1887. ] of the Quaternion Analysis. 167 
we have 
s,’ (Ss,s,8,)° + 8,” (Ss,8,8,) + 8,’ (Ss,s,8,)° + ,° (Ss,8,8,)" 
— 28s,s, Ss,s,5,Ss,8,5, + 2Ss,s,88,s,8, 58,88, — 28s,8,Ss,8,8, 55,58, 
— 28s,s,Ss,s,5,Ss,5,8, + 2Ss,s,Ss,s,s,Ss,s,8, — 2Ss,s,Ss,8,5, Ss,s,5, = 0. 
Supposing the origin to lie within the tetrahedron ABCD formed 
by the four planes s,, s,, s,, s,, though it no longer necessarily co- 
incides with the centroid, and adopting the notation of the pre- 
ceding article, we have 
T?+77+T +7) =2T,T, cos(A, B)+2T7,T, cos(A, C) 
+ 27 T, cos(A, D) + 2T,T, cos(B, C) 
+ 27,7, cos(B, D) + 27,7, cos(C, D). 
19. If we square the equations 
V.Vs,s,Vs,8, = 8,Ss,8,8, — $,S8,8,5, = 8 yS8,8,8, — 8S5,8,8,, 
we obtain 
—(TV. Vs,s, Vs,s,)" = 8,’ (Ss,s,8,) + 8," (Ss,8,8,) — 2Ss,s, Ss,8,8,88,8,8, 
=s, (Ss,s,8,) + 5,7 (Ss,s,8,)” — 2Ss,s,Ss,s,8,98,5,8, - 
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 TVs,s,=sin(C, D)/p,p,. Hence, if (AB, CD) denote the angle 
between AB and CD*, we have 
TV. Vs,s,Vs,s,=sin(A, B) sin(C, D) sin(AB, CD)/p, p, p, p,; 
and our equations become | 
sin’(A, B) sin’(C, D) sin*(AB, CD) = 7? + T,? —2T.T, cos(A, B) 
= 7+ 77 -27,T, cos(C, D). 
20. Consider next the equation 
Ss,s,Ss,s,s, — Ss,s,Ss,s,8, + Ss,s,Ss,s,8, — Ss,s, Ss,s,s, = 0. 
Take the origin within the tetrahedron ABCD formed by the 
planes s,, s,, s,, 8,, and let the transversal plane s, be denoted by 
the symbol (4). Then we have 
T, cos(A, X)+T, cos(B, X)+,.T, cos(C, X) + T, cos(D, X) = 0, 
where, as in all former cases, for the dihedral angle between two 
planes is to be taken that particular dihedral angle within which 
the origin lies. 
* The order of the letters is understood to indicate direction. 
VOE wn, Cetin, 11) 
