TRANSACTIONS OF SECTION A. 641 
which nominally is of degree ” in ss One root, however, is zero, because the term 
P 
independent of : vanishes, on account of the relation 
p 
Ietlpr...+472%=1; 
there are therefore »—1 values of p thus determined, which may be denoted by 
Py +++) Pn-1 Further, the 2—1 directions from 2,, . . ., 2% along the surface F=0, 
in which the normal is met by the normal at a consecutive point, are given by taking 
any x —1 of the equations 
1_0b\7, _S Oba, 
°= (Fan) & aa 
t=1 
and substituting, in the ratios determining the directions, the —1 values of p in 
succession. Denoting by do, the arc along any one of these directions, which can 
be regarded as directions of principal curvature, by dy the angle between the 
two consecutive normals, and by p, the corresponding distance (say the correspond- 
‘ing radius of curvature), we have 
do,= pray, 
‘so that dyy,, for each value of 7, is unchanged for the parallel surface. 
In order to build integral invariants upon the invariantive differential element 
represented by the infinitesimal angle between consecutive normals, we proceed 
from the two simplest cases, viz. parallel curves in plane space, and parallel 
‘surfaces in three dimensions. 
I. Parallel Curves in plano. 
For a given oval plane curve without singularities, two characteristic magni- 
tudes are its perimeter and its area. We shall compare their values with the values 
of the corresponding magnitudes of the parallel curve, which is the outer envelope 
of a circle of diameter a rolling on the outside of the given curve. Denoting the 
perimeter of the given curve by L, and its area by A, and the corresponding mag- 
nitudes for the parallel curve by L’ and A’ respectively, we have 
A’-A=3/{(p +4)?—p°}ay, 
L’=|(p+a)ay, L=[pdy, 
where p is the radius of curvature at the given point; each of the integrals is to 
be extended through a range 27, Thus 
A’—-A=alL+7na’, 
L’-L=2ra; 
and therefore 
1 
Ss Ree 
A’- = 
4 Ar 
for all values of a; that is, the quantity 
lie 
A-7L 
7 
is invariantive for parallel curves, where A is the area enclosed by any one of 
them and L is its perimeter, 
1899. : TT 
