24! jM, J. Plateau un the Equilibrium-jigures of Revolution 



of a property discovered by M. Delaunay, by means of the cal- 

 culus, and afterwards demonstrated geometrically by M. Lamarle. 

 In another series I propose to speak of these analytical and geo- 

 metrical resources ; in the present one I arrive at a knowledge 

 of the forms of the meridian curves, and of all their modifica- 

 tions and details, by means of experiment, and with the aid of 

 simple reasoning applied to the relation between the radius of 

 curvature and the normal which the equation of equilibrium 

 establishes. This research, in which experiment and theory 

 always march side by side, may, moreover, be considered as a 

 verification of the latter. 



I commence by demonstrating the following principle: amongst 

 all the equilibrium-figures of revolution, the sphere is the only one 

 whose meridian curve meets the axis ; amongst the spheres is 

 included, of course, the plane which is their limit. 



To prove this I remark, in the first place, that the meridian 

 curve of a surface satisfying the above general equation cannot 

 attain the axis except in a direction perpendicular to the same ; 

 for if it cut the axis obliquely or touched it, the normal N would 

 be zero at the point of intersection or contact, and consequently 



the quantity iy + ny' which equilibrium requires to be constant, 



would become infinite at this point, whilst it would be finite at 

 neighbouring points. 



I assume then a non-circular curve meeting the axis perpen- 

 dicularly; I conceive its curvature to increase on leaving the 

 point on the axis; and I take a small arc of the curve, with one 

 extremity on the axis, such that the curvature increases through- 

 out its length ; lastly, through the extremities of this arc, I draw 

 a circle having its centre on the axis. By reflecting upon this 

 construction the following conclusions will easily be deduced : — 

 First. Since the arc of the curve and the arc of the circle both 

 leave the axis in the same direction and, after immediately sepa- 

 rating, rejoin each other at their other extremity, it follows that 

 the curvature of the first is at the commencement less, but after- 

 wards greater than that of the circular arc ; at the point where 

 the two arcs rejoin each other, then, the radius of curvature M 

 of the are of the curve is less than the radius of the cireu.lar arc; 

 the same is the case, too, with the normal N, for the latter evi- 

 dently cuts the axis at a less acute angle than does the radius. 

 Secondly. If upon the arc of the curve and upon the circular arc 

 two equally long portions be taken, having a common extremity 

 on the axis, and so small that at the other extremity of the first 

 the curvature has not yet ceased to be less than that of the cir- 

 cular arc, the normal N, corresponding to this extremity, will 

 evidently be longer than the radius of the circle, and, in con- 



