see Mécanique Analytique, Avertissement, t. I. p. Y. [Ed. 3, p. vil] : only in this latter 
case V stands for the disturbing function, the principal forces vanishing. 
> = Ap & + Bq $ + Or £, Aup-rBq + jM-); 
dp dq dr d/c 
if Tf df 
{Ap ' ~ pBq ' + ^ ~ l Bqv ' + 1 CrtL ' ~ % (Ap * vBq+ ' xCr) - 
Also d I= Ap t +Bqd I +Cr ^ 
. . (d dT dT\ 
and hence 
= i {Ap'-vBq’ + yGr') - - Bqv' + - Cry! + -(.4jj= + Bq'- + Ch*) - *, (A p - vBq + yCr). 
K, K/ K> K> ^ 
Substituting for A', y, v, k , after all reductions, 
i^ t ~-^) = lUAp' + (C-B) q r}-v{Bq' + (A-C)rp}+yiCr + (B-A)p q }]-, 
and, forming the analogous quantities in y, v, and substituting in the equations of 
motion, these become 
dV 
{Ap + (G-B) qr) - v [Bq' f (A - C) rp) + y {Gr' + (B — A) pq] = \k , 
dV 
v [Ap' + (C — B) qr] + {Bq + (A — G) rp) - A {Gr f + (B - A)pq) = \k, 
dV 
y {Ap +(C — B) qr) + A {Bq + (A - G) rp) + {Gr + (B - A) pq) = £/c ; 
or eliminating, and replacing p, q, r, by ^^^, we obtain 
A% + (C - B )qr = \{(\+K)^ + (\y + v )d ^+{vX-y/-^), 
B^-KA-C)rp = i\(\y-v)^+(l+y^ + (yv + \)^), 
G % + (B ~ A)pq= * { ( " x+/l) s; + ^ ~ x) % + (1 + " !) aï} ;