1887.] of the Quaternion Analysis. 161 
9. An interesting property of the mean plane follows from 
the equation 
T?(ms + nt) =m’. T’s + n?. T’t — 2mnSst. 
Let (X) be the mean plane, with respect to O, of the given 
planes (A) and (B) for multiples m and n respectively; and let 
OL, OM, ON be the perpendiculars from O on (X), (A), (B) 
respectively. Then we have 
2 
(m+ny mm fo Oe se 
OFM TOM TONEY SOMAOM 
where (A, B) is that dihedral angle between the two given planes 
within which O lies. 
It will be easily seen that this theorem admits of extension. 
In fact, if (X) be the mean plane, with respect to O, of the n 
qlames/(CAn), (AG), 3... ,(A,) for multiples m,, m,,...... ,M,, Tespec- 
tively; and if OL, OM,, OM,,...... , OM, be the perpendiculars from 
Orannexe), (Ap), (A) ee , (A,) respectively, then we have 
(m, + Mm, + vee sO : 9S 
cos (A, B), 
Oy mm, 
om? OM. 0M, Au 4s). 
Another interesting property of the mean plane, closely con- 
nected with this, follows from the equation 
T? (M8, + MS. + eevee + m,8,) + m8 .s,(m,s, + m,8, + ....-. +™,S,) 
+ mS . 8, (m8, + M8, + ...+- +m,S,) + &e. = 0. 
rom! 2 -draw LN. IN, ... , LN, respectively perpendicular 
to OM,, OM,, ....-. ,OM,. Then we have, by aid of the above 
equation, 
ape Oe +m ON) +m, + +m 
OTE te On aia Pes ee ane a 
or as it may be written 
m a +m M,N, ae m MN, 2/0) 
OM OMe » OM, 
10. Suppose that we have three planes s,, s,, s,, and let PA, 
PB, PC be the respective intersections of s, and s,, of s, and s,, 
and of s, and s,. Draw a sphere of unit radius, having its centre 
at P, and let it cut PA, PB, PC at A, B, C respectively. We 
have the theorem 
i a 
S.Vs,s,Vs,s, = s,'Ss,s, — Ss,s,8s,8,, 
and from this we will deduce a theorem concerning the spherical 
triangle ABC. Now the perpendiculars from the origin on V. s,s, 
and V.s,s, are respectively parallel to PC and PA, and we have 
