276 M. G. Van der Mensbrugghe on the 



OS' 



The laws we have deduced from the principles of statics 

 are in complete accordance with the consequences drawn from 

 the calculus of variations in regard to the present question ; in 

 fact, if the following question be proposed — to find on a given 

 surface a curve closed or passing through two given points on a 

 fixed curve, and such that it contains in itself, or makes with the 

 fixed curve, the greatest possible area, — we arrive at this result, 

 that if be the angle made at any point by the radius of curva- 

 ture p of the desired curve with the normal to the surface, we 

 have always* 



sin 



= constant. 



P 



But if, as in the present case, the radius of curvature is every- 

 where in the tangent plane to the surface, we have clearly 



sin = 1 ; 



whence p = constant, as has been shown above. 



We will now examine how far the theory just laid down is 

 confirmed by experiment. 



We have seen, in the first place, that on a plane film the silk 

 thread is arranged either as an arc of a circle or as a complete 

 circumference, according as its extremities are fixed to two points 

 of the solid skeleton or to each other : in this case, therefore, the 

 law of the constancy of the radius of curvature is satisfied. On 

 the other hand, it is clear a priori that the tension of the thread 

 is the same in each of its points, considering that perfect sym- 

 metry prevails all along the curve. Moreover it is clear that this 

 tension is independent of the length of the part immersed, pro- 

 vided the radius of curvature of the arc remains unchanged ; for 

 the equilibrium existing in any entire circumference will neces- 

 sarily exist in any arc of the latter, if at the extremities of the 

 arc a force be allowed to act equal and opposite to the tension 

 of the circular thread. 



As to the third law, which expresses the equality between the 

 force of traction of the liquid and the ratio of the tension of the 

 thread to its radius of curvature, the following is the mode in 

 which we have been able to verify it : — 



Let us take a large square of iron wire abed (fig. 4), having 

 sides 20 centims. in length, and supported by a fork; attach at 

 any point (m) of the skeleton one of the ends of a silk thread, 

 while a small weight (a wax pellet for instance) is fixed to the 

 other; this being done, and the horizontal liquid film having 

 been constructed, let us moisten a small portion of the thread, 



* Vide Legons de Calpul des Variations, par MM. Lindelof et Moigno. 

 Paris, pp. 292-296 ; or V 'Expose geometrique du Calcul Differentiel et In- 

 tegral, par M. Lamarle, part 3, pp, 567-570. 



