320 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
so that in the two tetrads each line intersects the four lines of the other tetrad, but 
it does not intersect the remaining three lines of its own tetrad. The intersections 
are four points corresponding to £= — a 2 , being the imaginary umbilici in the plane 
X = 0: four to £ = — b 2 , being the real umbilici in the plane Y — 0 : four to £ = — c 2 , 
being the imaginary umbilici in the plane Z = 0: and four corresponding to £ = oo, which 
may be called the umbilici at infinity ( x ). 
Sequential and Concomitant Centro-curves. Art. No. 7. 
7. Consider any particular curve of curvature ; the normals at the several points 
thereof successively intersect each other in a series of points forming a curve ; and we 
have thus, corresponding to the particular curve of curvature, a curve on the centro- 
surface, which curve may be called the sequential centro-curve. Again the same normals, 
viz. those at the several points of the particular curve of curvature, are intersected, 
the normal at each point by the consecutive normal belonging to the other curve of 
curvature through that point; and we have thus, corresponding to the particular curve 
of curvature, a curve on the centro-surface, which curve may be called the concomitant 
centro-curve. If instead of a single curve of curvature we consider the whole series, 
say of the major-mean curves of curvature, we have a series of major-mean sequential 
centro-curves, and also a series of major-mean concomitant centro-curves; and similarly 
considering the series of the minor-mean curves of curvature we have a series of 
minor-mean sequential centro-curves and also a series of minor-mean concomitant 
curves; the configuration of the several curves will be discussed further on, but it may 
be convenient to remark here that the centro-surface may be considered as consisting 
of two portions, say, 
(A) locus of the major-mean sequential centro-curves; and also of the minor-mean 
concomitant centro-curves; 
(B) locus of the minor-mean sequential centro-curves, and also of the major-mean 
concomitant centro-curves. 
Investigation of expressions for the Coordinates of a point on the Centro-surface. 
Art. Nos. 8 to 13. 
8. Consider the normal at the point (X, Y, Z). Taking in the first instance 
(x, y, z) as current coordinates, the equations are 
x—X y — Y z — Z 
Y~ = Y ~ = ~Z ’ = x suppose, 
a 2 b 2 c 2 
1 According to Salmon, Solid Geometry, [2nd Ed. 1865], p. 229, the number of umbilici for a surface of 
the n th order is =n (10m 2 - 25n +16) ; viz. for n — 2, this is =12, as in the ordinary theory, not recognizing 
the umbilici at infinity. But whether properly umbilici or not, the 4 points which I call the umbilici at 
infinity do in the present theory present themselves in like manner with the 12 umbilici.