1887.] of the Quaternion Analysis. 157 
Thus the equation of the locus of £ is 
2=m, (u, cosa+v, cos 8 +w, cosy)+...... 
+m, (u, cosa+v, cos B+ w, cos y) ; 
smu =mv, mw 
1.6. Pere eee asim — i — 0) 
We shall speak of this as the mean plane, with respect to O, 
of the n given planes for the multiples m,, m,, ...... , m, Yrespec- 
tively. 
3. Let the equation of a plane be given in the form 
ux +oy+wz—1=0, 
where w,v, w are the reciprocals of the intercepts on the axes of co- 
ordinates. We shall denote the position of this plane by the 
expression 7w+ju-+kw, where i, 7, & are symbols obeying the 
laws 17 =7° =k’ = yk=—1. Sometimes we shall find it convenient 
to replace the expression iu +7v + kw by the single symbol s. 
It is clear that any plane parallel to the given one will be repre- 
sented by a scalar multiple of the expression which denotes the 
given plane. The plane at infinity will be represented by zero, 
and any plane through the origin will be represented by an infinite 
vector. We shall still, for the sake of convenience, continue tu 
speak of an expression of the form w+ jv+kw as a vector 
although it is no longer taken to represent the relative position of 
two points. 
4. Let uatoytwz—-1=0 and ua2#+v,y+wz—-1=0 be 
the equations of two planes. Consider the plane 
m (ue +v,y+wz—1)+nu,e+v,y + w,z—1)=0, 
or as it may be written 
MU, + NU, pie mv,+ nN, mw, + NW, 
z—1=0. 
m+n mtn m+n 
Let this be equivalent to wz + vy + wz—1=0, then we have 
(m+ n)u=mu, + nu,, 
(m+n) v= mv, + n0,, 
(m+n) w=muw,+ nw,. 
Thus if we write 
s=w+tjut+kw, s,=1u,+jv,+hw,, s,=w,+jv,+kuw,, 
then we have (m+n)s=ms, + ns,. 
