A RIGID BODY IN ELLIPTIC SPACE. 
313 
xy(x°-\- u°) -\-xy(y 2 -\- it?) -\-xyz? — xyi? 
=xy(x 2 -\-y 2 -\-z^+u 2 ) 
-xy. 
All the other terms may be simplified in the same way, and therefore finally 
y 2 = (i)-f(x l -\-u 9 ) +oj 3 3 (3/ 2 +m 2 ) + (u 3 2 (z 3 +m 2 ) 
+ "r% 3 + z3 ) +co 5 3 (z 3 4-^ 3 ) +uZ(x 2 +y 2 ) 
+ 2xy((0 z (0 s —0) 5 0) 6 )-\-2zx(a) 5 0 ) 1 — (0 & 0)^ J r 2 xy((o 1 co . 2 — co i w 5 ) 
-p 2 xu(co 3 o ) 5 — <y 3 co 6 ) "h ^yu[yi x (o G — &> 3 oq) -f- 2 zu (&) 3 &q—aq&q) 
Next multiply the equations by a> l5 co 2> . .. co G and add; the same expression as before 
appears on the right, and therefore 
that is 
,xK/+ co. 2 g + o)Ji -pco 4 « + o) G b -p w 6 c ] — y 2 , 
y = ~p -p co^b -f- OJgC. 
Hence y is the velocity of the point under consideration, and the above expressions 
for ya, yb, . . . are the velocities of the point ( x , y, z, u) resolved along the six edges 
of the tetrahedron. 
47. Multiplying them by m the mass of the particle, and taking the sum for all the 
particles of the body, we find expressions for the linear momenta of the body resolved 
along the six edges of the tetrahedron. For brevity, let A=Sm(ar-{-M 3 ), &c., and let 
P y =tmyz, P 2 =%mzx, &c. Then if (g) L3)3i4i5i6 denote the angular momenta about the 
edges 23, 31, 12, 14, 24, 34, or the linear momenta along the edges 14, 24, 34, 23, 31, 
12 respectively, 
<Zl= Aoq +P 3 <y 3 +P 2 <y 3 . — P 6 w 5 -f P 5 &> 6 
2b— IV'T + B&u +P 1 <U3+P 6 «4 • — P^fi 
c h— 
f h— • +P6 w i — P o w 3+Fw4 — P 3 W 5 —P 3 W 6 
q h —— V 6 w 1 . -(- P> 3 —P s w 4 + <Jco 5 — Pi&jg 
q 6 = P 5 W 1 +P 4 W 2 • — P 3 w 4 —PiCOg+Hwg 
Also taking \%my 2 we get the kinetic energy, viz.: 
2T=Awp -j- Bo) 3 3 Co) 3 3 -p F oj 4 2 -p G&q 3 -pH oj 6 3 
+ 2P 1 (coo w 3 — &> 5 a) 6 ) -p 2Po((y 3 a> 1 — w 6 w 4 ) + 2 P^aqajg — &q(u 5 ) 
+ 2P 4 (w 3 w 5 — w 3 w 6 ) + 2P 5 (w 1 w 6 — togoq) + 2P 6 (<y 3 aq—aqcug) 
2 s 
MDCCCLXXXIV. 
