A RIGID BODY IN ELLIPTIC SPACE. 
305 
<q— . + 0 6 %— 0 b a 3 . + d s a- 0 — 0 2 ci 6 
«2 = — ^fi«l • +^#3 —^3% • + 9 i a 6 
Of 2 ft O 1 9 2 • [~ fhcq, • 
a 4 = . + $3«o — #o« 3 . +$6 a 5 — ^5 a 6 
%=— 1 03«1 • +^ 1 « 3 —^G«4 • + ##6 
« 6 = % r i — ^i«2 • + Oy^—Oyi- 
34. Let 34 , q 2 , q s , q±, q 5) q 6 , be any quantities which obey the same law of resolution 
as has been proved for angular velocities ; we can now find the rates of change of 
these quantities, referred to axes moving with the body. Let Q 1} Q 3 , . . . Q 6 , be the 
rates of change of these quantities. Now cqcq+cqg'gfi- oqg'gfi- cq^-fi a h q b -f a 6 q Q is the 
component of the q s in a fixed direction. Hence we must have 
d 
—(ui^i+ay^-h . . . + a 6 ? 6 ) — a iQi+ a 2Q3+ • • • + a 6 Q 6 
Differentiate out the left side, and substitute for cq, cq, cq, . . . cq, in terms of the 
angular velocities of the moving axes about fixed axes instantaneously coinciding with 
them, i.e., in terms of the 0’s. Then 
.cqQi+a 3 Q 3 + . . . +a- 6 Q 6 
= a iQi - 9 s<h + e s<h ~ e z<h + 
similar terms in cq, cq . . . cq, 
and there are similar equations with b, c, . . . substituted for a. If we multiply these 
equations by cq, b } , <q,. . . , and add, we find an expression for Q x . In this way we 
arrive at the equations 
Ql = 2 l • -te+%3 • -%5 + ^6 
Q 3 =( ? 2+%1 • -%3 + %4 • ~%6 
Q 8 = ?S“^5?l + ^45 , 8 * ~ 9 ^+ 9 lTo 
Q*=& • -% 2+^ 3 • -%,+%6 
Q 5 = ^5 + %1 • “%3 + te • -%6 
Q 6 = 2 ' 6 -%l + ^2 • -%4+%5 
Theory of screws. 
35. It has been shown that any state of motion of a rigid body may be reduced to 
a translation along a line and a simultaneous rotation about it. Such a motion is 
MDCCCLXXXIV. 2 R 
