Motion of a System of Bodies. 287 
moment relative to the axis of x; also B and C are the moments 
relative to the axes of y and z, and c?, c’*, ce’? are the squares of 
the cosines of the angles which these axes make with the axis of 2’; 
hence, generally, if we multiply the moment of inertia relative to 
each of the principal axes passing through any given point, by the 
squares of the cosines of the angles which they severally make with 
any other axis drawn through the same point, and add the products, 
we shall have the moment of inertia relative to that axis. If g= 
=k, (w) become identical independently. of the angles 1, 9, 4, 
*. every axis drawn through the origin of the codrdinates is a prin- 
cipal axis; and by (x) Sz’*dm=g=const. whatever may be the di- 
rection in which’ the axis of 2’ is drawn, .*. S(2’?-+-y/? )dm=2g. 
If we put sin. =0, the first and second of (w) are satisfied, and the 
planes of 2’, y', and a, y coincide ; also the axes of 2’ and z coincide 5 
and by (0) a”, b”,c are each =0, .*.the third of (w) becomes 
abg+-a'b‘h=0, and by (f) ab+a'b’=0 or a'b'=—ab .. ab(g —h) 
=0, and by (f) a2+b?=1; these equations are satisfied by making 
b=0, a=-1 which make the axes of 2’, y’ to coincide with those 
of x,y; the above equations are also satisfied by making a=0, 
b=+1, which indicate that the axis of x coincides with that of Ys 
and the axis of y’ with that of x; on these suppositions we there- 
fore have no new system of principal axes: but if g=A the above 
equations are satisfied, and as a, 6 are indeterminate, every axis 
drawn through the origin in the plane 2, y is a principal axis, and 
we have an infinity of systems of principal axes, the axis of z being 
common to them all. 
Again, by (x) when g=h, Sz’?dm=g-+-e?(k—g); .°. Sz/?dm= 
const. in whatever direction the axis of z’ may be drawn, provided it 
always makes a constant angle with the axis of z; .". S(a’? +-y'2)dm 
= const. when the axis of z’ makes a constant angle with the axis of 
2; also, (as before,) all the axes drawn through the origin in the 
plane x, y are principal axes. If no two of the quantities g, h, k 
are equal, then no two of A, B, C are equal; let A be the greatest 
and C the least of them, then (y) is easily put under the forms 
S(x’? +y/?)dm = A—e?(A—B)—c”?(A—C) =C+4e2(A—C)+ 
¢?(B—C), which show that S(a’?+y'*)dm is less than A, and 
greater than C, whatever may be the direction in which the axis of 
2’ is drawn; .*. Ais a maximum, C a minimum, and B neither a 
maximum nor minimum. Put o=X-+ a, y¥=Y¥+,y, 2° =Z+,z, 
m= to the mass of the solid, and suppose fsa X, Y, Z are the co- 
