322 
PROFESSOR DONKIN ON THE 
given in terms of ii, ... ;ji, ... p, q, r, ... from (82.). 
dy\i ‘ d^i 
We shall then have 
( 84 .) 
where O is in general a function of 
... jJi, ... p, q, r, ... p', q', r', ... and ^ ; 
and the differentiations with respect to t]; are performed only so far as those vari- 
ables appear explicitly in <!>. But after these differentiations, we may introduce the 
actual values of jo, ... y, ... in terms of the variables and their differential coefficients. 
It is obvious that the original variables will not, in general, have been eliminated 
from the system (84.) ; but of course the elimination may be afterwards completed*. 
Similar considerations apply to that particular case of transformation which we have 
called a transformation of coordinates (art. 68.). We have then 
P= ('^1)3/1+ ••• 
and the relations connecting with ... may contain p, p, r, ... ; so that 
(a’l), &c. are functions of jo, r, ... as well as of ... 
We might have deduced the preceding conclusions from the following simple con- 
sideration. Since p, q, r, ... are actually functions of t, though unhnown functions, 
we may imagine them to be hnown, and to be expressed explicitly in terms of t-, and 
then the case resolves itself into that of art. 62, &c., so far as the demonstration is 
concerned. But as a doubt might possibly have arisen whether any fallacy was 
involved in the circumstance that p, q, r, ... involve (when supposed to be expressed 
in terms of t) the arbitrary constants of the problem itself, it seemed best to refer to 
the original reasoning; the most important part of which is that contained {mutatis 
mutandis) in art. 6. (For the “ mutanda” see the beginning of art. 63.) It is then 
apparent that this circumstance is perfectly immaterial with reference to the con- 
clusions in question, though it may be important in other points of view. 
72. This being premised, we will proceed to an example of transformation more 
interesting than the former, namely, the 
Transformation from fixed to moving axes of coordinates. 
Let X, y, z, u, v, w have the same signification as in art. 70, and let x,y, z, u, v, w 
be the new variables, where x, y, z are rectangular coordinates, referring to a system 
of moving axes of which the origin always coincides with that of the original fixed 
axes of X, y, z. 
Let the direction cosines of the new (moving) axes with respect to the old be 
(■^■'05 5 ^45 (^15 •'l 5 ^2J (^23 *^23 thUS'|' 
* The final equations in this case will not in general have the canonical form. 
t 1 do not know who first used this convenient way of indicating the nine direction cosines by a diagram, 
but 1 first saw it in one of M. Lame’s works. 
