A RIGID BODY IN ELLIPTIC SPACE. 
315 
77 2 =:a 3 (ar + u 2 ) + b 2 {y 2 -\-u 2 ) + c 2 (z 2 + u~) 
+/% 2 + * 2 ) +r/( z ' + ;« 2 ) + h~{oc~+y~) 
+ 2 yz {be —gh) + 2zx ( cci—hf) + 2xy(ab —fg) 
+ 2 xu(cg — bh) + 2 yu(ah — cf) + 2 zu (fb — ag). 
Making use of the previous notation the expression for the moment of inertia 
becomes 
Mfc 3 = Aa 2 +B6 2 +C c 2 + Ff~ +G# 2 +H/F 
+ 2P 1 (6c —gh) + 2P 3 (ca—hf) + 2P 3 («6 — fg) 
+2P fcg - bh) +2P S (ah-cf)+2¥ ( .(fb—ag). 
If we refer this to the principal tetrahedron of inertia, all the P’s vanish, and it 
reduces to 
m 2 =Aa 2 +Bb 2 +Cc 2 +¥f 2 +Gg 2 +m 2 . 
It must be noticed that the quantities A, B, C, F, G, H are not independent; 
for since 
X~ + if + z 2 + u 2 = 1, 
we have 
.a+f=b+g=c+h=m, 
where M is the mass of the body. 
51. We are now in a position to write down the most general equations of motion 
of a rigid body, under the action of any forces, referred to a quadrantal tetrahedron 
moving in space, in any manner. For in § 34 we obtained formulae for the rates of 
change of any quantities, obeying the usual law of resolution and composition, rela¬ 
tively to axes moving in any manner. Applying these formulae we can obtain the rate 
of change of momentum relative to any moving axes. Equating the rate of change of 
momentum to the impressed forces, we get the equations of motion. Let q x , q. 2 , . . . q Q 
denote the six components of angular momentum about the six edges of the tetra¬ 
hedron, and let Q 1; Q 2 , . . . Q 0 denote the total impressed couples about these lines. 
The expressions for q x , q 2 , . . . q e , have been given in § 47. The equations of motion 
are, therefore, 
C ll • 'h^6?2+^5 ( Z3 • — 03<76+02(76 = Ql 
0 6?1 • + • -%6 = Q2 
?8“”05?l + 04?8 ’ • —Q3 
c h ' ~ ^ 2 + ^2^3 • — + —Qr 
?6 + ^l?l • • “^G=Qo 
• -0- Q q±+0±q 5 • =Q e 
2 s 2 
