of Edinburgh, Session 1870-71. 317 
Id fact, if we apply the members of the general equation above 
to €, we have 
This leads to the two equations 
~ ^ e = 0 , 
~i[) e = 0 > 
which, belonging to two cones of the second degree, give in general 
four values of e. 
7. The interest of the general question before us, from the 
analytical point of view, lies mainly in the determination of the two 
unknown linear and vector functions x and 0 from the equations 
X + 0 = <p, 
xo = ^ , 
each of which is in general equivalent to nine or in certain cases 
six (not, as in ordinary quaternion equations, four, or as in vector 
equations three) simultaneous scalar equations. They have also a 
where 
2 9i = m 
(g\ — 
4m„ 
The values of g being found, -a- is given by the expression above. 
A similar process may easily be applied to the general equation of (6), but 
it may be well to exhibit the present simple case in its Cartesian form. 
Let S iai — p x , S iaj = p 2 , S iuk — p 3 , 
S joi ~ q x , Sjaj = q, , S qak = q 3 , 
S kui— r x , S kaj— r 2 , S hah — r s 
Also let 
where 
— ccSz -p /SS j -p , 
« = ix l + jx 2 + kx 3 , 
a = iy x + jy 2 + ky 3 , 
y ~ iz x + jz % + kz 3 , 
then the problem reduces itself to the determination of the nine scalars 
x, y , z, &c., from nine equations of the second degree, of which we write only 
the first three :—viz. 
d* y 1^2 d~ Z X %3 — P\ » 
r 2 x x -P ypc % -P — Pz > 
x 3 x x + ypc 2 -p z 3 x 3 — p 3 . 
