ON THEORETICAL DYNAMICS. 35 
dV _ 
fe Ay eae 
where the z quantities a, ... are new arbitrary constants. 
65. Section 5 of the memoir relates to the theory of the variation of 
the elements considered in relation to the following very general problem: 
viz., Q,... P,... being any functions whatever of the 2m variables q,... 
p,-.. and ¢; it is required to express the integrals of the system 2n dif- 
ferential equations 
dq_y Gp_ 
Pam ia 
in the same form as the integrals (supposed given) of the standard system 
dq_d dp__ a 
dé dp’dt dg 
by substituting functions of ¢ for the constant elements of the latter system. 
And section 6 contains some very general researches on the general pro- 
blem of the transformation of variables, a problem of which, as the author 
remarks, the method of the variation of elements is a particular, and not 
the only useful case. In particular, the author considers what he terms 
a normal transformation of variables, and he obtains the theorem 8, which 
includes as a particular case the second of the two theorems in Jacobi’s 
note of 1837, in the ‘Comptes Rendus. This theorem is as follows : 
viz., if the original variables g,..p,.-. are given by the 2n equations 
—. SS I 
a and if the new variables n,...@,... are connected with the original vari- 
__ ables by the equations 
s dK dK 
ied ila 
4 
_ where K is any function of 7,.++ p,+++ which may also contain ¢ explicitly, 
then will the transformed equations be 
dy_ do da__de, 
di da’ dt dn’ 
in which @ is defined by the equation 
: o=H-F 
and is to be expressed in terms of the new variables, the substitution of the 
r new variables in oS being made after the differentiation. In particular, if 
K does not contain é explicitly, then ria and =H, so that, in this case, 
the transformation is effected merely by expressing H in terms of the new 
-yariables. There is also an important theorem relating to the éransforma- 
‘tion of coordinates. To explain this, it is necessary to go back to the 
generalised Lagrangian form at 
