262 WITHDEAWN FEOM THE ACTION OF GEAVITY. 



tion of the conditions of equilibrium of the small mass of junction has led to 

 this consequence, that the two films and the partition must intersect one an- 

 other under angles of exactly or very near 120° ; and it is evident that this ne- 

 cessity of intersecting each other under angles of 120° may equally serve to 

 determine the radius of the partition. Now, no relation between the two prin- 

 ciples which serve as the bases of these two determinations is to be seen a priori, 

 and it may be asked whether the two results coincide ; this I propose to examine. 



In order to avoid the complications which would arise from the small irregu- 

 larities mentioned in § 3, I will suppose two films forming originally two com- 

 plete spheres, spheres which have afterwards partially penetrated each other so 

 as to give rise to a partition, and shall imagine this whole system intersected 

 by a plane passing by the centre of the two films : it is clear that the centre of 

 the sphere to which the partition pertains will be found on the right line which 

 contains the two above centres. 



This being premised, it is plain that if the angles under which the two films 

 and the partition meet are of 120°, the radii of the two films brought to a point 

 of the line of intersection of the latter will form between them an angle of 60°, 

 and it will.be readily seen that the radius of the partition brought to the same 

 point will also form an angle of 60° with that one of the two others to which 

 it is nearest. Let^ i^^^- H) be one of 

 the two points at v/hich terminate the 

 three arcs, along which the two films and 

 the partition are cut by the plane in ques- 

 tion, a plane which Ave shall take for that 

 of the figure, and let pc=i phe the radius 

 of the larger film. Draw the indefinite 

 lines 2-'^ ^^^ P^ i'^ such manner that 

 the angles cjim and ?npn shall be each of 

 60°. On pm let us take pc' equal to p' — that is to say, to the radius of 

 the smallest film; let us join cc' and prolong the right line till it meets, at 

 d, with pn. The three points, c, c', and d, will evidently be the three centres, 

 and-pd will be the radius r of the partition, so that if from these three centres 

 and with these radii we trace three portions of circumferences, terminating 

 on the one hand at the point p, and on the other at its symmetrical q, we 

 shall have, as the figure shows, and still on the hypothesis of angles of 

 120°, the section of the system of the two films and of the partition. Let 

 u§ seek now to determine the radius of this partition in a function of the 

 two others. For this, take pf z=: pc' and join c'J", the angle cpc' being 

 60°, the triangle Jpc' will be equilateral, and we shall consequently have 

 /6-' =:pc' --=: p' ; for the same reason, the emglefc^p will be 60°, like the angle c'pd, 

 whence it follows that the right linesyb' and pd will be parallel ; we may 



therefore assume -— = — r ; by then substituting, in this formula, ioxpd,fd, and 



pc their respective values r, p' and jo, and observing that^c is equal to p — p', 



pp^ 



we shall deduce r = ; being identically the value given by the first 



P — P 

 method, (§ 7;) thus two laws, apparently independent, conduct to the same 

 result. 



It does not follow, however, that the three angles are strictly equal, for what 

 we have just demonstrated has no necessary converse ; in other words, the 

 radius of the partition may have the above value in the case of angles differing 

 from one another ; but since these angles, if not identical, must approach ex- 

 ceedingly near to identity, as I have shown, and since, on the other hand, their 

 rigorous equality implies the theoretic value of the radius of the partition, it 

 must be regarded as highly probable that this perfect equality exists. 



