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II. the Attractions of an extensive Class of Spheroids . I 3 y 
J. Ivory, A. M. Communicated by Henry Brougham, Esq. 
F. R. S . M. P . 
Read November 14, 1811. 
In this discourse I propose to investigate the attractions of a 
very extensive class of spheroids, of which the general de- 
scription is, that they have their radii expressed by rational 
and integral functions of three rectangular co-ordinates of a 
point in the surface of a sphere. Such spheroids may be cha- 
racterized more precisely in the following manner : conceive 
a sphere of which the radius is unit, and three planes inter- 
secting one another at right angles in the centre ; from any 
point in the surface of the sphere draw three perpendicular 
co-ordinates to the fixed planes, and through the same point 
in the surface likewise draw a right line from the centre, and 
cut off from that line a part equal to any rational and integral 
function of the three co-ordinates : then will the extremity of 
the part so cut off be a point in the surface of a spheroid of 
the kind alluded to ; and all the points in the same surface 
will be determined by making the like construction for every 
point in the surface of the sphere. The term of a rational 
and integral function is not to be strictly confined here to such 
functions only as consist of a finite number of terms ; it may 
