286 Motion of a System of Bodies. 
=(h’sin. L +k cos.-L).(k/sin. )—h’ cos. .L)?, or [(g —A) sin. | cos. 
+g7/(cos.?.) —sin.2.)] x [(k'g — h’g’ — k’k) sin. b+ (Wk+K'g’—Wh) 
cos. | ]=(A’sin. b+’ cos. L).(k’ sin. | — A’ cos. L)?5 put u=tan.t, 
and we have (h/u+k’).(k/u—W’)? +((g—h)u+g'(l—u*)) x ((Kk+ 
hig’ —k'g)ut+-Wh—k’g' —Wk)=0, (v). Since (v) is a cubic equa- 
tion, it has (by the theory of equations,) at least one real root; .°..% 
is a real quantity; .*.. becomes known, and thence tan. 4 is found 
by the second of (u); .°.4 is known: having found J and 4, we can 
easily obtain p, for multiplying the first of (r) by dm, taking the inte- 
gral and putting S2’y/dm=0, we have tan. 2p= — = M and N be- 
ing known rational functions of sin. ), cos. 1, sin. 4, cos. 4, g, A, ete. 
-".p becomes known; hence the position of the axes of 2’, y’, 2’ is 
determined so as to satisfy (p); and it may be observed that the 
axes thus found are called the principal azes of the solid. It may 
be observed, that u=tan. |= the tangent of the angle made by the 
axis of x with the line of common section of the planes a, y, 2’, ¥'3 
but it is evident that (p) will exist if we change y’ into z’, and 2’ into 
¥ thatis, if we change the plane 2’,y' into x’,z’, and 2’,z/ into 2'yy’5 
*. (v) will give another value of u, which will be the tangent of the 
Aa made by the axis of x with the line of common section of the 
planes x,y, and a’,z’; and in a similar way it may be shown that (v) will 
give another value of u, which will be the tangent of the angle made 
by the axis of « with the line of common section of the planes @, y; 
y', 2; -'. the three roots of (v) are real, and they appertain, gener- 
ally, only to one system of axes: hence a solid has, in general, but 
one system of principal axes passing through any given point. 
Again, if Srydm=g'=0, Sxzdm=h’=0, Syzdm=k’=0, the axes _ 
of x, y, z will be principal axes: in this case, every term of (v) will 
=0, but (u) become sin. é cos. (g sin.? J + hcos.2 —k)=0, 
(g —A)sin.4 sin. L cos. L=0, also S.x’y/dm = abgta'b'h+-a"bvk= 0, 
{w) ; and (¢) becomes Sz’/?dm= g sin.?4 sin.? + A sin.2¢ cos.24+ 
kcos.26= ge? + he’ + ke’?, (x); also S(a/? -y'? +2/2)dm= =: §(a'? 
+y'? )dm+S2’?dm=S(x? +y? +27 )dm=gthtk=(since c? +¢?+ 
e/¥==1,) c2(A+h)+e2(g +h) +e (g-+h) + ge? +h’? + he’, or by 
(x), S(a'? +’? )dm=c? (h+k)+e(g+k)+e'2(g+h) ; put hth 
=A, g+k=B, g+h=C, and we have S(a'? +y'2 )dm=c?A+¢?B 
+¢//?C, (y). It may be remarked, that the first member of (y) is 
the moment of inertia relative to the axis of z’, and that A is the 
