412 Mr.J. J. Waterston on a Law of Liquid Expansion 
angular velocities and two measurements of diameter; hence the pro- 
portionate differential of the diameter and the correction to be applied 
to the angular velocities to reduce them to the same radius. We 
might thus obtain the velocity, the increment of velocity, and incre- 
ment of distance expressed in terms of r, the radial distance of the 
moon from the earth at the middle epoch. Now since ar 2Qvdy, if 
x 
we had these same quantities for different values of p or 7, and pro- 
jected the, different values of a as ordinates tothe correspon- 
vdv 
ding values of r, the points would converge in a straight line to the 
zero of r; and if an approximate parallax was obtained, the point 
corresponding to the value of 2vdv at the earth’s surface would fit in 
and confirm the propricty of the projection. If it is a question what 
should direct us to this particular projection, it might be answered 
the increment of square velocity is a square quantity, and the inverse 
form of function is applicable to a power depending on distance. 
Note C. § 3.—The longest series of observations on the expansion 
ofa liquid that I have met with is that of M. Muncke, on sulphuric 
acid from —30° to +230°C. Ihave been enabled to put them to the 
test by the following equation for the tension of its vapour, viz. 
Sa) = p (English measures). In this the value of g=354°7 
is assumed to be the same as that for steam, and for the vapours of 
several hydrates of sulphuric acid observed by M. Regnault, and 
referred to in § 1. of paper in the Philosophical Magazine for March 
1858. ‘The value of 4(=1288) is derived from the boiling-point. 
seal aeunh ea = is thus 1°33. The inverse volumes being com- 
p hk 
puted to this power, and laid off as ordinates to the temperatures, 
were found to range well in a straight line above 30°. The line 
drawn through 45° and 220° is expressed by the equation 
(1433°-2—2)v5= 14361 C. A. 
The differences between the temperatures computed from the volumes 
by this equation and those observed are laid off in fig.1 (PI. VI.) as 
ordinates to the temperatures, the scale vertical to horizontal being 
10 to l. 
The law of continuity is evidently broken at about 40°, the de- 
flection being similar to the other cases referred to in §§ 14, 15, and 
probably due to the same cause. 
Fig. 7 (Plate VI.) represents the differences of Muncke’s observa- 
tions on petroleum projected in the same way. ‘The equation is 
(489°°5 —2) v*14=4895 C., 
in which Liia.ge has been deduced from Ure’s observations on the 
Pp 
