96 
PROFESSOR DONKIN ON THE 
r-\-t. Consequently, for every such integral we shall have, by (36.), 
[gr, A]=0, (37.) 
since ^=0. 
GT 
This particular consequence of the formula (36.) follows also immediately from (32.), 
art. 22, since the equation g'=0 gives, by (32.), [Z, g]=0, and in this case Z—h , so 
that [Z, g~\ = [h, g~\. In this manner the theorem expressed by (37.) has been already 
obtained by M. Bertrand. 
Examples of the preceding Methods. 
27. I shall now exemplify the principles which have been explained, by applying 
them to two of the most familiar as well as important problems of dynamics. First 
then let it be required to obtain in a normal form the integrals of the differential 
equations which determine the motion of a material point, acted on by a force ema- 
nating from a fixed centre and depending only on the distance. 
Taking the centre of force as the origin of a system of rectangular coordinates, let 
m be the mass, and x, y, z the coordinates of the moving point. Then 
T=\m(x ,2 +y' 2 +z' 2 ), 
and U (see art. 17.) is a given function of r, say <p(r), where r 2 =x 2 -\-y 2 -\-z 2 . Let us put 
dT 
dx l 
— U. 
dT 
dy' 
=*\ 
dT 
ds' 
=w. 
so that, referring to the notation used in the preceding pages, we have 
x, y, z instead of x x , x 2 , x 3 
u, v, w instead of y x , y 2 , y 3 . 
Moreover, u=mx', v=my', w==mz'. 
Hence we obtain Z=(T) — \]=m~ l {u 2 -\-v 2 -Yw 2 ) — p(r), 
so that the integral of vis viva, or Z—h, becomes 
^(u 2 +v 2 +w 2 ) — <p(r) = h ; 
and the three integrals which express the conservation of areas become 
yw — zv=c x 
zu — xw — c 2 
xv — yu=c 3 . 
These integrals are immediately seen to satisfy the conditions 
O2, ^3]] c,, lc„ cj = c 2 , [cj, c 2 J — c 3 ; 
from which it follows (see art. 25, the result of which is obviously unaffected by the 
negative signs), that if we take &=(Ci+C 2 +c^, the condition [c 3 , &] = 0 will be satisfied 
