640 REPORT—1899. 
one-fifth to those at 10? and 90°, and one-third to those at 50° and 95°, Then 
unite the tops of these centiles with a free-hand curve, 
4, A System of Invariants for Parallel Configurations in Space, 
By Professor A. R. Forsyru, Se.D., L.R.S, 
There is one class of invariants appertaining to parallel configurations which I 
have not seen noticed; they arise in spaces of tivo, three, and any number of 
dimensions. 
It is known that, for a plane curve parallel to a given plane curve, the normals 
at corresponding points are the same in direction, and therefore the angle 
between corresponding consecutive normals is the same, so that this infinitesimal 
angle is an invariantive element. Moreover, the centres of curvature are the same, 
so that the difference between the radii of curvature is the diameter of the rolling 
circle, the two enveloping curves of which are the given curve and its parallel. 
Similarly, in the case of parallel surfaces, it is convenient to consider the 
principal directions of curvature at each point. They are respectively parallel to 
one another at corresponding points; the corresponding normals are coincident in 
direction ; and the centres of principal curvatures for the two surfaces are the 
same. Hence the difference between a principal radius of curvature and the 
corresponding principal radius of curvature of the parallel surface is equal to the 
diameter of the rolling sphere, the other envelope of which is the parallel surface ; 
and this holds for each of the two principal radu. 
Likewise, in space of x dimensions. To render the explanations clearer, we 
consider a configuration 
1 Gai egy cay OF 
which, as for two and for three dimensions, will be consid»red devoid of special 
singularities. Let 
= Pea, een 
where 
2 ne 6F 9 
ae se) ° 
Tf, then, a distance p be measured inwards along the direction indicated by 
Z,+++%, (say along the normal to the surface =0), the coordinates of the 
extremities are given by 
&,=2,—ply 
If this point be a point of intersection with a consecutive normal at Uy POL. sce 
Uy + dy, then ' F 
€,=a,+dx —p(l,+dl), 
aay, Cer 
Q=(eS 5 pe 5 
(ae) ear Casta 
the term in dz; being omitted from the summation on the vight-hand side, and 
the equation holding fors=1,.... The possible values of p are given by the 
equation 
fows=— 1)", 0.75 thatiis, 
ae ol, ol, al, | O 
gna” On. eer ihe 
OES 101, Ol, 
One p On,” OF, ia | 
Oy Be 1_ 8, 
Ox, Our,’ i Ox, i | 
