326 
PROFESSOR DONKIN ON THE 
be performed before or after the substitution for y„ z,, in terms of z,. If then 
we put 2=0 and 
R: 
-m, 
1 xx, + yy ,\ 
8 rf 
we shall have, from the fourth and fifth of the transformed differential equations*, 
X 
ff ^oc I 
y' 
dx 
dy 
( 87 .) 
from which it is evident, that, assuming the motion of m^ to he known relatively to the 
new axes, the variations of the four elements of the orbit of m which determine the 
dimensions of the orbit, its position relatively to lines fixed in its own plane, and the 
time of perihelion passage, will be expressed in terms of the differential coefficients 
of R in the same way as if the plane of the orbit were fixed. But the motion of the 
node of the orbit upon the fixed plane of xy, and its inclination to that plane, must be 
determined by means of the last of the differential equations, as follows : that equa- 
tion gives 
co{U — a,,V-(^^ (2 = 0 ) ; 
or if we put — H for the term multiplied by mm^ in the value of U given above, 
and 2 is to be put =0 after the differentiation, which reduces the above to 
fda\ 
.,u-co,v=[-^-y 
Let col-{-ai[=a^, SO that oc, is the angular velocity of the plane of the orbit about the 
radius vector; then (observing that u=mx', See.) we have 
VCUq — WCOj 
'm{xy' — ody)^ 
whence V(Xq — uu^——{xy' — x'y) 
and therefore mcc— / 
—e^)\dzj 
which gives a in terms of the four elements referred to above, and of t. And if we 
put / for the inclination of the orbit to the plane of xy, v for the longitude of the node 
* These equations (87.) have been obtained in a dliFerent way by Mr. Bronwin. Camb. Math. Journ. 
vol. iv. p. 24.5. 
t See below, art. 80. 
