of the Potential Energy of Liquid Surfaces. 435 



sheet is thin, even, and transparent in its central part, and 

 presents towards its contour the aspect of an annular zone 

 or aureola covered with numerous circular striae connected 

 by radial ones. In proportion as the charge is lessened the 

 diameter of the sheet increases, the aureola becomes more 

 transparent, narrower, and entirely disappears when the pres- 

 sure at the orifice does not exceed 60 centims. The sheet then 

 attains its maximum diameter (80 centims.), and affects the 

 form of a wide cap, the concavity of which is turned down- 

 wards. 



To the whole of these phenomena the propositions enun- 

 ciated at no. 4 are applicable. 



In proportion as the charge decreases, the sheet gradually 

 becomes less in diameter, and at the same time curves back 

 upon itself at its lower part, going toward the stem that sus- 

 tains the disk ; at a pressure of about 32 centims. the sheet 

 closes up entirely, assuming the form of a solid of revolution 

 with a perfectly even surface. 



The formation of the closed figure is due, as has already 

 been pointed out by M. Plateau, to the effect of capillary 

 pressures in the lower portion of the sheet. 



6. We now come to the truly singular transformations ob- 

 served by Savart. 



Directly after the closing of the sheet its dimensions dimi- 

 nish, at first gradually, simultaneously with the charge .; when 

 this no longer exceeds 10 centims. the shape of the sheet 

 abruptly changes : the upper part suddenly becomes concave, 

 rising above the plane of the disk ; then, after an extremely 

 short time, the former shape reappears ; and these instanta- 

 neous changes of aspect are periodically repeated seven or 

 eight times, until the sheet entirely vanishes. 



Savart, who most carefully studied these abrupt changes, 

 vainly endeavoured to penetrate the cause of them. Since 

 then M. Boussinesq has essayed to give the mathematical 

 theory of the formation of the even and closed sheets ; but, 

 like M. Hagen, he regards the capillary constant as remaining 

 the same in every part of the surface. His calculations, too, 

 are not in accordance with experiment, and do not exhibit the 

 cause of the instability of the sheets under certain conditions. 



The following are the propositions to which my theory has 

 conducted me, and which, for the most part, I have verified 

 by direct observation: — 



(a) To every quantity of energy of motion destroyed cor- 

 responds necessarily, as in the case of the plane sheets, an 

 equivalent amount of potential energy, the seat of which is 

 the whole of both faces of the upper portion of the sheet, 



