156 Mr Brill, A new Geometrical Interpretation [Nov. 28, 
(4) A new Geometrical Interpretation of the Quaternion Analysis. 
By J. Brit, M.A. 
1. In the September number of the Messenger of Mathematics 
I described a new method for the geometrical representation of 
complex quantities in which the complex quantities were repre- 
sented by lines instead of points, tangential co-ordinates being 
substituted for cartesians. 1 now propose to apply the same ideas 
to the quaternion analysis, and thus to obtain a three-dimensional 
analogue of my former method. As in the former case, and for 
the same reason, it will be found that the analogy of the new in- 
terpretation to the old is not quite complete. 
2. In the paper alluded to in the preceding paragraph, in 
order to lead up to the question of the addition of lines, I intro- 
duced the idea of the mean line. It will be found that the theorems 
connected with this line have their analogues in geometry of three 
dimensions, and that these furnish us with material for dealing 
with the question of the addition of planes. Thus we have the 
following theorem : 
O is a fixed point, and through O a straight line is drawn 
meeting n fixed planes in the points 1,, 7,, ...... , 7, A point £ is 
taken on this straight line so that 
M, + M, +... ag LO me TOF mM, m,, 
OR Or, Or, = om 
The locus of & will be a plane. 
To prove this take a system of three rectangular axes through 
O, and let the equations of the m planes referred to this system be 
UZ+v,y +w,z—1=0, 
ue + v,y + w,z-1=0, 
U,c+U,Y +w,2—-1=0. 
Then, if a, 8, y be the angles made with the axes by the line 
through O, we have 
I 
Or = tx COS 4 + Y, COS B + W, COS ¥y, 
1 
1 
Dp = tz 008 a +, cos B + w, cos ¥, 
i i ee ee er ery 
Il 
Or. = Un COS & + 2, COS 8 +w, cos ¥. 
