THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 277 



surface of the liquid of the vessel. These considerations, it is clear, would 

 apply also to the case in which the ring, while being withdrawn, is oblique or 

 vertical instead of horizontal, and would alike apply to the case where the wire, 

 instead of being circularly curved, should be arranged in any polygon : there 

 would always be formed, by the same causes, a film between it and the- surface 

 of the liquid ; if, then, we plunge one of our frames into soap-water or glyceric 

 liquid, the iron wires which compose it will, in proportion as they afterwards 

 emerge from the liquid, be connected with the latter by hlms, as experiment will 

 evince. 



§ 24. Let us examine, in the second place, the manner in which the films 

 which are to constitute the system arrange themselves while being withdrawn 

 and immediately after withdrawal. "We will begin with the case of a prismatic 

 frame which is withdrawn with the bases held horizontally. When the upper 

 base emerges from the liquid, each of the solid edges of which it is composed 

 will be followed, as just shown, by a film. Now, if the angle comprised between 

 two adjacent lateral faces of the prism is equal or superior to 120° — that is to 

 say, to one of the equal angles which three films terminating at the same liquid 

 edge form between them, the films, which will proceed from all the edges of the 

 base must, as we shall see, remain attached to the vertical solid edges so long 

 as the inferior base shall not have emerged from the liquid. 



Let us take as an example the frame of a regular hexagonal prism, a prism 

 for which the angle of two adjacent lateral faees is 120°, and consider it when 

 it is onl}'- partly without the liquid. Suppose, for an instant, that the films 

 which proceed from the edges of the superior base re-enter towards the interior 

 of the frame, in which case other films will necessarily proceed from the vertical 

 edges to terminate at the liquid edges which will unite the former. All these 

 films will be attached to the liquid of the vessel by small masses raised along 

 their lower borders, (§ 3,) masses which will present, in the direction of their 

 height, strong concave curvatures. Let us, at present, direct our attention to 

 such of these small masses only as furnish the base of the films proceeding from 

 the edges of the base of the frame ; if these masses are curved in the direction 

 of their length, this longitudinal curvature will always, as is readily understood, 

 be very feeble in relation to the above transverse curvatures, so that the influ- 

 ence of these last will much predominate ; it will be necessary, therefore, for the 

 capillary equilibrium of these small masses, that the transverse curvatures of 

 their two surfaces should be sensibly the same, which evidently requires that 

 the films which rest on their crests should terminate there in a direction very 

 nearly vertical. The films in question, those, namely, which proceed from the 

 edges of the base, must consequently be inflected in descending towards the 

 liquid, and thus will be, in the direction of their height, convex towards the in- 

 terior of the figure. But as they will be in contact by their two faces with the 

 open atmosphere they can exert no pressure on the air, and thence it results 

 (5th series, § 12) that their mean curvature will be null, or, in other terms, that at 

 each of their points, the curvatures, in two rectangular directions, will be equal 

 and opposite; therefore, since the films in qiiestion are convex in the direction 

 of their height, they will be concave in the direction of the r breadth. Now, for 

 the two-fold reason of this concavity and of their re-entering direction towards 

 the interior of the frame, our films will necessarily form between thim, two by 

 two, angles superior to those of the faces of the prism, and consequently superior 

 to 120'-', a thing which we know to be impossible; hence, these films must 

 remain, as 1 before stated, adherent to the lateral solid edges so long as the 

 whole frame is not out of the liquid. This deduction is fully confirmed by ex- 

 periment : when the frame of a regular hexagonal prism is withdrawn, in the 

 position indicated, from the glyceric liquid, we simply obtain, until the lower 

 base has emerged, plane films occupying all the lateral faces. 



The thickness of the wires, a thickness which, in my frames, approximates to 



