SOLIDS u\ -i BE U'ES. I j 5 



it has a centre, or the summit or a poinl in the axis when it is 

 without a centre, is at the same time its centre of gravity. 

 Let the equations of the two surfaces be respectively 



V(5+S+.S)-> = «. 



n/( 





where the constants are the semi-axes of the figure. Or, what 

 will be more commodious in the presenl instance, lei these 

 equations be presented in the forms 



^(Ax' + AW + AV) — i = o, 

 Wtf+Stf + C^-i = »■ 



Where the constants arc the reciprocals of the squares of the 

 semi-axes, and . / of course not to be confounded "with the ./ 

 used before. These equations, although apparently only in- 

 tended for ellipsoids, spheroids <>f revolution, and spheres, will 

 answer for all surfaces of the second degree whatever, provided 

 the following changes be made in the results to which the 

 above would had. 



1. For a single-napped hyperboloid, change the sign of the 

 square of the semi-axis of the ellipsoid corresponding t<> the 

 imaginary axis. 



i. For a double-napped hyperboloid, change the signs <ii 

 the squares of the fcwo semi-axes of the ellipsoid which corre- 

 spond I" the two imaginary axes. 



;. I'm- an elliptical paraboloid, diminish, in the results, the 

 coordinates parallel to the figure's axis by the corresponding 

 semi-axis of the ellipsoid j then make all the semi-axes infinite, 

 but so thai the two third proportionals to the firsl mentioned 

 semi-axis and each of the other two, shall remain finite and 



\ OL. III. — i \ 



