TRANSACTIONS OF SECTION A. 



519 



between them. The object of the paper is to investigate the general relation be- 

 tween these quantities when the mean curvature is a maximum or minimum, if the 

 changes in the form of the film take place subject to the conditions that the 

 diameter and distance of the rings are constant. 



It has been recently shown by Professor Remold and the author that, if these 

 conditions hold, 



(a«E - /3'F + a^Ai cot ^i)8a + a-(F - E + Aj tan (/)i)8/3 = 0, 



where (f)^ is the upper limit of the integrals and A, = \/ 1 - sin^ 6 sin^ (p^. 



Writing this in the form ASa + BS^ = 0, it is proved that the curvature has in 

 general a critical value when A — B = ; so that 



2E -F(l + cos' 6) + 2Ai cot 2^^ = 



is a condition which must be satisfied by 6 and (f)^. 



To find values of ^j corresponding to given values of 6 the equation must be 

 solved by trial ; but it is proved that, if a pair of corresponding values is given when 

 6 lies (say) in the first quadrant, the values of <pi can be at once found whicb 

 correspond to tt - ^, tt + ^, and 2Tr—6. 



The values of cp^ corresponding to 6 and tt — 5 are equal, and, if <^j be the value 

 corresponding to tt + ^ and ^ir - 6, it is given by the equation 



tan (/>! tan (tt — <^j) = sec 6. 



By means of these equations a curve can be drawn, showing the relations between 

 <^i and 6, and thence are found the values otpjY, Xjp and X/Y, where aY, 2X, and 

 2p are the diameter and distance of the rings and the magnitude of the principal 

 diameter. 



If we now conceive the two rings gradually to approach or recede from each 

 other, and the principal diameter to be altered so that the condition of critical 

 curvature is always fulfilled, it is proved that the changes in its form would be as 

 follows : — 



Beginning with the cylinder the distance of the rings is (as has been shown 

 by Maxwell, Art. Capillarity, ' Enc. Brit.') half their circumference. As the 

 diameter increases the rings move apart, and the distance between them is a 

 maximum when 6 = 64-2°, being 17 per cent, greater than in the case of the cylinder. 

 When 6 = 90°, the figure is a sphere, and the distance between the rings is about 

 4 per cent, less than in the case of the cylinder. The sphere has a larger diameter 

 than any other figure of critical curvature. The surface next becomes a nodoid, 

 and the distance between the rings diminishes till when 6 = 180° they touch, and 

 thus the surface reduces to a circle. In the next quadrant the rings separate, but 

 the figure is now dice-box-shaped, and the pressure exerted by the film is out- 

 wards. When 6 = 270°, the figure is the cateuoid. The principal ordinate is then 

 less than that of any other figure of critical curvature, and the radius of the rings 

 is a mean proportion between this minimum ordinate and the maximum which 

 was attained in the case of the sphere. The same relation holds between the 

 principal ordinates of any two figures which correspond to values of which differ 

 by 180°. In the fourth quadrant the figure becomes an unduloid, the pressure is 

 inwards, the rings continue to separate, and the ratio of the distance between the 

 rings to the principal ordinate is a maximum when 6 = 298°. 



In the paper tables and curves are given to illustrate the ' march ' of these 

 functions. 



To secure continuity the problem is discussed without reference to the question 

 as to whether the surfaces are in stable equilibrium. 



In conclusion it is shown that by means of the curves we can solve a number of 

 problems with sufficient accuracy for practical purposes. Thus, if any two of the 

 three quantities, the diameter of the rings, the distance between them, and the 

 diameter of the surface of critical curvature, are given, the third can be found. 



4. A Mercurial Air-pump. By J. T. Bottomlet, M.A., F.B.8.E. 

 The primary object of the pump described in this paper is the removal of the 



