DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
91 
since the expression for Z must reduce itself identically to h when the values of y x ...y n 
obtained from the integrals are substituted in it. Hence 
and therefore X=— Af+V, 
V being a function not containing t explicitly. We have then so that V is 
to be found from the n expressions 
Lastly, the n remaining integrals will be 
dX_ dX__, 
dh T ’ da t 
(r representing the arbitrary constant conjugate to h ) ; and, substituting in these the 
above expression for X, we obtain 
dV 
dh 
(29.) 
The function V now satisfies, and may be defined by, the partial differential equation 
where f(x „ ... x n , y x , ... yj) is the expression for Z in terms of the variables. 
This, in dynamical problems, is the case in which the so-called “ principle of vis 
viva ” subsists. I shall, in the rest of this paper, use h exclusively in the above signi- 
fication, and call it, whether actually referring to a dynamical problem or not, the 
“constant of vis viva," whilst the integral Z =h may be called the “integral of vis 
viva" 
20. When the 2 n integrals of the system of differential equations (I.), art. 14, are 
expressed in the manner which has been explained, it follows from the conclusions of 
former articles, that when these integrals are put in the form 
at— <pjx x , ..., x n , y u ...y„ t) 
&i — •••» yi> '"Vm 
the conditions \a i: b^\ — 1, [a h bj] = 0, [b iy bi\— 0 will subsist, as well as [a ; , a y ]=0. I 
shall call any system of 2 n integrals in which these conditions are fulfilled, a “ normal 
solution,” or a system of “normal integrals,” whilst the 2 n arbitrary constants con- 
tained in such a system may be called “normal elements.” Any pair a { , b t , may be 
called (as before) conjugate elements. In the case considered in art. 19, h and r are 
conjugate elements, these letters being used instead of a, b, merely from obvious 
motives of convenience. 
It has been one principal object of these investigations to ascertain what advantages 
could be gained — either for the actual integration of a system of equations of the 
