560 
two smith’s prize dissertations. 
[553 
of the forces at corresponding positions is satisfied ipso facto; and the condition as to 
magnitude is 
r n in' s' t 2 
km x m*. m s t' 2 ’ 
that is, 
k' in' m x m« s'r' n t- 
k mfmf m sr n f- 
_m ' n+1 i® 
~m' t'*’ 
or, say 
t! 
t 
's'\ 2 / km y 
.*/ \k T m') ’ 
In the case n = 2, the present theorem (applying however only to the case of two 
elliptic orbits of the same eccentricity) agrees with Kepler’s third law, or say with 
the theorem 
lira 1 
WY 
that is, 
where observe that the /x, or mass in the sense of the formula, is the km, or 
attractive force on a unit of mass, of the theorem as above written down. 
2. In a family of surfaces F (x, y, z, p) = 0, containing a single variable parameter p, 
there is through any given point of space, a surface or surfaces of the family; or (if 
more than one, confining the attention to one of these surfaces) we may say that 
there is, through any given point of space, a surface of the family. 
Considering now two other families of surfaces, there will be through any given 
point of space, three surfaces, one of each family; and if (for every given point of 
space whatever) these intersect each other at right angles, we have a system of 
orthogonal surfaces. 
Supposing the equations of the three families to be 
F {x, y, z, p) = 0, 
(x, y, z, q) = 0, 
^ {x, y, z, r) = 0,