DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
325 
Now since we are supposing the equations to belong to a dynamical problem, we 
have u=mx', v=my', w=mz', and if we substitute on each side of these equations 
the values derived from (85.), art. 72, we find easily 
u=m{s’ — oj^y), v=^m{y' w=m{7l -\-o)^y — a^x), 
relations which are true on all suppositions as to the motion of the axes ; but the 
assumptions {u.) reduce them to 
u—mx', v—my', w—mz'. 
The further assumption z = 0, which involves z'=0, gives m?= 0, and also (by the 
equations (co.)) ^^=0. Thus the equations (86.) are reduced to 
/ dZ 
dZ 
dx 
, dZ 
dZ_ 
dy 
Od^Zt ^ d^Zi I 
= -r-5 0= COfP, 
dw dz 
where inland ^5 z and 10 are to be put =0 after the differentiation. It is to be 
dio dz 
observed also, that the values of u, v, w above given reduce the three first of the 
equations (86.) to the form u-=m v—m w — m ^5 from which it is evident that 
^ du dv dw 
where U is the original force-function, expressed in terms of the new variables. This 
depends only upon the conditions (w.) ; but in the case now considered we have also 
w;=0. 
Let the origin of coordinates be the Sun ; m a planet disturbed by another planet 
Ml whose original coordinates are x^, y^, and new coordinates x^, y^, ; also let 
x^+y^^+z^=r^, and (x,— x)^+(y,— y)^ 
+ (z,— z)^=S^; we have evidently r^=r^, rf=r^, and 
{x^-yf-\.{Y-yy-\-{z-zf = {x,-xy-\-{y^-yf-\-{z,-zy, 
and y.-%^-\-yY^-\-zz^z=xx^-\-yy^-\-zz^. 
We have then originally 
U=!f+mm,Q-52‘±S^'), 
where |«;=7n+mass of Sun ; and it is evident that this expression preserves the same 
form when expressed in terms of the new coordinates, and also (which is essential to 
the validity of what follows), that &c. are the same whether the differentiation 
CLOC 
2x2 
