224 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
Using the preceding notation, the component of this in the direction 
indicated by X is 
T' a(a\ + Pfi + yv), . 0 N 
m + oVY* (aX + ^ + yv) 
(A 2 4 -p 2 fx 2 + q 2 v 2 ) 
M 
= 3^[Aa(l - a 2 ) - Ba/3 2 - Cay 2 ] 
= 3^[A-(Aa 2 + H/? 2 + C 7 2 )]a, . 
( 11 ) 
Fig. 1. 
and the similar expressions for M T 0 , vT'o can be at once obtained by cyclical 
interchange of A, B, C ; a, y. Hence, from the condition 
L' 0 = - M[(A + B + C) - 3(Aa 2 + B/3 2 + Cy 2 )] 
along with (8), we have 
aT 0 ~ ( aL o “ L o) a 5 
0 = ( juU 0 — L o)0 5 
vT o = ( vUo ~ L o/7 • 
These results indicate a very simple geometrical construction for the 
principal components of the transverse force by means of the surface (10), 
whose radii have the values L 7 0 . Draw the spheres of radii A L/ 0 , M L' 0 , „L' 0 , 
which are the principal radii of (10), and draw the radius- vector of (10) in 
the direction (a, /3, y). The differences between the radii of the spheres and 
the radius-vector of (10) give, when projected respectively upon the prin- 
