204 
MR. LUBBOCK ON THE PENDULUM. 
sine C C' G 2 {0 sin S cos S' — y sin S'} 3 + /3 2 sin 8' 3 + y 2 cos 8 3 cos S ' 2 
F + r 
cos CC'G = 
/3 sin 8 cos S' + y sin S' 
7 W+7 2 ’ 
if G C = a' 
jS = a' sin X, y = a! cos X, X being a small angle 
sin C C' G 2 = { sin X sin S cos S' — cos X sin S'} 2 -f- sin X 2 sin S' 2 + cos S 2 cos S' 2 
cos C C' G = sin X sin S cos S' -j- cos X sin S' 
neglecting sin X sin S and sin X 2 sin S' 
cos CC'G= sin S', sin CC' G = cos S' 
G C = G" C" cos S' — t sin S' 
Let g, g', g" be the angles which the line 0,r makes with the coordinate axes 
G x t , G y t , G z t ; and A, B, C the moments of inertia of the pendulum about 
these axes : then by a well knoAvn theorem, if G C = a 
the length of the simple pendulum 
_ M a 2 + A cos s 2 + B cos s' 2 + C cos s" 2 
Ma cos e, 
cos g = cos S' cos S, cos g f = cos S' sin S, cos e" = sin S' 
S and S' may be considered as the deviations of the knife edge in azimuth 
and altitude. 
C being the point in axis O x where a perpendicular from G cuts it, the 
index at foot indicates the knife edge, the length of the simple pendulum if 
£/ = 0 
r , , A cos Sj® + B cos s 12 + C cos e!\ 2 
~ U Ll H M GCj 
let A — MW 2 , B — MW 2 , C = M W' 2 , if the knife edges (1) and (2) are iso- 
chronous 
^ . F F sin e 2 - F 2 cos e 12 - F 3 cos e" 2 
G Ll + G C, “ 
G C t 
— G C 2 + 
F 
G Co 
lc 3 sin b 2 — k ’ 2 cos e f 2 2 — F 2 cos e " 2 2 
GO 
whence 
GC 
k 2 = G C, X G C 2 + g cJ- G C ~ W sin g / 2 “ U 2 cos 6 *' 2 “ U’ 2 cos g", 2 } 
— {A 2 sin g 2 2 — A-' 2 cos g' 2 2 — A" 2 cos eV} 
