378 



i \PILLABY ACTION. 



be completely defined in terms of the positions of its principal normal sections 

 and their radii of curvature. 



Before going farther we may deduce from equation 9 the nature of all 

 the figures of revolution which a liquid film can assume. Let us first deter- 

 ^fc^ ^|j e nature of a curve, such that if it is rolled on the axis its origin 

 will trace out the meridian section of the bubble. Since at any instant the 

 rolling curve is rotating about the point of contact with the axis, the line 

 drawn from this point of contact to the tracing point must be normal to the 

 direction of motion of the tracing point. Hence if N is the point of contact, 

 NP must be normal to the traced curve. Also, since the axis is a tangent 

 to the rolling curve, the ordinate PR is the perpendicular from the tracing 

 point P on the tangent. Hence the relation between the radius vector and 

 the perpendicular on the tangent of the rolling curve must be identical with 

 the relation between the normal PN and the ordinate PR of the traced curve. 

 If we write r for PN, then y = r cos a, and equation 9 becomes 



pr 



This relation between y and r is identical with the relation between the per- 

 pendicular from the focus of a conic section on the tangent at a given point 

 and the focal distance of that point, provided the transverse and conjugate axes 

 <>f the conic are 2a and 2b respectively, where 



T F 



a = , and 6 = . 

 p Trp 



Hence the meridian section of the film may be traced by the focus of such 

 a conic, if the conic is made to roll on the axis. 



ON THE DIFFERENT FORMS OF THE MERIDIAN LINE. 



(1) When the conic is an ellipse the meridian line is in the form of a 

 series of waves, and the film itself has a series of alternate swellings and con- 

 tractions as represented in figs. 8 and 9. This form of the film is called the 

 unduloid. 



