510 
SOLUTIONS OF A SMITH'S PRIZE PAPER FOR 1871. 
[537 
we have 
c 2 z 2 
—r i" = 1, 
7' a- 
the required locus, which is thus an ellipse. 
If the point (X, 0, Z) is an umbilicus, it is clear that the corresponding point 
of the locus will be a point of the evolute of the principal section; and to prove 
that the locus touches the evolute, it is only necessary to show that the tangent of 
the locus is also the tangent of the evolute; or what is the same thing, that the 
tangent of the locus passes through the umbilicus. 
Now for the umbilicus we have 
the corresponding values of x, z being 
yX otZ 
Take £, £ as the current coordinates of a 
a?x£ c 2 z% 
7 2 + oX 
or, substituting for x, z the foregoing values, 
%l_ZK 
7 a 
and these should be satisfied by f, £=X, Z\ 
point on the tangent of the locus, we have 
= 1, 
= 1, 
viz. we ought to have 
7 a 
and this equation is in fact true for the values of X, Z at the umbilicus; viz. for 
these values we have 
a 2 c 2 
~0 + 0~ ‘ 
that is, /3 = c 2 — a 2 , which is in fact the value of /3. 
There is obviously a contact in each quadrant, that is, there are four contacts 
(in the present case all real) of the locus with the evolute. 
The same theorem holds good in regard to the other principal sections; only for 
these, the umbilici being imaginary, the points of contact of the locus with the 
evolute are also imaginary. 
Remark. There is a great convenience in questions relating to the ellipsoid, in 
the use of the foregoing notations a, /3, y.