312 
ME. R, S. HEATH OH THE DYNAMICS OF 
Multiply these equations by v lf v 2 , . . . v 6 and add; then 
tm(v 1 v 1 +v i v i + . . .)=S(X 1 v 1 + . . .), 
and therefore integrating, 
i'Zmv 2, =% 
(Xi^id- . . . )dt , 
which is the equation of conservation of energy. 
45. We must now express the velocities of all the particles of the body in terms 
of the given angular velocities &q, oj 2 , co 3 , oq, co 5 , &q. 
Let a, b, c,f } g, h be the line through ( x , y, z, u) along which this point begins to 
move. The line will therefore join the two points ( x , y , z, u) and {x-\-x^t, y-\-yht, 
z-\-z§t, u-\-uht). Hence 
a :b : c :f: g : h 
=y(z+zht)—z(y+yht) : . . . 
—yz—zy : zx—xz : xy — yx : xu—ux : yu—uy : zu — uz 
But 
yz—zy=zu(D. 2 —yuco 3 +(y z -\- z 3 ) oq —xy oq— zxco 6 
and we have similar expressions for the other terms. Hence 
yf=(x 2 +u 2 )u 
1 +xyoi 2 
-\-zxw 3 
ZU(D- +yuco 6 
3 
H 
II 
cn 
+ {y' 2 +u 
2 ) w 2+2/ Zft, 3 
-{-zu(o± . — xucoq 
yll—ZX oq 
+ (z 3 +w 2 ) 
&> 3 — yuio± xilco^ 
ya— 
-\-zucj 2 
— yuw 3 
+ (r+ 2 2 ) — xyo) 5 — ZXOJg 
yb = —ZWoq 
• 
XUg)q 
- zy&q + (z 3 +X 2 ) 0) 5 - yZaq 
g,C =?/Waq 
— xmo 2 
—— yz<o 5 + {x 2 +y 2 ) 
where y is some common factor yet to he determined. 
46. Square and add these expressions. The left side reduces to y 2 ; on the right, the 
coefficients may be simplified by virtue of the relation, £ 2 +y 3 +z 2 + u 2 = 1. Thus the 
coefficient of oq 3 is 
x l -j- 2x~u° +« 4 + x 2 y 2 + z 2 x 2 + z 2 u~ + y 2 u 3 
= (x 2 + u~) (x 2 -j- y~ +z 2 + ir) 
=x 2 -\-u 2 . 
Again, the coefficent of 2aqaq is 
