420 On the rolling Pendulum. 
the two cylinders will take place when they are parallel to two 
axes possessed of equal momenta of inertia, and when their di- 
stances from the centre of gravity satisfy the equation (1). The 
sum of these distances diminished by twice the radius of the ev- 
linder will then, by equation (2), give the length of the simple 
pendulum that oscillates in the same time. 
Of all the infinite number of cases in which the same body 
may oscillate in the same time upon two equal cylinders, there 
is one, and one only, in which the length of the simple pendu- 
lum of isochronous vibration, is equal to the distance between 
the surfaces of the cylinders; and that is when the axes of the 
cylinders are parallel, and situated in the same plane with the 
centre of gravity. We may add that the equation (2) does not 
take place, unless the equation (1) be satisfied ; which excludes 
an infinite number of cases of oscillation in equal times, when 
the cylinders are placed at equal distances from the centre of 
gravity, parallel to one another, or to axes that have equal mo- 
menta of inertia. 
It is an indispensable in all that has been said, that the cylin- 
ders roll without sliding: the value of the ordinate y involves 
this condition. leurty: 
Postscript.—Having, in the last Magazine, investigated the 
expression of the refraction in Cassini’s hypothesis, it may be 
worth while to deduce from it a formula for the case when a 
star is not very near the horizon, for the sake of a comparison 
with the similar formula used in the construction of the French 
Tables. For this purpose nothing more is necessary than to ex- 
pand the terms in the value of 7, p. 344, retaining only the 
quantities multiplied by 6, 28, 8?: thus we get 
sin A -, sinA sin3A 
i, Pasa = 08 cos3A + 28? cossA? 
SA 8 | 
2cos 2A 2 sr 
The French formula, Méc. Céleste, vol. iv. p. 268, is this 
2i—B 
r=ftmA }1—-——— +8}; 
which exceeds the first value of r by the quantity % 6? tan A. 
Now 6 =:0002938 ; and hence 
$ 6? tan A= 0"-027 tan A. 
At 70° from the zenith the difference will therefore be 0"*074 ; 
and at 80°, which is beyond the proper limit of the formule, it 
will amount to 0154. We are therefore warranted in saying 
that, when the same elementary quantities are used, there is, 
practically speaking, no difference in point of St ae 
the 
or, r= BtanA}1— 
