TRANSACTIONS OF SECTION A. 645 
and 
I’m = (Tm; Im—1, +» + Sy, 1) (1, @ + €)™ 
=a CS ay J mays ss Sm Ly ye)"; 
for all values of m. There, accordingly, are combinations of the quantities J which, 
by the theory of binary forms, are invariantive for these linear transformations; as 
the transformations are characteristic of the parallel surfaces, it follows that these 
combinations of the quantities J are invariants of the system in question. The 
simplest are 
J Dar J v, 
J, — 33,3, + 23,5, 
J, — 43,3, + 3J,*, 
and so on ; in fact, every principal seminvariant gives an invariant of the system 
of parallel surfaces. 
By way of verification, we take x = 3; then © = 47, L = 20J,,8 = oJ, 
V = 40J, in our former notation ; and then 
3 1 1 
si aS) ig Ren Ot ee 
J, — 83,5, + 23,5 = 5 (Vv SES + ah ) 
It may be pointed out that the single invariant, for a system of parallel plane 
curves, vanishes for a circle; and that both the invariants, for a system of parallel 
surfaces in space of three dimensions, vanish for a sphere. 
5. On the Notation of the Calculus of Differences.' 
By Professor J. D. Everert, £.2.S. 
In conjunction with the ordinary symbol A defined by 
AYn = Yn+1— Yny 
the author empioys another symbol 6 defined by 
8 Yn = Yn — Yn-1. 
This gives the relation 
Ad=A-5, 
leading to a number of developments, such as 
(GY = G+ ay =14na +2@—) ars be; 
eet 8y = yi lA ea) ee 
(5) = (1-8) =1-nd+"¥ 8 - &e,; 
[SS Se oe (MAD) icgn 
(4) =a-38) =1+nd4+ 2707084 &,; 
(yor 
of which some express well-known properties, and others are believed to be new. 
By performing the operation Am" on any one of the entries in a table of 
1-nait@t) we _ ge; 
1} The Paper will be published in the Messenger of Mathematics. 
