334 
PROFESSOR DONKIN ON THE 
The complete variations of the elements are then found by adding to the terms 
just written the expressions given in art. 25. 
It is easily seen that the expressions (95.) might have been deduced from geome- 
trical considerations alone, if we had been at liberty to assume beforehand that the 
mechanical and geometrical parts of the variations might be calculated separately ; 
the former as if the axes were at rest, and the latter as if there were no disturbing 
forces. It would not, I believe, be difficult to establish by a priori and simple 
reasoning the validity of such an assumption, and then the above results would only 
serve as a verification of the method which has been employed to obtain them. 
80. In order however that no obscurity may rest upon the interpretation of the 
formulae obtained in the last articles, it is necessary to consider the physical (or rather 
geometrical) meaning of the elements a, e, &c., which we have so far only defined by 
means of their expressions in terms of the variables x, y, z, u, &c., and to ascertain 
what relation they bear to the elements similarly defined by means of the original 
variables x, y, &c., which refer to axes whose directions are invariable. 
The relations between the variables ((85.), art. 72.) give immediately 
«^=x^-l-y^-|- z® 
-f- = U^ + + W^ 
t^a?-l-P3/-l-2ic=ux-l-vy+zw ; 
and if we put yw — zv^-A, zu — xw — B, XV — yu= C 
yw— zv=A, zu— xws=B, xv— yu=C, 
we find, by virtue of the relations — &c., the following equations : 
A — ^0 A + + *'oC 
5 = X ] A -j- 1(/; 1 B 1 
C— ^2 A -|- l'2C 
and A^--\-B'^+ 0= A^-f B^-4- 
Now A( = yw— zv=wi(yz'— zy')) is the projection on the plane of yz of the areal 
velocity of m (relative to fixed space) multiplied by the mass, and B, C have analogous 
meanings; hence it is evident from the above equations that A, B, C are the pro- 
jeetions on the three moving coordinate planes of yz, zx, xy of the absolute areal 
velocity of m relative to fixed space, multiplied by the mass. (The projections of the 
areal velocity relative to the moving axes would he yz' — zy\ &c., which are not pro- 
portional to yw—zv, &c., since u, v, w are not the same as mx' , my' , mz', except on a 
particular hypothesis as to the motion of the axes. See art. 73.). 
Inasmuch as the definitions of the elements a, e, / involve the variables only in the 
forms A, B, C, it follows that these three elements are 
ye^pect\ve\y the semiaxes, excentricity, and inclination to the plane of xy , of the absolute 
osculating ellipse of the orbit in fixed space. Thus the instantaneous ellipse, relatively 
to the moving axes, is of the same dimensions and in the same plane as the true 
