DIFFERENTIAL, EQUATIONS OF DYNAMICS, ETC. 
101 
The three integrals which express the conservation of areas, namely, 
Aap -{- Ba'q + C a"r = e, 
Abp-\-Bb'q-{- C b"r = f, 
Acp -|- Bc'q + Cc"r = q, 
become, after simple reductions, 
, sin 4/ 
— u cos v — (v— w cos 6\ — e 
T sin 9 v y 
, COS 4/ . as 
— u sin (v — w cos 0)==/ 
iv = q. 
Let e 2 -f/ 2 +y 2 =A 2 ; we have, adding the squares of these three equations, 
(v — w COS 0) 2 72 
tt H-*tr 
sin 2 6 
✓ 
(ii.) 
and we may take the three equations (i.), (ii.), and 
w=(j (iii.) 
as three normal integrals; the conditions 
[g, h] = 0, [h, A] = 0, [A, $r] = 0 
being obviously satisfied. 
These three equations determine u, v, w as functions of 6, <p, ; and supposing the 
three former variables to be explicitly expressed in terms of the latter, we should 
have at once the three partial differential coefficients ^ ~ ; the determination 
1 d9 dip d4> 
of V would therefore depend upon simple integration, and the remaining integrals 
would be given by means of the three equations 
dV dV d\ 
dh— t ~T~ T ’ dff— Cl » dk ~ C 2 ’ 
r, c 15 c 2 being new arbitrary constants. 
In the general case, however, the algebraical solution of the equations (i.), (ii.), (iii.) 
is impracticable, since the elimination of v and w leads to an equation of the fourth 
degree in u ; nor does it seem possible to .evade the difficulty by choosing a different 
combination of integrals, since it may be shown that the necessary conditions cannot 
be satisfied unless two at least of the combinations chosen are of the second degree 
in u, v, w. 
32. Mr. Cayley has given* a solution of this problem, which, though differing 
totally in form and method from the above, resembles it in arriving exactly at a cor- 
responding point. For in Mr. Cayley’s equations (27.), (28.), and v are to be 
expressed as functions of v; but this requires the algebraical solution of the system 
(18.) for p, q, r, and is therefore impracticable. (The two equations (i.), (ii.) of the 
* Cambridge and Dublin Mathematical Journal, vol. i. p. 167. 
