104 On the Depression of Mercury in the Tube of 
ber of divisions. The extent of the curve, beginning from the 
vertex, is the angle which its termination makes with the hori- 
zon, amounting in this case to 52 decimal degrees: this has 
been divided into twelve equal parts, and the depression of the 
mercury supposed successively equal to 4°5, 4, 3°5, 2, 1-5, and 
1 millimetre: then for each tenth down to °1 and “05. milli- 
metre. The depression being always inversely proportional to 
the radius of curvature at the vertex of the curve, the first radius 
was immediately known. This radius gave the values of the 
absciss and ordinate corresponding to vie first division, consi- 
dering it as the are of a circle, the absciss being always taken 
upon the axis of the curve, beginning from its summit. These 
first values, being substituted in the expression for the radius of 
curvature, gave the second radius, and by means ef this the in- - 
crements of the absciss and ordinate in the second division were 
determined, considering the curve again in this part as a circular 
are described with the second radius of curvature. In this man- 
ner the second values of the absciss and ordinate were obtained, 
by means of which a third radius of curvature was determined. 
Proceeding in the same manner to the last division, the last value 
of the ordinate becomes equal to the semidiameter of the tube 
corresponding to the supposed depression. But for a depression 
below eight-tenths of a millimetre, the radii of curvature near 
the vertex are so large,. that it was necessary to divide the ex- 
tent of the are into smaller portions; the calculation has there~ 
fore been made for every two decimal degrees, as far as 12; and 
for the first six degrees, by the assistance of converging series, 
which I have derived from the differential equation of the sur- 
face of a liquid, when the greatest inclination is inconsiderable, 
The formule and the series which have been employed for 
the first six decimal degrees are these. Let V“ be the inelina= 
tion of the end of the curve at the lower extremity of the rth 
division; let x? and w™ be the absciss and ordinate corre- 
sponding to the same extremity; let b@ also be the radius of 
curvature at the same point, and J at the vertex of the curve ; 
] 
the differential equation of the curve will give —~ = 4 + 
p 
2ax oy os 
1 pe Rtg, b : 
> +Sin V3 @ being a constant coefficient equal to 
uw 
ah sane ; 4 
Ee? when the millimetre is made unity. We shall then have 
rq 1 
aor yO + 2b sin mee T) any Vor) . COS 5 (VOsp + 
yo); zrtyD— PAP aL 21 sin. s (Vor 1) —Vm) sing Vert 1) + 
¥™); and for the first division uw = b,sin VO, nid 2) se 
2b 
