DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
331 
y^,), Z(i). Then instead of the original system of diflFerential equations of the second 
order, we have the following system of the first order : 
^^1 dZ(i)_ dZ^i) 
du, 
ii) 
Yar 
dv(i 
(i) 
dw, 
(i) 
u:,+g'=o, v;,,+^=o, w;.,+gi.=o, 
(i) 
in which 
Z(i) — 
These equations are not of the canonical form, because is not the same for all the 
planets. But it is easy to put them in the pseudo-canonical form (art. 74.), a process 
which is not necessar'y, but saves trouble by bringing them under the operation of the 
general rules of transformation established in former articles. 
In fact, if we take 
Z=2 
11^ -I- -A- 
2/n(0 r(i) 
/ X(i)Xo-) + + Z(i)Z(y) 
V 
U) 
+ 7(8)70-) + Z(oZ(i) 
P 
(t-) — 
, dZ dQ 
dxf. 
( (X(8) - xo))2 + (7(8) - 7(,-))2 + (Z(.-) ■ 
where the summation in the first term extends to all the planets, and in the second 
to all their binary combinations, and a horizontal line placed over any letter indicates 
exemptim from differentiation, we shall have 
dZ^_d^^ (91.) 
f(i) dXi^i'j 
with similar equations for /(.), Zj,,, v'^^), w^^). 
dOi 
[The terms &c. are only written for the sake of uniformity, being really =0, 
since Q does not involve Uf^), Vj^j, w^^).] 
In these equations Z and Q are the same for the whole system, and the differentia- 
tions of Z are total ; but those of Q are restricted to the quantities not marked by the 
«?(ol 
dRii 
(i) 
horizontal line, so that is really the same thing as 
78. Let us now refer the whole system to new (moving) rectangular axes, whose 
position at any instant with respect to the original axes is defined, as in art. 72, by 
the variable direction-cosines &c. Let be the new coordinates, and 
M(,), v^i^, W j) the new’ variables conjugate to them. The transformation will be effected, 
as in art. 72, by taking for the modulus 
P = 2((Xoa?(i) -f -f U(i) -f- (^o-^(,.) -b/A, -1- V(.-) + ( V(i) + + «'22;(8)) W(i)), 
and the result will be as follows : put 
n = Q -f ^o2 — 3/(i)M^(i)) 4- 2 (^(i)«^(i) — S(,:)M(8)) + «22 (3/(0«(i) — •i?(i)«^(,-)) 
MDCCCLV. 2 Y 
