398 
Proceedings of the Royal Society 
given since. The last of these, due to Professor Tait, rests on purely 
quaternion methods, and is so elegant and concise that I was led to 
reinvestigate the whole subject by ordinary methods in the hope 
that the analysis might have some points of interest. Two methods 
of arriving at Minding’s result are given, and a variety of other 
conclusions are arrived at by means of the second method sufficient 
to indicate the course of a full investigation of the complex formed 
by the central axes, and of the congruency formed by the single 
resultants of Minding’s system. 
First Method. 
The components of force and couple are found in terms of the 
Rodrigues co-ordinates Xgv, which determines the position of the 
rigid pencil representing the direction of the forces. 
The equations to the single resultant are then found in terms of 
two constants g and h , and the parameters Xgv. 
Equations are then deduced for the values of Xgv corresponding 
to a ray passing through a point xyz. Eliminating g and v a biqua- 
dratic is found for A. The system of resultant rays therefore 
forms a congruency of the fourth order. 
This biquadratic becomes wholly indeterminate for points on the 
real focal conics of the ellipsoid 
xr yi Z z 
f + T? + W + g* = 1 ’ 
(A) 
Some farther discussion leads to the conclusion that the resultant 
rays of Minding’s system is identical with the congruency of rays 
that intersect the two focal conics of (A). 
Second Method. 
If ijrjt, be the co-ordinates of the feet of the perpendicular from 
the origin on any ray whose direction is (A, g, v), and p the length 
of that perpendicular, it is shown that 
P 2 = pV + ...(B) 
p 4 + g 2 v 2 + h tp = g y2 * AV . . (C) 
(B) is true for central axes generally, and determines a complex 
of the second order which they form. Both (B) and (C) are true for 
