DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
335 
osculating ellipse ; and it only remains to show that it coincides with the latter in 
position, for which purpose we must prove that it touches the true orbit. (It does 
not in general touch the relative orbit, because mx\ wy', mz' are not in general the 
same functions of the elements and t that u, v, w are.) 
81. If we suppose the coordinates x,y, z oi m expressed in terms of the elements 
a, e, &c. and t, and denote by ^5 &c. their differential coefficients taken so far as t 
• dx du dz 
appears explicitly in these expressions, then — are proportional to the direc- 
tion cosines, relatively to the axes of x, «/, z, of the tangent to the (relative) instan- 
taneous ellipse. And therefore the direction cosines of this same tangent referred to 
the fixed axes of x, y, z, are proportional respectively to 
. dy , - dz 
dx , dy , dz 
dt 
h 
dx , dy , 
dt ' ^ dt ' 
dz 
On the other hand, the direction cosines of the tangent to the absolute instantaneous 
ellipse, referred to the fixed axes, are proportional to ^ (the differential 
coefficients of x, y, z, taken so far as t appears explicitly in the expressions for those 
variables in terms of the elements of the absolute ellipse). The identity of the two 
tangents will therefore be established, if we can show that 
dx . 
dx . dx , . dy , . dz Q 
Now is that part of x' which does not depend upon the disturbing function ; i. e. 
(equations (92.), art. 78.) we have identically 
dx c?Z dy d7i dz c?Z . 
dt du dt dv dt dw 
and, in like manner, 
dx dTi dy dZ dz dZ 
dt dt dY dt dw 
where Z is the same as before, but expressed in terms of x, y, z, u, v, w, instead of 
x,y^z,u,v,w. Now let the latter set of variables be expressed in terms of the former 
hy the formulse of art. 72, and we have 
dZ dZ du 
du du c?u 
dZ dv dZ dw 
dv dw' dw 6?u’ 
1 du dv div 
but ((85.),art. 72.),&c., 
and therefore 
as was to be proved. 
82. It follows, then, that the mode of treating the problem adopted in the pre- 
eeding articles is equivalent to representing the motion of each planet by means of 
the true osculating ellipse of its actual orbit (relatively to the sun) in fixed space. 
