1887. ] of the Quaternion Analysis. 159 
We are now in a position, with the aid of this formula, to discuss 
the question of the multiplication of two or more planes. Thus, if 
s, and s, denote two planes, we have 
iD ae 
SiS. =a (2.cos a, +7 cos B, +k cos y,) (2 cos a, +7 cos 8, + k cos ¥,) 
2 
1 
Bip {— (cos a, cos a, + cos 8, cos B, + Cos y, COs ,) 
1f£ 2 
+1 (cos 8, cos y, — cos B, cos y,) +7 (COs ¥, COS a, — COS ¥, COS @,) 
+ k (cos a, cos 8, — cos a, cos 8,)}. 
Let 6 be that dihedral angle between the two planes within 
which the origin lies, then 
Sei and TV (s,s,) = sind f 
PiP Lee 
1-2 
1f£ 2 1f 2 
The vector portion of s,s, is a plane, and, if we write s/c=V.s,s,, 
we have p=p,p,/csin@. Further, the direction cosines of the 
perpendicular from the origin on s are proportional to 
cos 8, cosy, — cos B, cos y,, 
COS Y, COS a, — COS ¥Y, COS G,, 
cos a, cos 8, — cos a, cos B,. 
Thus the plane s is perpendicular to the line of intersection of s, 
and s,, and its distance from the origin is p,p,/c sin 0. 
Since V.s.s,=—V.s,s,, we see that V.s,s, and V.s,s, denote 
a pair of parallel planes at equal distances from the origin on 
opposite sides of it, their orientation and the distance of each from 
the origin being given by the preceding paragraph. It only re- 
mains to determine which is which. To specify this, let O be the 
origin, OL the perpendicular from it on s,, and OM that on s,. 
The perpendiculars from O on V.s,s, and V.s,s, will have to be 
measured off on a line drawn through O perpendicular to OL and 
OM, the perpendicular on V.s,s, beg measured in such a sense 
that a right-handed screw motion about it would tend to turn OL 
towards OM. The perpendicular on V.s,s, would have to be 
measured in the opposite sense, t.e. so that a right-handed screw 
motion about it would tend to turn OM towards OL*. 
If the two planes be at right angles to each other we have 
S.ss,=0. Thus the product of two planes at right angles to 
one another is a plane at right angles to both of them, and, if 
* T have taken the fundamental system of axes to be a right-handed screw 
system. 
