162 Mr Brill, A new Geometrical Interpretation [Nov. 28, 
also 7'V (s,s,)=sin C/p,p, and 7'V(s,s,)= sin A/p,p,, and conse- 
quently S.Vs,s,Vs,s,=— sin C'sin A cos b/p,p,"p,. Thus we have 
sin C'sin A cos b = cos B+ cos C cos A. 
Comparing this with the formula that can be deduced from the 
same analysis by means of Sir W. Hamilton’s method of interpre- 
tation*, we see that each can be deduced from the other by means 
of the theory of the polar triangle. 
11. We have the equation} 
V.s,Vs,s, = 8,Ss,8, — 8,8s,8, 
and squaring this we obtain 
—(TV.s,Vs,s,) = s,'(Ss,s,)" + 5, (Ss,8,) — 28s,8,8s,s,8s,5,. 
Now the perpendicular from the origin on V.s,s, is parallel to 
PA, and TV(s,s,) = sin A/p,p,; therefore 
TV.s,Vs,s, = sin A sin 6/p,p, D5; 
where @ is the angle between PA and the normal to s,._ Now, if 
p, be the are of a great circle drawn through A perpendicular to 
the side BC of the spherical triangle ABC, we have sin 6 = cos p,. 
Thus we deduce the formula 
sin’.A . cos’p,= 2 cos A cos B cos C+ cos*C'+ cos*B. 
This formula may be written in the form 
sin’ A.sin’p, = 1 —cos"A — cos’ B—cos’C — 2 cos A cos B cos C. 
12. Sir W. Hamilton gives the following equation 
8,885 8. at (Ss,s,Ss,8,)° = (Ss,s,Ss,8,)” ar (Ss,s,Ss,s,)° 
+ 2s,*Ss,s,Ss,sSs,s, + 2s,°Ss,s,Ss,s,5s,8, + 2s,'Ss,s,Ss,5,8s,s, 
+ 25,'Ss,s,Ss,8,Ss,8, = 2S8s,s,Ss,s,Ss,8,Ss,8, + 28s,s,Ss,8,55,5,58,S, 
+ 28s,5,9s,s,Ss,s,Ss,s, + 8,8, (Ss,s,)” + s,s," (Ss,s,)° + 8,78,” (Ss,s,)? 
we sys. (Ss,s,)° oF 8,'3, (Ss,s,)° ae Sy 8, (Ss,s,)°. 
Suppose that the four planes s,, s,, s,, s, form a tetrahedron 
ABCD, enclosing the origin, A, B, C, D being the vertices respec- 
tively opposite to s,, s,,5,,8,. We shall represent the dihedral 
angle contained by the faces respectively opposite A and B by the 
symbol (A, B), and similarly for the other dihedral angles, it being 
understood that the internal dihedral angles of the tetrahedron 
are meant. Then the above formula becomes 
* Tait’s Quaternions, p. 56. 
+ Hamilton’s Elements, p. 316; Tait’s Quaternions, p. 44. 
+ Hamilton’s Elements, p. 347. 
