DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
97 
(as is easily found to be true) ; and since neither of the integrals c 3 , k contain t ex- 
plicitly, the conditions [h, c 3 ]=0, [h, &] = 0 will subsist also (art. 26.). Hence it 
follows that if we solved algebraically the three integrals h, c 3 , k so as to express u, 
v, iv in terms of.r, y, z, their values would be the partial differential coefficients of a 
function V, from which the three remaining integrals could be found (arts. 12 and 19.). 
But it is more convenient to adopt a different system of coordinates. Reverting 
then to the primitive form of the three integrals which we have chosen, and writing c 
instead of c 3 , we have 
T-U =h (i.) 
m(xy' —yx')=c (ii.) 
m 2 (r 2 (x 12 +y 12 -\-z 12 ) — r V 2 )=k 2 ( i i i . ) 
28. Let us now employ, instead of x, y, z, the three coordinates g, 0, z ; where 2 is 
the same as before, g is the projection of r on the plane of xy , and 0 is the angle 
between g and the positive axis of x. We shall thus have 
g 2 -\ -z 2 =r 2 , x=g cos 0, ?/~£sin0, 
and 
T=^m(g ,2 i-g 2 0 l2 +z' 2 ). 
Let 
dT 
d T 
dT 
—- — u, ——V, -t-.—VO 
dJ ’ dd' ’ dz' 
(where u and v have now a new signification), then 
d—H, q'—JL, z'=- • 
? m mg 2 m ’ 
and the three integrals at the end of the last article become, after obvious reductions, 
^(w 2 +p+^ 2 ) =h+(p(r) (i.) 
v=c (ii.) 
(gw — zu) 2 -\- r - iv 2 =k? (iii.) 
The conditions [ h , c]=0, \h, A'] =0, [c, A-]=0 continue to subsist with reference to 
the new variables ; the two former necessarily , because (ii.) and (iii.) do not contain 
t (art. 26.), and the third actually, as is seen on trial ( not accidentally , as will be 
shown hereafter). 
We know, therefore, that the values of u, v, w, found from these equations, will 
be the partial differential coefficients with respect to g, 0, z of a function V of these 
latter variables. 
c 2 
The two first give M 2 -f-^ 2 =2m(A+®(^))~p ; 
and if we multiply this by f+z 2 =r > , and subtract (iii.), we obtain (introducing the 
condition (ii.) 
( g u + z w) 2 = 2 mr 2 ( h + <p (r ) ) — k 2 . 
MDCCCLIV. 
O 
