644: REPORT—1899. 
say, symbolically, so that dQ represents the element dy,....dW,1; thus 
© is an absolute invariant for the system of parallel surfaces. Also, let 
1,= {p,do, 
where p,= 2p; ... ps, the summation being for all the combinations of s of the 
n—1 quantities p; and the integral extends over the whole of the original surface. 
Then if I’, denote the same magnitude for the parallel surface, we have 
V'.-Ie=[[3{(, +4) pH pe) pi ans PIG: 
Late (a1 9) 
Welly | eG 128)1 (0 -ye" 
where only the first term occurs when s=1; in particular, 
l’",-],=(n—1)ae, 
Aine 
es om es +(n—2)al,, 
1’, -1,- G2 De=2)(a=3) —— Opry esta ee) oy = 8) 221, +(n—8)aly 
and so on. Now let 
ue (n—-1)! 
I, =0J« Tere ih 
then 
Se Pes! i? Ss, 
that is, 
J’,=(J,, Ip) es Jy, 1) ql, a). 
Further, we have 
WIA ling + pnt, teeet 1 Plat aly—13 
n n—1 2 
and, therefore, writing 
V=20J, 
we have 
n-1 n! 
Teenie SS aad, 
1 n—s!s! 
that is, 
Send apes are eda ea Lei 
The quantity J, is of dimension s; so that these expressions conform to the condi- 
tions of homogeneity. 
Moreover, they conform to the expressions obtained as by linear transformation 
of binary quantics. To verify this statement, we note that if, in 
Jn = (Jn, In-1) ++ J,,1) Gd, @", 
a be replaced by a + c, we have (say) 
J”, = (In, In.» J,, 1) (1, @ + ©) 
=(J"n, J'n-1,...9',, 1) A, ©): 
