1^22 PHILOSOPHICAL TRANSACTIONS. [aNNOI737. 



Corol. — Instead of the circle kl, if there was a certain ellipsis, or any other 

 curve line, which should differ very little from a circle, by the same arguments 

 as in the note, it is easily gathered that the foregoing proposition would always 

 hold good. 



Theorem I. Let AEae, fig. 14, be an elliptic spheroid, the axis of revolu- 

 tion being Art ; then the attraction this spheroid exerts on a corpuscle at n, is 

 the same as that attraction, which any spheroid exerts whose pole is n, its axis 

 of revolution ntz, and its second axis the radius of a circle, having the same 

 superficies as fg, the elliptic section of the ellipsoid AEae, by a plane erected 

 perpendicularly on fg, its conjugate diameter. 



To -demonstrate this, conceive innumerable elements kl, parallel to the 

 ellipsis FG, that is, all erected on ordinates to the diameter, it is evident that 

 the spheroid AEae will differ from the aforesaid spheroid only in this, that in 

 the first all the elements make with cn an angle differing infinitely little from 

 a right angle, but in the second all the elements make a right angle, without 

 any difference, while in both spheroids the elements have the same superficies. 

 But, by the preceding proposition, the attraction of every element kl on n, is 

 considered as the same in both cases ; but as to the thickness of the elements 

 kA/l, we may take hA for the perpendicular hi, because of the smallness of the 

 angle jAh ; therefore the total attraction of both spheroids may be taken the 

 one instead of the other. 



Prob. II. To find the attraction of the spheroid AEae on a corpuscle at any 



point N. 



Let AC = a, CE = b, cn = r, CG the conjugate diameter to on will be 



- since a and b have very little difference; from the preceding proposition, 



find the attraction of the spheroid, whose greater axis is r, and the less 



.abb 7 ,a 



\/— or by/ J. 



For this, we must apply the formula which we found in prob. 1, viz. ^c — 



tVJC, or ^pr rt^pr, putting pr for c, but in this formula instead of a sub- 



stituting '■-~^'^~ = 1 — ^ /p or \n — 7n, putting a-\-ma for b, and a -j- Tza 

 for r and in the computation neglecting the second powers of n and m. 



If therefore \n — mhe. put instead of a, the aforesaid formula will become 

 ^pr _ ^prn -\- -J^prm, or fjba — -^pan + -^pam ; which expression is the re- 

 quired attraction of the spheroid on n. 



If n = 0, then we have ^pa + -^pam for the attraction on the pole a. 



But if n = m, then we have \pa -f- -hpa^ for the attraction at the equator. 



Theorem II. Let AEae, fig. 12, as before, be a spheroid, whose axes differ 



