316 
MR. R. S. HEATH ON THE DYNAMICS OF 
We may write the values of the ^’s in the form 
where T has the value given to it in § 47. 
52. If we suppose the tetrahedron fixed in the body, we must put 9 1 =&> 1 , 6- 2 =Cl> . 2 .. 
and if further, the tetrahedron be the principal tetrahedron of inertia, the equations 
will correspond to Euler’s equations. They then become 
A&q — (B — H)« 2 <y 6 — (G— C)w 5 (U3=Q 1 
Bw.,-'(If-Aj&Jg&p-(C -P Q-2 
Cwo —(A —— (F — B )m 4: oj. 2 = Q s 
Faq— (B — C )(o 2 oj 3 — (G — H)w 5 <y 6 = Q i 
Gap— (0 —Ajopoq— (H — F)a> 6 &q=Q 5 
Ha> ( ;• (A —B japan—(F —G)w 4 6) 5 =Q 6 . 
53. If the body be moving under the action of no forces, there are two invariable 
complexes, viz. : 
qj+ q%g +#+7i«+#+ q Q c =o 
and its conjugate complex. For since the equations of motion express the fact that 
the q’s do not change, these complexes will be fixed in space. The complex whose 
equation has just been given is the locus of lines along which there is no linear 
momentum. If no forces act, if we multiply the equations of motion respectively by 
op, op, . . . oj t; , and add, we get by integration 
A<y j 3 d~ Bq) 3 3 -p C&> 3 3 +F <u 4 2 -F G<a 5 2 +H<u 6 2 = 2T 
which is the equation of conservation of energy. 
If again we multiply the equations of motion by Aoq, Bap, . . . , and add, we arrive, 
at another integral of the equations, 
A 3 Wl 2 +BV+C 3 a) 3 2 +FV+GV+HV=K 3 , 
which expresses the fact that the sum of the squares of the q’s is constant. 
One more integral may easily be found. For let 
A =ma z , B —mb 2 , . . . 
then 
/==l-a 2 g~ = l—b' 2 h*= 1-c 3 
and 
G —C=M(1 — 6 2 —c 2 ) = H—B, &c. 
Hence the first three equations may be written in the forms 
