332 
PROFESSOR DONKIN ON THE 
(where Q is expressed in terms of the new variables, and (the angular velo- 
cities of the moving system of axes about the three moving axes themselves) are 
marked with the horizontal line to show that these quantities are exempt from diffe- 
rentiation in forming the following system of differential equations, though they may 
be functions of the variables) ; then the system (91.) is transformed into 
, dZ dfl , dZ \ 
with similar equations for yl, &c. 
In these equations Z is to be expressed in terms of the new variables ; and it is 
evident from the original form of Z and Q, that when so expressed, these two quan- 
tities are the same functions of the new variables that they were of the old, and in- 
volve (see art. 74.) the quantities exempt from differentiation in the same way*. 
Thus the transformed system (92.) contains no terms explicitly depending upon the 
motion of the axes, except those introduced by the three terms multiplied by 
in the value of Vl given above ; and the addition of these terms constitutes the only 
difference between the form of the old and of the new system. 
79. We may now apply the method of the variation of elements to the system (92.) 
as follows ; — 
The system obtained by omitting the disturbing function O, namely. 
dZ 
, dZ , 
dZ ■) 
dii\i) 
dZ 
dZ 
, , (IZ 
o 
+ 
t;i 
"ll 
Oj , — 0 
' dZi^i) 
(93.) 
is canonical, and consists simply of the aggregate of the equations representing, for 
each planet, undisturbed elliptic motion about the sun (relatively to the new axes 
of coordinates). 
The integrals of these equations may therefore be expressed in any of the usual 
forms. We will suppose that the elements chosen are 
a, e, (s), /, V, 
with significations corresponding to those given to the same symbols in art. 55. 
These letters unaccented will apply to the planet m, and e^, &c., a^^, &c., , 
&c., to the planets ni^^, &c. 
The definitions of the elements a, e, sr, &c. are their expressions in terms of the six 
* Since the direction cosines X,,, &c. are exempt from differentiation in forming the equations connecting the 
old and new variables from the modulus P, they continue exempt throughout. (Theorem X. art. 74.) Hence 
W6 ~ - _ 
^^i + yyt + ^^, = {\x + \y-\-\.^s){\x^ + \y, + \.yZ,)-\-kc.=xXi-\-yy, + zz^, 
and similarly for the rest. 
fu'^ + v’^ + w® fimX 
t Z = S( —j, the summation extending to all the planets. 
