TRANSACTIONS OF SECTION A. 643 
It may be noted in passing that if, in the single invariant of parallel curves, 
we write 
A=43.278, L=2na, 
a(B— a’), 
that is, the invariant effectively is 8—a?. Also if, in the two invariants of parallel 
surfaces, we write 
it becomes 
V=44ry, S=4r8, L=2.4ra, 
the invariants become 
4n(B—a*), 44n(y—3a8 + 2a'), 
that is, the invariants effectively are 
B—a*, y—s8aR+ 2a°, 
The similarity in form to the leading coefficients of the simplest covariants of a 
binary quantic is obvious. 
Ill, Parallel Surfaces in n Dimensions, 
The geometry has been introduced solely to simplify the description of the 
analytical results, which may be regarded as invariantive relations among certain 
definite multiple integrals. Denoting by V the volume enclosed by the surface F = 0, 
and by V’ the volume enclosed by the parallel surface, which is the envelope of a 
sphere of diameter a, say 
(X,—2,)? +... + (Xn—2,)? =, 
rolling on the outer side of F=0, we have 
Vv v=| ay, \\- Gee hei. Kinin ealeaby i. dyes 
Let o= |... (aby... dyn 
= {aa, 
w,, a8 can easily be seen from the consideration of the surface of a sphere; and 
really dy, dy, is the elementary solid angle subtended at the centre of a sphere of 
radius unity by the two perpendicular arcs dy,, dW, on the surface; so that 
| | dy, dy,=the whole solid angle 
= 4. 
Similarly, in the case of m dimensions, the quantity @ (with the [notation adopted 
below) is the hypersolid angle subtended by the surface of a hypersphere at its 
centre. With the notation indicated, we have 
oe: az Sr hae's 
= 2a, 
ain 
tine 1)! 
which is 
when 7 is even, and is 
Q} (+1) gp} 2-1) 
1.3.5......2%—2 
when 7 is odd. 
TT2 
