wa 
ON THEORETICAL DYNAMICS. 33 
riables % .. +, and of the 2 constants 4,..., and that the 2 variables p,..., 
and the z constants a,..., are determined by the conditions 
dV 
dq ee! - (1) 
dV 
Py re | see (2) 
so that in fact by virtue of these 2n equations the 2n variables X, qg,... 
p»--- may be considered as functions of ¢, and the 2n constants 0,...a,... 
(hypothesis 1), or conversely, the 2” constants 0,... eel +.) nay be con- 
sidered as functions of ¢ and of the 2x variables gq, . .. (hypothesis 2). 
Theorem 2 is as follows: viz., if from the 2n caustions (1, 2) and their 
total differential coefficients with respect to ¢, the 2m constants be elimi- 
nated, there will result the following 2 simultaneous differential equations 
of the first order, viz. :-— 
dq_ dH 
dts dp 
dp dH 
Seay 
where H is a function of Q,+++ p,++» (which will in general also contain ¢ 
explicitly), and is given by the equation 
ep 
dé’ 
where, on the right-hand side, the differential coefficient a is taken with 
respect to ¢, in so far as ¢ appears explicitly in the original expression for V 
in terms of g,...5,... and @, and after the differentiation, b,..., are to be 
expressed in terms of the variables and t, by means of the equations (1). 
__ Theorem 3 is, that there exists the following relations, viz. :— 
db dp da dp" 
Vee ee 
db dgda dq 
where (p,q) are any corresponding pair out of the syéteri »,...and 
G++ and , a) are any corresponding pair out of the systems 8,... and 
@,..., So that the total number of equations is 477: in each of the equations 
the left-hand side refers to hypothesis 1, and the right-hand side to hypo- 
_ thesis 2. 
_ To these theorems should be added the supplemental theorem contained 
in article 50, viz., that there subsists also the system of equations 
db_ dH 
: Re 
da_ du 
dt db eee 
where the left-hand sides refer to hypothesis 2, while the right-hand sides 
dV 
refer to hypothesis 1, as before H=—~7)’ but here H is differentially ex- 
pressed, being what the H of theorem 3 becomes when the variables are 
expressed according to hypothesis 1. 
1857. D 
