160 Mr Brill, A new Geometrical Interpretation [Nov. 28, 
this plane be denoted by s/c, its distance from the origin will be 
P,P,/¢- 
If the two planes be parallel then we have 
V.ss,=— V.s.s,=0; 
and thus the two products s,s, and s,s, will be entirely scalar, and 
each of them will be equal to —1/p,p,. Further, if we make 
s,=s,, we have 7's, =1/p,. Thus we see that the equation 7's = 1/a 
may be taken as the tangential equation of a sphere of radius a 
having its centre at the origin. 
6. Before proceeding to discuss the multiplication of three 
planes we will seek to obtain the interpretation of some quaternion 
formule with the aid of the preceding article. 
If Ts = Tt, then we have 7 (ms + nt) = T (ns+ mt), and from 
this we easily deduce the following theorem : 
Let (A) and (B) be two tangent planes of a sphere having the 
point O for its centre; and let (X) and (J) be the mean planes, 
with respect to O, of (A) and (b) for multiples m, n and n, m 
respectively. Then (X) and (Y) are tangent planes of a sphere 
concentric with the former. 
7. If Zs= Tt, then we have S(s —#) (s +t) =0, from which we 
deduce the following : 
Let (A) and (B) be two tangent planes of a sphere having its 
centre at the pomt O. Through O draw a plane parallel to (B), 
and through the intersection of this with (A) draw a plane parallel 
to that containing O and the line of intersection of (A) and (5). 
This last plane will be perpendicular to the polar plane of O with 
respect to (A) and (B). 
8. We will next consider the equation V (s—¢)(s +t) =2Vst. 
Let O be a given point, and let p, and p, be the perpendiculars 
from it on two given planes (A) and (B). Through O draw a 
plane parallel to (B), and through the intersection of this with (A) 
draw a plane (X) parallel to that contaming O and the line of 
intersection of (A) and (B). Then, if (Y) be the polar plane of 
O with respect to (A) and (B), the above equation becomes 
UO OD = 15 (GCA: 
Thus, if g, and g, be the perpendiculars from O on (X) and (Y), 
we have 
sin (A, B) = sin (X, Y) ::p.p, : ¢,9.; 
where (A, B) and (X, Y) denote the dihedral angles between (4) 
and (B), and between (X) and (Y), respectively. 
