340 Mr. A. M. "Worthington on Laplace's 
only of their masses and the distance between them, and that 
this function vanishes when the distance becomes sensible, he 
points out that any elementary length dz of the filament at a 
sensible distance below the surface is attracted by the sur- 
rounding liquid equally in all directions, and that to establish 
hydrostatic equilibrium in the canal it is only necessary to 
show that the pressure on the liquid of the canal at V exceeds 
the pressure at by the weight of the column reaching from 
Oto the level of V. He therefore examines the action of the sur- 
rounding liquid on the topmost element of the canal at V, and 
finds that it can be expressed by a certain integral, which he 
calls K, and is of the nature of a pull downwards, producing a 
pressure K on the base of the element dz which is transmitted 
along the canal. At the other extremity of the canal 
there is the same action K downwards due to the liquid 
below the tangent-plane KOI, and at the same time an 
action upwards due to the meniscus NKIM, which latter 
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action he finds can be expressed by y, where H is a second 
integral and b the radius of the curved surface (supposed in 
the first instance spherical). And he points out that this 
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second quantity -r is very much smaller than the quantity K; 
and he argues that the elevation of the liquid in the canal 
Z above the level V is due to the attraction of the meniscus. 
This proceeding is perfectly legitimate. It explains the 
equilibrium of one portion of the fluid, namely the infinitely 
thin filament, considered on the supposition that the surround- 
ing liquid is in equilibrium both with itself and with all por- 
tions of the filament at a depth greater than dz below the 
surface. But it does not account for this latter equilibrium. 
Nevertheless Laplace goes on as if he had now explained the 
elevation of the whole of the liquid in the tube. In doing so 
he tacitly assumes that the argument is equally applicable (or 
at least applicable in kind) to all the filaments of which the 
liquid in the tube may be regarded as composed. It is here 
that his theory is incomplete. 
In considering that the equilibrium of the filament is deter- 
mined by the pressures parallel to its length produced by the 
action of the surrounding liquid on the topmost element at 
each end, Laplace assumes that all elements at a greater depth 
below the surface are in equilibrium with the liquid around 
them. If this assumption is made for one filament, it must be 
made for all. In other words, he really starts with the assump- 
tion that all the liquid at a depth greater than dz below the 
