OBSERVED BY DR. YOUNG. 581 
_._4 2 B 2, 2 wpe_pot_ 2 t] 
yee Re RH) + TT +5 (T-1)*-S 7 
A 2 2 2 Ms) 
a yA) 3 “ 
h= = A, gdje (T+) + fay TH SPT —5r7} 
_ which give the heights / and fh, (above the level of the external 
_ fluid) of the two centres or points on the axis of the upper sur- 
faces in which the small column of each fluid terminates, in 
_ terms of the tensions TT,I'I',, of the respective densities of 
_ the two fluids A and A, of the radius « of the tube, and of the 
~ volume of the upper fluid expressed by 7 as. 
Now observing that we have 
a dx dx, 
; dt dt 
1) cos a = — — cos a, = — Jo" 
dx dz? 
iL i 
v1 + dt? Vv a dt 
equations (2.) become 
dz dx? 
— =.0 
' dt +1Vy oF ade 
ip G2 . dz 
eR, 4 +75r=%, 
and will hold for ¢ = «, where « is the radius of the tube. 
The signs to be prefixed to the radicals in these equations are the same as 
d 
_ those of the preceding equations. If then we make ¢ = @ and eliminate aH 
dz 
i 
Ped 
and 7d 4 between these four equations, we shall have 
hy 
2g hf ztdt—ea® + eek 10 
ap lags le dla aie agli iil a et SP (3.) 
4 29(A,—A) [2 tdt+ e+ 2a0=0. 
a 
Let h and h, be the vertical ordinates of the centres of the two capillary sur- 
faces, that is to say the respective values of z and zx, when t= 0. At these points 
the two radii of curvature of each surface are equal and of the same sign, If 
then we put e=¢'=¥; o,=¢/=y, when ¢=0, we shall have from equations (1.) 
2T 
MPN on. Ey WARS sts (4.) 
Piet f 
g(A,—A)h, +o = —. 
YI 
gAh—c= 
Now for a first approximation, which will be sufficient, we may suppose that 
the capillary surfaces coincide with their osculating spheres at the points where 
they intersect the axis of their figure, the coordinates of which are respectively 
handh, In this case we shall have 
ssh+y—vV//— ?#, e%=eht+y—-Vy77—@. 
The radicals having respectively the same sign as the values of + and y,, viz. 
positive or negative according as each surface turns its concavity upwards or 
