DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
321 
and the transformed equations of motion are therefore 
mr'—u, —vr m<p'=w{rs\ri 0)' 
mu'=r ®(^;^+^^;^(sin 
t / ‘ ^ ■> A , , dU 
mv—{rsinu) w cos3-\-m-^> w =-^- 
The preceding process is not, of course, given for the sake of the result, which may 
easily be verified directly, but in order to illustrate the meaning of the theorem on 
which it depends. It is hardly necessary to add, that if the problem involved the 
independent rectangular coordinates of any number of material points, the trans- 
formation to polar coordinates would be effected in the same way, by merely adding 
to P analogous terms for each point. 
71 . Before proceeding further there is an important remark to be made. 
It has been hitherto assumed that the modulus of transformation P was a function 
of no other quantities than the 2n variables |i, ... |„, 3 / 1 , and t. But if every step 
of the demonstrations of the theorems of transformation which have been given (art. 
62, &c.) be examined, it will be seen that they continue to hold good in the following 
more general form. 
Take 
P=/(li 5 ^ 2 , ^1,3/2, p, fj, 
where p, q, r, ... are any functions of any or all the variables, old and new, with or 
without t. 
Let the equations connecting the old and new variables be, as before. 
(83.) 
with the condition that p, q, r, ... are exempt from differentiation in forming these equa- 
tions. 
Then take T= — with the following signification; ( 1 ) denotes the 
differential coefRcient of P with respect to t, so far as t is contained explicitly, and 
also through the variables m p, q, r, that is to say. 
d{'P) dP dP , dP , 
UfD u/D 
(where x[-\-Slc. &c., but this substitution is not to he made at this stage) ; 
( 2 ) denotes the result of substituting in the above expression the values of 
yi,...yn in terms of yn,...}in,p, q, r,... from (83.), so far as contains 
explicitly {i. e. not involved in p, q,r,...). 
Lastly, take <I>=(Z)-f-T, 
where (Z) denotes the result of substituting in Z the values of the old variables as 
