252 THE FIGURES OP EQUILIBRIUM OF A LIQUID MASS 



to a plane surface. It is, therefore, evident that the bases of our liquid cylinder 

 must neces^sarily be convex, as is shown to be the case by experiment ; for as 

 equilibrium requires that the pressures should be the same throughout the 

 whole extent of the figure, these bases must produce a greater pressure than 

 that which corresponds to a plane surface. 



Our plane figure, then, fully satisfies theory; but veMficatiou may be urged 

 still further. 'J'heory allows us to determine with facility the radius of those 

 spheres of which the bases form a part. In fact, if we represent this radius by 

 X, the formula (1) of paragraph 4 will give, for the pressure corresponding to 

 the spheres iu question, 



P+A.-. 



X 



Now, as this pressure must be equal to that correspondhig to the cylindrical 

 surface, we shall have 



PH-|.l = P + A.l, 



2 / X 



from which we may deduce 



x = 2X. 



Thus the radius of the curvature of the spherical segments constituting the 

 bases is equal to the diameter of the cylinder. 



Hence, as we know the diameter, which is the same as that of the solid 

 rings, we may calculate the height of the spherical segments ; and if by any 

 process we afterwards measure this height in the liquid figure, we shall thus 

 have a verification of theory even as regards the numbers. "We shall now 

 investigate this subject. 



40. If we imagine the liquid figure to be intersected by a meridional plane, 

 the section of each of the segments will be an arc belonging to a circle, the 

 radius of which will be equal to 2A, according to what we have already stated, 

 and the versed sine of half this arc will be the height of the segment. If we 

 suppose the metallic filaments forming the rings to be infinitely small, so that 

 each of the segments rests upon the exact circumference of the cylinder, the 

 chord of the above arc will also be equal to 2X ; and if we denote the height of 

 the segments by h, we shall have 



7i = ,l(2-A/3) = 0.268.>l. 



Now, the exact external diameter of my rings, or the value of 2A, correspond- 

 ing with my experiments, was 71.4 millimeters, which gives 7^ = 9.57 millime- 

 ters. But as the metallic wires have a certain thickness, and the segments do 

 not rest upon the external circumference of the rings, it follows that the chord 

 of the meridional arc is a little less than 2A, and that, consequently, the true 

 theoi-etical height of the segments is a little less than that given by the pre- 

 ceding Ibrmuhu To determine it exactly, let us denote the chord by 2c, which 

 will give 



A = 2/1 -a/ 4/2 -c2. 



Now, let us remark that the meridional plane intersects each of the rings in 

 two small circles to Avhich the meridional arc pf the spherical segment is tan- 

 gential, and upon each of wdiich the chord of tliis arc intercepts a small circular 

 segment. The meridional arc being tangential to the sections of the wire, it 

 follows that the above small circular segments arc similar to that of the spheri- 

 cal segment ; and as the chord of the latter differs but very slightly from the 

 radius of the circle to which the are belongs, the chords of the sijiall circular 

 segments may be considered as equal to the radius of the small sections, which 

 radius we shall denote by r. It is moreover evident that the excess of the ex- 

 ternal radius of the ring over half the chord c is nothinj? more than the excess 



