SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
509 
537] 
viz. if U is of the form (a, b,...){x, y) m with arbitrary coefficients, then we have thus 
a series of equations giving the required value of but if (a, b, ...) are arbitrary 
coefficients contained in any manner whatever in the function U, then we have a 
series of equations satisfied by the values x, y which belong to the double root. 
9. The normal at each point of a principal section of an ellipsoid is intersected 
by the normal at a consecutive point not on the principal section: show that the locus 
of the point of intersection is an ellipse having four {real or imaginary) contacts with 
the evolute of the principal section. 
The principal section is for convenience taken to be that in the plane of zx; 
the coordinates of any point thereof are therefore X, 0, Z where 
X 2 £ 2 
a 2 c 2 
= 1. 
Consider the normal at a point X, Y, Z of the ellipsoid; taking x, y, z as current 
coordinates, the equations of the normal are 
x — X y — Y z — Z 
X Y Z 
a? b 2 c 2 
Writing herein y = 0, we have 
= X 1 
z = Z(l--„ 
c 
viz. x, z are here the coordinates of the point where the normal meets the plane 
of xz; and observing that the point in question lies on the normal at the point 
X, 0, Z, it is clear that x, y, z will be the coordinates of the intersection of the 
last-mentioned normal by the normal at the consecutive point not on the principal 
section. 
Writing for shortness 
a = b 2 — c 2 , /3 = c 2 — a 2 , y = a 2 — b 2 , 
(a + 8 + y = 0, a and 7 positive, /3 negative) the values are 
yX aZ 
X= a?’ Z = 
wherefore 
X _ ax Z cz' 
a 7 c <x ’ 
or, substituting in 
X 2 Z 2 , 
~Zo 3 I9 = 1>