642 REPORT—1899. 
Il. Parallel Surfaces in Three Dimensions. 
For a given closed oval surface without singularities, three characteristic 
magnitudes are its volume, its superficial area, and the surface-aggregate of the 
mean! of the curvatures at any point. We shall compare their values with the 
values of the corresponding magnitudes of the parallel surface, which is the outer 
envelope of a sphere of diameter a rolling on the outside of the given surface. 
Denoting the volume by V, the superficial area by S, and (twice the) surface- 
ageregate of the mean of the curvatures by L, and the corresponding magnitudes 
for the parallel surface by V’, S’, L’ respectively, we have 
V-V= Ke +2)(p,+2)2dy,dy,, 
S'= (fot Qraaya, S=[lopdhays; 
Prults 3 
ee 
(~ saath = {Je +p )dy,dy., 
= [Jt +a+p,+a)dy,dyy, 
all the integrals extending over the whole of the original surface. Thus 
, ies 4 
av —~V=aS+50°L + gm 
S’-S=alL+4ra?, 
L’—-L=8na ; 
and therefore 
r_ lye 1 L2= ling 1 ys 
nid y Speekichi peat a em mm pat ar 
ne a! 5 “pel ae 
= B =§=—2L? 
: 16x l6r ’ 
for all values of a: that is, the quantities 
i 
Sau 3 
lér ’ 
1 1a 
4 8 Toa 4 
are invariantive for parallel surfaces, where (for any one of such surfaces) V is. 
the volume it contains, S is its superficial area, and L is twice the surface area of 
the mean of the curvatures.* 
1 The surface-aggregate of the Gaussian measure of curvature is a pure 
constant, for 
[faveye = | { ap inta= Am, 
1P2 
and is therefore an invariant. 
2 In evaluating integrals such as V’, 8’, L’, care must be exercised in regard to 
the range of y, and y,. As amatter of fact, the range of p, is affected by that of 
eed 
