2.3... q)(B . S=I 8=2 • • • S-?-l)D ? 



the successive numbers 8, 28—2, 38— 6, &c. being the succes- 

 sive sums of the arithmetical series 8, 8—2, 8—4:, 8—6, &c. 



