038 PROFESSORS A. W. REINOLD AXD A. W. RUCKER 
there has only been one maximum or minimum value of y in the part of the curves 
under consideration. 
Again, when if < —a/3, which can only be the case when /3 is negative, the curve is 
convex towards the axis of revolution, and the pressure is exerted outwards. This is 
also a case we have not practically realized, so that it will be convenient to take the 
square root as affected with the negative sign. 
If then, with Beer, we assume 
y~ =./3~ sin 3 </) -f a 2 cos 2 </>.(4) 
k~= (a 2 —/3 2 )/a 2 and A = \/l — k~ sin 2 </>, 
we get from the differential equation 
.r=aE+/3F, 
where F and E are elliptic integrals of the first and second kinds respectively. 
As the lower limit of the integrals is zero, the origin is assumed to lie on a 
maximum ordinate, for y—dn.a when (j>= 0. 
When it is convenient to transfer the origin to a minimum ordinate, (/> is >777 2 for 
positive values of x ; and, taking this as the upper limit, we may write 
x=a(E—E 1 )+/3(F—F 1 ), 
where F : and E : are the complete integrals taken between the limits 77/2 and 0. 
Again, if Y is the volume included between the plane through the origin 
perpendicular to the axis of x and any other parallel plane, 
V/»=(^+/3a= + h/3=)E-~a^F + fa(a S -^) A sin 2<f> .(5) 
This expression has been given by Beer. If the origin lies on a minimum ordinate 
we must, as before, write E —E : and F —F : for E and F respectively, remembering 
that </> is >7r/2. 
Let us suppose that two rings of radius Y are arranged with their planes 
perpendicular to the straight line on which their centres lie, the distance between 
them being 2X. 
Let the rings be joined by a liquid film, the maximum or minimum ordinate of 
which lies half-way between them. It is convenient to speak of this as the principal 
ordinate. Let </> 1 be the value of </> which corresponds to Y, and A L the value of A 
when </> =:</q. 
If then the form of the film be altered, but so that it remains a surface of revolution 
of constant curvature, the quantities a, /3, and must vary subject to the conditions 
that X and Y are constant. 
Hence, differentiating the expressions for X and Y given above, we have 
