362 ON THE MOTION OF 



permitted to take any of the lines 0,B, for the axis of z,. For 

 the sake of greater brevity we may call the points B, the bal- 

 lancing points and the lines 0,B, the natural verticals of the 

 body. 



The phenomena of the motions of the body immediately 

 about its state of equilibrium will manifestly depend upon the 

 configuration of the surfaces or areolas as we may term them 

 in the immediate vicinities of the two points B and B , the 

 former denoting any of the points of the sustaining surface 

 with which B t may be in contact when the body is at rest. 

 From the established theory of contacts, it follows that every 

 point, not singular, of any surface whatever may be brought 

 into a contact of the second order with some curve surface of 

 the second degree. Dupin, in particular, has shewn, in his 

 excellent supplement to the Analytical Geometry of Monge, 

 that every plane section of any curve surface parallel to a tan- 

 gent plane and infinitely near to it is a conic section, indicating 

 all the characters of the curvature around the point touched 

 by the tangent plane. It is easy to infer from this, that for 

 all phenomena depending upon the curvature of the areolas 

 al B and B t these points may in all cases be regarded as the 

 summits of paraboloids, elliptical, hyperbolical or intermediate. 

 This proposition, which is fundamental, might be also proved 

 thus. Let x, y, z denote the coordinates of either areola 

 reckoning from B or B l along the tangent plane and normal. 

 The most general equation of the areola will then be 



z = Ax-[-Bxy-^€y\ 



the condition of a tangent plane requiring that z should be of 

 two dimensions in x and y, and the condition that the point is 

 not a singular one excluding fractional and negative exponents. 

 As the direction of the axes x and y in the tangent plane is 

 arbitrary, the term Bxy may be made to disappear, and the 

 equation becomes simply 



r = d'x' + Cy, 



