314 
MR. R, S. HEATH ON THE DYNAMICS OF 
48. The last article suggests the idea of moments of inertia. We may define the 
sum of the products of the masses of all the particles of a body into the squares of 
the sines of the perpendicular distances from a given plane, as the moment of inertia 
of the body with regard to the given plane. Similarly the product of inertia with 
regard to two planes is the sum of the masses multiplied by the product of the sines 
of the perpendicular distances from the planes. The moment of inertia about a line 
may be defined in a similar way. The moment of inertia of a body about any plane 
(l, m, n, p) is 
%m{lx-\-rny-\-iiz -\-p w 2 ) 2 
i.e., 
l~%mx 2 -f- m^my 2 -f- n 2 Xmz 2 -f- p 2 %mu 2 -\-2lm'Zmxy -fi . . . 
Thus the equation 
l 2 %mx 2J r . . . -\-‘2mn%yz J r ... =0 
is the tangential equation of a surface of the second order, which has the property 
that the moment of inertia about any tangent plane is zero. Professor Clifford calls 
this the null surface of the body. The property just mentioned is independent of the 
system of coordinates chosen. 
49. By an orthogonal transformation of such an equation of the second degree 
we can rid the equation of the products m, n, . . ., and at the same time keep 
Z 2 -f-m 3 -f -n 2 -\-p 2 unchanged. The surface now becomes 
/ 2 2 nix 2 +m 2 2my 2 + ir'Zmz 2 +p 2 2 mu 2 = 0. 
The planes of the particular quadrantal tetrahedron chosen may be called the four 
principal planes of the body, and the edges the six principal axes of the body. 
The envelope of planes which give a constant moment of inertia, MK 3 , is given by 
the equation, 
l 2 %mx 2 ^m 2 ’Zmy 2 -\-n 2 Xmz 2 -\'p 2 Smii 2 =WK 2 [l 2 -\-m 2 -\-7i 2 -\-p 2 ). 
This is a surface of the second order. Now 
l 2 -\-m 2 -\-n 2 -\-p 2 =Q 
is the tangential equation to the Absolute quadric. Hence, this surface has the same 
common tangent planes with the Absolute as the null-surface has. In other words, 
this surface and the null-surface stand in the same relation to each other as confocal 
quadrics in ordinary geometry. Hence the planes which give a constant moment of 
inertia envelope a quadric confocal with the null-surface. 
50. We next consider the moment of inertia about a line. 
Let a, b, c, f g, h, be the coordinates of any line, then it was shown in § 17, that 
if try be the sine of the perpendicular distance from any point (x, y, z, u) to the fine, 
