320 
PROFESSOR DONKIN ON THE 
and u, V, w the variables conjugate to them ; so that, putting T=^m(x'^+y'^+z'^), we 
have 11 =^, &c.; whence T=~(u^-{-v^+w^), and the equations of motion are x'=^5 
u'=—^,&c., where Z= 5 |^(u^+v^+w^) — U, and U is a given function of x, y, z, with 
or without t. 
Now let r, 0, <p be polar coordinates of P, so that 
x=rsin^cos(p, y=r sin ^sin z=rcos^. 
Let u, V, w be the variables conjugate to r, 0, (p. Then the ordinary process of trans- 
formation would be as follows : — 
(1) to express x', y', z' in terms of r, p, r', <p’, and thus transform T into a 
function of the latter quantities ; 
(2) to define u, v, w by the equations 
dT dT 
and by means of these relations to express r', 6', p' in terms of u, v, w, r, 6, <p, so 
that x', y', z', and therefore, finally, T and Z, might be expressed as functions of 
the six new variables. 
Instead of this, let us adopt the method indicated by the theorem at the beginning 
of this article. 
We have then, for the modulus of transformation, 
P=(x)u-f (y)v-l-(z)w, 
in which x, y, z are to be expressed in terms of r,0,<p; so that the proper form of P is 
P=ur sin 0 cos <p-j- vr sin 0 sin ip-j- wr cos d, 
and the equations (corresponding to (art. 69.)) which define the new variables 
w, V, w, are 
These give 
dP dP 
^^=dd’ 
w = sin ^(u cos ^d-v sin <p)-l-w cos 0, 
v=r cos ^(u cos V sin p) — rw sin 
— r sin ^(u sin p— v cos <p). 
from which the values of u, v, w are easily obtained in terms of the six new variables. 
But in order to effect the transformation of T, we have only to square each side of 
these equations, and add them, after dividing the second by and the third by 
(rsinOy; we thus obtain 
r^'{r sin 
