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,