Stability of thin Films of Liquids, 207 



overcome with wonderful sagacity and rare good fortune by M. 

 Lamarle. 



He begins by establishing, more definitely than I had done, the 

 above-mentioned principle of the minimum; and then, starting 

 from this, he discusses the case of films meeting at the same 

 liquid edge. He supposes any number of plane films starting 

 from solid edges and all uniting in one common liquid edge, and 

 imagines the whole cut by a plane perpendicular to this latter. 

 The section consists of straight lines starting respectively from 

 fixed points and all meeting at the same point ; and he shows 

 first, by elementary geometrical considerations, that if the straight 

 lines are three in number, their sum will be a minimum when 

 they make equal angles with each other. If the straight lines 

 are more numerous, he shows, by considerations equally simple, 

 that, in order that their sum may be the smallest possible, the 

 single point of junction must be replaced by several points con- 

 nected together by additional straight lines in such a way that 

 at each of these points there are only three straight lines making 

 equal angles with each other. Lastly, the diminution in the sum 

 of the straight lines beginning with the commencement of these 

 modifications (that is to say, for example, in the case of there 

 being more than three straight lines, as soon as the point of 

 junction divides so as to give rise to new straight lines and 

 points), it follows that the demonstration applies equally to 

 curved lines, for these may always be replaced by their tangents 

 in the immediate neighbourhood of the point of junction. 

 M. Lamarle then shows that all these results extend to the films 

 themselves, all of which, whether plane or curved, are cut by the 

 plane in question ; that is to say, the minimum of the sum of 

 the areas requires that these films should meet three by three 

 under equal angles in each liquid edge. 



My first law — namely, that in every stable system of liquid 

 films more than three films never meet in the same edge, and 

 that these always make equal angles with each other — is thus 

 completely demonstrated, and appears as a consequence of the 

 principle of the minimum. 



M. Lamarle next considers the question of liquid edges meet- 

 ing at the same liquid point. In dealing with it, he supposes a 

 number of plane liquid films all meeting at the same point in the 

 interior of the system, and he investigates the conditions which 

 these films must satisfy in order that they may meet three by 

 three,forming equal angles with each other, conformably with the 

 foregoing law. He considers the point which is common to all the 

 planes as the centre of a sphere, which is therefore cut by them 

 along arcs of great circles : we have thus a number of hollow 

 pyramids whose summits lie in the same point, and whose bases 



