DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
357 
without recurring to the formula (D.). And since each equation of condition gives, 
if differentiated totally with respect to t, with the substitution of ^ for an equa- 
tion such as 
t?L . d'Zt . 
^ dt~^ doc-^ dy^ "1“ • • ”l~ dyn 
(L.) 
the m equations {ri.) with the r equations L (in which last &c. must be expressed, 
after the differentiation, in terms of |, &c.), give n equations by means of which 
3/„ ... may be expressed in terms of li, &c., rii, &c., and can be so expressed only in 
one way. 
Lastly, the value of (see equations (G.)), may be obtained as follows; — 
Since ^= — (W)-l-(|',)jji-i-...-l-(D>Jm 
and Z = — [W] + [a?;] ?/,+ ... -f- 
and (W) is only [W] differently expressed, we have, without reference to modes of 
expression. 
On the other hand, since P can contain t explicitly only through (a?,), &c., we have 
dt dt 
and also 
%(x,y,) = ^yP§]+^, 
d^ 
hence, observing that '^=ni, we obtain 
d^h ' ’ 
dV 
dt * 
and therefore, finally, 
dV 
so that Z — ^ must become identical with '9', when expressed entirely in terms of 
the new variables. 
These results may be stated in the form of the following theorem. The system (E.) 
is transformed into the system (G.) by the following substitutions : — 
(1) x^, ... are expressed in terms of li, ... by means of the m equations which 
define the latter variables, together with the n—m equations of condition 
L, = 0, ... Lr=0 {r=n—m). 
dV 
(2) ni, ... rim are defined by the m equations where the modulus of trans- 
formation P is given by the equation 
P= + (^2)3/2+ •••+ 
(Xi), &c. being expressed in terms of li, &c., so that P is explicitly a function of ...5™, 
yi ’••yn> with or without t. 
2 B 2 
