158 Mr Brill, A new Geometrical Interpretation [Nov. 28, 
The plane s is what we have called the mean plane, with 
respect to the origin, of the planes s, and s, for multiples m and n 
respectively. If s be the mean of the planes s,, s,, ...... Sa On 
multiples m,, m,, ....+. , m,, respectively, then we have 
(On, FB sebode +m,) $= M,8, +M,8, +... + ,8n. 
If we make m, = ™m,=...... =m,, then the above equation be- 
COMNES 109 = G, 4G AP occbor +s,, and s coincides with the polar plane 
of the origin with respect to the n given planes. Thus, to obtain 
the plane which is the sum of m given planes, we have to draw the 
polar plane of the origin with respect to these planes and then to 
draw a plane parallel to this at one-nth the distance from the 
origin. 
If we have 7) -- 54>... +m, =0 then s will denote some 
plane through the origin, and it will be seen that cases such 
as s,—S,, s, +8, —2s,, &c. will need a separate investigation. We 
will, however, only trouble ourselves with the case s,—s, as all the 
rest may be reduced to this. Thus, if s, be the polar plane of the 
origin with respect to s, and s,, we have s,+s,= 2s, and therefore 
S, +5, — 28, =2 (s,—8,). 
The equation of the plane s, — s, is 
(u, —U,) ©+(v, — 2) y¥ +(w,—w,) z—-1=0. 
This is obviously parallel to 
(uw, —u,) & + (v, — U,) y + (w, — w,) z=, 
the plane which contains the origin and the line of intersection of 
s, and s,. Further, the equation of s,—s, may be written in the 
form uc+yy +w,z—1—(ue+v,y + w,z) = 9, 
which shews that it passes through the intersection of s, with a 
plane drawn through the origin parallel to s,. Thus, to construct 
the plane s,—s,, we draw through the origin a plane parallel to s,, 
and through the intersection of this with s, we draw a plane 
parallel to that which contains the origin and the line of inter- 
section of s, and s,. 
5. Let p be the perpendicular from the origin on the plane 
ua + vy + wz —-1=0, 
and let a, 8, y be the angles that it makes with the axes of co- 
ordinates. Also, let a, b, c be the intercepts of the given plane 
upon the axes. Then 
p=acosa=bcos 8 =c cosy, 
and therefore 
: : ra Pye | 1 aie Lt 4 
dae aaah eR aur ry ces gst meOS Shar VCO 9) 
