MR. LUBBOCK ON THE PENDULUM. 
203 
are the equations to any straight line (f) in space, the equations to a straight 
line perpendicular to this line, and passing through the origin, are 
$ax-\-by-\-cz=- 0 1 
\y (a y — b x) = (3 (a z — c x) J 
(ay — b x) = (3 (a z 
and the shortest distance from the origin to the given line 
_ / (<3 b — y cf + (3 a c 2 + y 9 a 3 
V fl 3 (a 2 + Z > 2 + c 2 ) 
The equation to a plane passing through the origin and the given line is 
y (ay — b x) — (3 (a z — c a) 
and the equations to the intersection of this plane with the plane (z y) are 
yy = (3 2 
x = 0 
If a' y = V x + (3' 
a! z = c' x + 7 f 
be the equations to any other straight line (§') in space 
aa' + bU + cc* 
COS £ § 
V cP + b 2 + c' V a 12 + V 2 + d 2 
Hence the cosine of the angle formed by the line § and the intersection of 
the plane 
y (ay — b x) = (3 (a z — c x) 
with the plane s y 
(3b + 7 c 
V cd + b 2 + c 2 V /3 2 + y 2 
the sine of the same angle 
= v /(^ 
V s/ a 
y c) 2 + /3 2 c 2 + y 2 a 2 
V a 2 + b 2 + c* */ ( 3 2 + y 2 
These equations being premised ; let C be the point in the axis or knife 
edge, O x , where a perpendicular let fall upon it from the centre of gravity G 
cuts it ; let C' be the point where the plane (z y) cuts the axis O x ; and let C" 
be the point where one of the surfaces of the pendulum, supposed a parallele- 
piped, cuts the same knife edge ; and let G" be the point in this surface where 
a perpendicular let fall from G cuts it. 
If half the thickness of the pendulum be called t 
GC = G" C" sin C C' G - t cos CC'G 
2 d 2 
