168 Mr Brill, A new Geometrical Interpretation [Nov. 28, 
21. A construction for the vector portion of the product of 
three planes can be obtained from the equation 
V .8,8,8, =8,88,8, — 8,58,8, + 8,5s,s,. 
Draw (X) the mean plane of (A), (5), (C), with respect to the 
origin, for multiples 
cos(B, C’)/p,p,, —cos(C, A)/p,p,, cos(A, B)/p, p,. 
Then draw a plane parallel to (X), such that its distance from the 
origin bears to the distance of (X) from the origin a ratio 
cos(B,C) _ cos(C, A) i cos(A, B) 
P2Ps PsP, PiP2 
Age 
22. As a final example we will discuss the scalar portion of 
the product of four planes, supposing, for the sake of fixing ideas, 
that the origin is within the tetrahedron formed by the planes. 
We have the equation 
S . 8,5,8,5, = SS,8, 95,5, — SS,8,58,8, + 98,5, Ss,5,, 
and from this, utilizing the notation of the preceding articles, we 
deduce 
Saag =) eee Jeos(B, @\co( Al D) = conyers Ceam 
P: Po Ps Ps 
+cos(A, B) cos(C, D)} : 
Other expressions may be given for S.s,s,s,s,: thus we have 
= SOS SS aro os OS, = Sega 4a/S), 6,5. 7.8... 
Now the perpendiculars from the origin on Vs,s, and Vs,s, are 
respectively parallel to CD and Ab*, also TVs.s, =sin(A, B)/p, p, 
and T'Vs,s,=sin(C, D)/p,p,. Hence, if (A.B, CD) denote the angle 
between AB and CD*, we have 
S. Vs,8, Vs,s, =— sin(d, B) sin (C, D) cos (AB, CD)/P, P2 Ps Pa: 
Thus we have 
SS 885, 
S.8,8,8,8, = a (A, B) cos(C, D) 
P:P2 PsP. | | 
in Bycin(C Dengan cD)| 
feos (B, 0) cos(D, A) 
1 
Fp PkpNps 
Sin(ByG) sin (DA) eon (BG! Day} 
* The order of the letters is understood to indicate direction. 
