58 



In (1) if b=a=^l, we have x^(l + z)^+y2(l — z)^={14-z)^(l— z)^, a surface 

 whose sections by planes perpendicular to the z axis give us between z=0 and 

 z=l ellipses of all possible eccentricities. A similar remark may be made of 

 equation (2). 



Among the deformations of which surface (1) is susceptible, one is worthy of 

 special attention. If the threads representing the elements be weighted below 

 the lower straight line directrix, and the upper directrix be then revolved until 

 ii comes into the plane of the lower directrix and the common perpendicular, the 

 surface will gradually close up until it becomes a plane, but in every position the 

 form of the Cartesian equation remains the same, while the axes of reference will 

 be the equi-conjugate diameters of the ellipse cut out by the x y plane. 



The Gravitational Attraction of a Hojmogeneou.^ Ellipsoid of 

 Revolution. 



[Abstract.] 



In this paper the following problem was discussed: Given an ellipsoid of 

 I'evolution of given mass, but of variable eccentricity; find how its attraction on 

 a particle at the end of the axis of revolution varies as the ellipsoid alters con- 

 tinuously from the infinitely prolate to the infinitely oblate form. 



It was pointed out that this was the only case in which the expression for the 

 attraction of a spheroid did not lead to elliptic integrals. An expression for the 

 attraction in the above case was found by direct integration without recourse to 

 the potential function. The integral took two forms according as the ellipsoid 

 was prolate or oblate. The ordinary process of finding the value of the eccen- 

 tricity corresponding to a maximum led to an insoluble equation. Hence the 

 position of the maximum was approximated to by trial and interpolation. The 

 conclusion was, that starting with the infinitely prolate form and passing through 

 the spherical stage to the infinitely oblate form the attraction increased continu- 

 ously, until that oblate stage was reached at which the axis of revolution was 

 seventy-two hundredths of the equatorial axis, then it decreased until when the 

 axis of revolution was fifty-one hundreths of the equatorial diameter, the attrac- 

 tion had fallen again to that at the spherical stage, from whence on it decreased to 

 zero. 



It was pointed out that this invalidates the common argument that the 

 weight of a body at one of the earth's poles must be increased by the polar flat- 

 tening. 



