MR. LUBBOCK ON THE PENDULUM. 
205 
The length of the simple pendulum is 
r* , /~i n , k 2 (sin g, 2 — sin s 2 2 ) — 7c7 2 (cos i\ 2 — cos g' 2 2 ) — k" 2 (cos g", 2 — cos £ f ' 2 2 ) 
Lr L/! + (j L/2 i GCj-G Cj 
G C = G" C" cos S' - t sin S' 
= G" C" 1 1 - 2 sin ~ j - t sin S' 
The apparent length of the simple pendulum = C", C" 2 
The true length of the simple pendulum is 
G" C'\ + G" C" 2 - 2 G" C\ sin ^-2 G" C" 2 sin ^ - t sin S', - t sin S' 2 
+ 
k 3 (sin £, 2 — sin e 2 2 ) — 7c' 2 (cos s', 2 — cos s 2 2 ) — 7c" 2 (cos e" 2 — cos e" 2 ) 
G C 2 - G C, 
The angle C", G" C" 2 = 
C," C a " = G" C," + G" C a " - 2Cl "c,“C/ 8 " G " { sin ^ 2 ~ } 2 
The true length of the pendulum is 
C,"C," + 
2 C,G x C 2 G 
| sin ^ 2 ^ | 2 — 2 G Cj | sin ^ | 2 — 2 G C 2 | sin ^ j 
+ 
C, C 2 
— t sin S', — t sin S 2 
k 2 (sin s, 2 — sin g 2 ) 2 — k' 2 (cos g, 2 — cos g' 2 ) 2 — k" 2 (cos e" x 2 — cos g" 2 2 ) 
G C 2 - G C, ~ 
The sign of the quantity t sin S' depends upon which surface of the pendu- 
lum the distance between the axes is measured, and might be got rid of by 
measuring the distance between the knife edges on each of the surfaces which 
are intersected by them, and taking the mean. 
I have endeavoured as far as possible to conform to the notation of M. Poisson 
in the Traite de Mecanique ; but this is rendered difficult, because M. Poisson 
sometimes takes the axis Ox to be vertical, (vol. ii. p. 113,) and sometimes the 
axis O z (as vol. ii. p. 185), and he uses the letters in two different acceptations, 
(vol. ii. pp. 119 & 185.) 
In the notation of the article on the Pendulum in the Supplement to the En- 
cyclopaedia Britannica 
s = X, s' = Y and e" = Z, a = h. 
