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'^, c"^ are the squares of 

 the cosines of the angles which these axes make with the axis of z'', 

 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 ^ = 

 h=k, {w) become identical independently of the angles 4-? 'Pj ^ 

 .". every axis drawn through the origin of the coordinates 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{x'^ +y'^)dm=2g. 

 If we put sInJ=0, the first and second of (w) are satisfied, and the 

 planes of x\ y', and x, y coincide ; also the axes of z' and z coincide ; 

 and by (0) a", h", c are each =0, .'. the third of {w) becomes 

 abg-{-ah'h = 0, and by (/) a6 + a'5'=0 or a'b'= — ab .'.ab{g-h) 

 =0, and by (/) a^-[-b'^ = l; these equations are satisfied by making 

 b = 0, a = ±l which make the axes of x', y' to coincide with those 

 of X, y ; the above equations are also satisfied by making a=0, 

 6 = zblj which indicate that the axis of x' coincides with that of y, 

 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=h the above 

 equations are satisfied, and as a, b are indeterminate, every axis 

 drawn through the origin in the plane x, 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-^c"'^(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{x'^ -i-y''^)dm 

 = const, when the axis of z' makes a constant angle with the axis of 

 z; 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 = K-&'{A-B)-c"'{A-C) = C + c'{A-C) + 

 c'2(B — C), which show that S{x''^-\-y'^)dm is less than A, and 

 greater than C, whatever may be the direction in which the axis of 

 z' is drawn; .'. A is a maximum, C a minimum, and B neither a 

 maximum nor minimum. Put x'='X-{-,x, y'-=Y-^,y, z'=Z-\-,z, 

 m= to the mass of the solid, and suppose that X, Y, Z are the co- 



