DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
353 
coincides, in substance, with that taken by Jacobi ; but the above mode of stating it 
may tend to make it clearer, and to call attention to a matter which, so far as I know, 
is not so much as mentioned in any of the elementary works usually in the hands of 
students of physieal astronomy. 
[^Addition to Appendix B.] 
Received March 1, 1855. 
The remark made above, that the symbol ^ is unmeaning in the case considered, 
is not of course intended to imply that a meaning may not he given to it ; but then 
such meaning is different from the ordinary signification of the symbol, which is a 
partial derived function. 
The whole matter may be strikingly illustrated by a simple example. 
Consider the movement of a rigid body about a fixed point. Adopting the nota- 
tion of art. 40 (Part I.), we have 
pdt— —cos ^dd— sin 0 sin (pd'^ 
qdt— sin pdO— sin d cos pd-ip 
rdt=dp-\- cos M-p. 
Let a, j 8 , 7 be three new variables, defined by the equations 
a=:^{pdt, qdt, y=Ardt-, 
0 ^0 ^0 
so that du—pdt, &c. Then the above equations give 
dd= — cos pd(x,-\- sin pd^, dp=d'y-\- cot ^(sin pda^- cos pd^) 
dp— — cosec ^(sin pdci-\- cos pd(3). 
Here a, (3, y are the sums of the elementary angles described about the axes in the 
course of the motion ; and no one would maintain that P p, p are functio?is of a, (3, 7 , 
for the values of the latter variables at any time do not determine the values of the 
former. If therefore we choose to write such equations as 
^=-cos?, ^=smp,&c., 
we must admit that^) &c. are not partial derived functions in the ordinary sense'. 
dd 
At most, ^ is the derived function of that function of a which & would become if /3 
and 7 were maintained invariable, i. e. if the motion were restricted to a rotation 
about the A-axis. A 
pret them as follows : 
about the A-axis. Again, if we admit such symbols as we must inter 
° ’ •’ d^da dud^ 
d^ d d& d cos _ . d<^ 
d^—d^ dP— ~ d^ — 
