SOLIDS ON SUHFACI'.s. 363 



,t paraboloid, elliptical, hyperbolical or intermediate, according 

 as C is positive, negative or nought, the constants .1 and (' 

 representing the reciprocals of the greatest and l< ast diameters 

 of curvature. In a similar manner it might be shown that 

 every areola whatever may be represented bj the areola around 

 the summil of some assignable hyperboloid with an arbitrary 

 vertical axis. — elliptical when the areola is coMcarwa/e, that is 

 with the curvatures of all its normal sections directed the sam< 

 way. — hyperbolical when discurvate, <>r with the curvatures 

 of its normal sections directed some one way and some the 

 opposite, — cylindrical when the curvature of the areola is in- 

 termediate as in the case of developable surfaces. 



It follows therefore, from what precedes, that in the prob- 

 lem of the small oscillations of supported bodies, the equations 

 (31) obtained above for surfaces of the second degree, with 

 the positions there proposed, will answer for all possible areo- 

 la- of contact, the arbitrary values of the axes a and y, ena- 

 bling us to avail ourselves completely of this simplification by 

 placing the centre of the osculating figure in the centre of 

 gravity of the body, at the same time thai we may take an\ 

 point at pleasure in the vertical through B for the origin of 

 the invariable axes. 



The hypothesis that, during the motion of the body, its na- 

 tural vertical declines hut very little from the position which it 

 would occupy if at rest, is equivalent to supposing that c and c 

 are at all times very small, and we shall regard them therefon 

 in tin' following calculations as infinitesimals of the fust order. 

 The hypothesis that the two areolas of contact are indefinitely 



small is analytically expressed by considering '• and ./ . ./ and // 



as quantities infinitely small. Tin preceding formulas will 

 n i\\ enable n> to ascertain what values the rest of the denoted 

 quantities acquire in consequence of these two hypotheses, 

 and the conditions of their legitimacy will appear in the equa- 

 tions of condition which arise in tin- course of the solution 

 of the problem. The fundamental relations (4) give us in 

 tie first place, neglecting all infinitesimals of higher orders 



than the hist, c =1. h = — 11. b = it. Tie values of 

 \ 01.. III. — J / 



