1887. ] of the Quaternion Analysis. 163 
1+ cos’(B, C)cos’(A, C) + cos’(C,A) cos’ (B,D) + cos’(A, B) cos’(C,D) 
—2cos (B,C) cos (B,D) cos (C,.D) — 2 cos (C,A) cos (C,.D) cos (A, D) 
— 2 cos (A,B) cos (A,D) cos (B,D) — 2 cos (A, B) cos (B,C) cos (C, A) 
= 2 cos (C,A) cos (A,B) cos (B,D) cos (C,D) 
+2 cos (A,B) cos (B,C) cos (C,D) cos (A, D) 
+2 cos (B,C) cos (C,A) cos (A, D) cos (B,D) 
+ cos’(A,D) + cos’(B,D) + cos’(C,D) 
+ cos’(B,C) + cos’(C,A) + cos’(A, B). 
13. We will next consider the equation 
S.s,(s,—8,) + S.s,(s,—8,) +S. s,(s, —8,) =0. 
Suppose that we have three planes (A), (B), (C). We can 
draw, by means of the construction given in Art. 4, three other 
planes (X), (Y), (Z) such that 
=(B)-(C), (Y)=(C)- (A), 4) =(4)-(); 
and we have 
S.(A)(X) + 8. (B)(Y) +8. (C)(4) = 
Hence if p,, 2,, Ps, 91> GW» 7, be the aie from the origin 
on (A), (B), (d), 0X5; (Y), (Z) respectively, we have 
cos (A, X) Hees (dss J) i cos (C,Z) _ 0, 
Ph Pole Pls 
it being understood that by the dihedral angle between any two of 
the planes that particular angle is intended within which the 
origin lies. 
14. As another example of a similar type we will consider 
the interpretation of the equation 
S. (s, + 8,)(8, + 8,) +8. (s, + 8,)(8, + ,) + S. (8, + 8,)(8, + 8,) 
=s,+8,'+ s,'+ 3 (Ss,s, + Ss,s, + Ss,s,). 
Let there be three given planes (A),(B),(C), and let (X),(Y),(Z) 
be the respective polar planes of the origin with respect to (B) 
and (C’), (C) and (A), (A) and (B). Then the above equation 
becomes 
4S. (Y)(Z) +48. (Z)(X) + 48. (X) (VY) 
= (A)’ + (B)’+ (C) + 3 {S. (B)(C) +8. (C)(A) + 8S. (A)(B)}. 
Hence if p,, P,, Ps: Ui» Io, J, be the respective perpendiculars from 
the origin on (A), (B), G, Ex), ( (Y), (2), we have 
