240 On the Oscillation of the Pendulum. 
For the case of oblique vibration (Fig. 2), call mz =z (x being 
an abscissa of the dial circle), and the ordinate zm/=y. | ; 
R”+zsin 
re Gi 
The point of the dial m/ will have a velocity = 
( being the latitude) and the difference of rotary velocities be- 
R” + ssin © ; 
R and the distance 
tween it and C will be v(1 
om/’ gained, in the direction of rotation, by the pendulum, will 
be, vi(1 Bai i) eae 
But the increment of abscissa om’ corresponding to the incre- 
ment of ordinate om” will be determined by the equation of the 
circle, these increments (being very small) being as the differ- 
entials of y and z. : 
ut 
ee \ ean 
Call r (= ] the radius of the dial circle, then 
Py ancl if Gigi cane 
om! : om’: :dx : dy: : V2rx-«? : r—zx; hence PEL BO nse i 
7-2 
R’+zsin \ Vv 2rx—2? R’/—R” 
, as seca \auice r= —— and hence 
1 
£ 
R’42rsin® r—z . V2rr—zr? 
8500 1 ag) ypLPEH 
Hence 
R/_-R” ‘ 
OT ie gS (R'-R’'-esin®)? 2 sin & jes ; 
mm!” =om" bom’ =V272 Re mR BEE or: 
which, being developed, reduces to 
(R’—R”)? 
2/2 
” FS 4; . 
V3e “he hence a 
eS 
an expression the same as obtained for the case of vibration in 
the plane of the meridian, and constant for all positions of the 
plane of vibration. : 
It is then proved that, at each vibration, the plane of vibration 
will shift on the dial through an angle whose arc, on the dial cit- 
cumference, will be equal to the difference between the ares of 
rotation described during the same time by the centre and eX- 
tremity of the meridional diameter of the dial. 
After a complete revelution of the earth, the length of the are 
of the dial passed over, or which measures the angular change of 
plane of vibration will be, evidently, equal to the difference be- 
