which are neither Spherical nor Cylindrical. 27 



constant on passing from one point to another unless M and N 

 either increase or decrease simultaneously. But if we were to 

 suppose that, xipon the small arc which forms the production of 

 the curve, the curvature increases more, or diminishes less 

 rapidly than upon the other small arc, a little attention will show 

 that at the extremity of the first the radius of curvature would 

 be smaller, whilst the normal would be larger than at the extre- 

 mity of the second ; and on the other hand, if we were to sup- 

 pose that beyond the minimum point the curvature increases 

 less, or diminishes more rapidly than on this side, we should 

 find that at the extremity of the first small arc the radius of cur- 

 vature would be greater, and the normal less than at the extre- 

 mity of the second. 



1 1 

 Thus the constancy of the quantity tt + iy necessitates the 



identity, or rather the perfect symmetry of the two small arcs ; 

 but this symmetry involves, as a consequence, that of the two 

 whole portions of the curve situated on opposite sides of the 

 minimum point. By repeated applications of the same reasoning 

 to both extremities of the realized liquid figure, I conclude that 

 the same reproduces itself indefinitely on both sides. The com- 

 plete meridian curve, therefore, undulates indefinitely alongside 

 the axis, from which it recedes and towards which it approaches 

 periodically ; its maxima and minima distances from the same 

 being all equal (fig. 1), so that the generated figure consists of 



Fig. 1. 



an indefinite series of equal ventral segments and nodes. For 

 the sake of brevity, and in consequence of the form of its meri- 

 dian curve, I call this figure the unduloid (ondulo'ide) . 



^Vhen a greater diameter is given to the solid cylinder in the 

 above experiment, the mass of oil remaining the same, this mass 

 spreads itself out more over the solid surface, the meridian cur- 

 vatures become feebler, and that the more, the lai'ger the cylin- 

 der. From this I conclude that, within certain limits, the undu- 

 loid may change its form, and that one of these limits is the 

 cylindrical form. 



On the contrary, with thinner solid cylinders and the same 

 mass of oil, variations of an o])posite kind manifest themselves ; 

 that is to say, the mass spreads itself out less in the direction of 

 the axis, and the meridian curvatures increase. In this manner 



