330 
PROFESSOR DONKIN ON THE 
function). The above method of deducing it is given as an additional illustration of 
the general theory of transformation. 
If, instead of the normal elements &c., See., we employ any other elements. 
Cl, C 2 , &c., which can be expressed as functions of the former, the formula (69.) of 
art. 54 will still be applicable, with the condition that p, q, r, &c. are exempt from 
differentiation with respect to Cj, Cg, &c. 
Section VII. — On the Differential Equations of the Planetary Theory. 
76. The differential equations which determine the motions of a system of mutu- 
ally attracting material points relatively to one of them, do not, as is well known, 
naturally present themselves under the canonical form (I.), art. 49. It is possible 
indeed to reduce them, by different artifices, to that form ; but it seems doubtful 
whether any practical advantage is gained by doing so. When the ordinary method 
is followed in the case of a planetary system referred to the sun, there is a distinct 
disturbing function for each planet ; but it is easily seen that the usual expressions 
for the variations of the elements hold good, not merely for each planet on the hypo- 
thesis that the motions of the rest are known, but as a complete and rigorous set of 
simultaneous differential equations involving all the elements of all the orbits, and 
their differential coefficients with respect to t (and containing of course also t ex- 
plicitly). It does not appear that we are practically farther from the attainment of 
the rigorous integration of this system, than we should be if it had the canonical form, 
as it might be made to have if it were derived from an original system of that form ; 
and so far as the development of the disturbing functions is concerned, the most 
troublesome part of them, which is that depending on the mutual distances of the 
planets, is not likely to be got rid of by any conceivable artifice. 
However this may be, all that I propose to do in the present section is to take the 
original differential equations in their ordinary form referred to r^tangular axes 
passing through the sun and parallel to fixed directions, and then to exhibit in a 
general manner the effect of a transformation to new rectangular axes, still passing 
through the sun, but changing their directions in space according to any arbitrary 
law. 
77 . Let M be the mass of the sun, and m, m^, m,^, Sec., the masses of the planets; 
and put ... And, referred to the 
original axes, let x, y, z be the coordinates of m, x^, y^ of m^, ..., z^) of 
Also let R, Rp ... R(i) have their usual significations, so that 
r> yf -f yy(i) + zzffl) 
“ l((X(i)-x)2+(y(,)-y)2+(Z(i3-z)2)^ (xf.^+y2^-i-z2^)t 
the summation extending to all the planets except m. 
Then if we put xfjj-l-yf;)-l-Z(,)=i%, the original differential equations of the second 
order are such as Let u-p Vj^p W(i) be the variables conjugate to 
^(i) 
