534] 
a ‘‘smith’s prize” paper; solutions. 
481 
C. VIII. 
61 
substituting these values in the equation of the ellipsoid, we have 
a 2 X' 2 b 2 Y 2 c 2 Z 2 
“ (a 2 + pY + (b 2 + p) 2 + (c 2 + pY ’ 
which determines p as a function of X, Y, Z. The tangent plane of the ellipsoid at 
the point (x, y, z) and of the parallel surface at the point (X, F, Z), are parallel to 
each other (or what is the same thing, the parallel surface cuts at right angles the 
normal of the ellipsoid), we have therefore 
x 
a 2 
dX + l 
dY + ~dZ=0, 
c- 
or substituting for x, y, z their values, this is 
XdX , YdY | ZdZ _ Q 
a 2 + p ' b 2 + p c 2 + p 
where p denotes as above a function of (X, F, Z) given by the equation 
a 2 X 2 Jf-Y 2 d z Z- 
~ (a 2 + pY + (b 2 + pY + (c 2 + pY' 
We have thus the differential equation of the parallel surfaces. It may be remarked, 
that the integral equation (involving k as the constant of integration), is found by 
the elimination of x, y, z, p from the foregoing equations 
x = 
a 2 X 
6 2 F 
c 2 Z 
a 2 + p’ V b 2 + p’ 2 c 2 + p’ 
* , r 
a 2 b 2 
+ 1 = 1 u- 0 X°Y + y_, 1 
+ C 2 “ 1 ’ k -P W + b* + <? 
or, what is the same thing, by the elimination of p from the equations 
k 2 _ X 2 F 2 , Z 2 
p 2 ~ (a 2 + pY + (b 2 + pY + (c 2 + pY ’ 
a 2 X 2 b 2 Y 2 c 2 Z 2 
(a 2 + pY + (b 2 + pY + (c 2 + pY ’ 
these may be replaced by 
X 2 F 2 Z 2 
a 2 + p + b 2 + p + c? + p p 
X 2 F 2 Z 2 _ № _ 0 
(a 2 +p) 2 + (b 2 + pY + (c 2 + pY ’ 
or, since here the second equation is the derived equation of the first in regard to 
the parameter p, the parallel surface is the envelope of the quadric surface 
X 2 F 2 Z 2 k 2 . 
+ p b- + p c 2 + p p 
where p is the variable parameter. Or analytically, we find the equation by equating 
to zero the discriminant in regard to p, of the quartic function