324 



NOTE III. 



THE whole calculation of Newton is founded on the 

 supposition that the earth is homogeneous. Taking this 

 for granted, it is possible to investigate without any long 

 analysis the form of the earth. We cannot easily prove that 

 the spheroid is the only form of equilibrium, but we shall 

 show that it is at least one of the forms of equilibrium. 

 That therefore if the earth had been originally fluid, it 

 might have assumed a spheroidal form ; or if not originally 

 fluid, yet if created in this form that its several parts 

 would have no tendency to break up. 

 A spheroidal mass of homogeneous fluid revolves round an 



axis with a uniform velocity; to determine if the equi 



librium of the fluid be possible, and if so what is the 



ellipticity of the surface. 



Let , b be the axes of the surface, e and e the eccen 

 tricity and ellipticity of the surface, then by definition 



Let the axis of revolution (b) be taken as the axis of z t and 

 let the centre be the origin. Let w be angular velocity, 

 p the density of the fluid. It is well known that the at 

 tractions of the spheroid on a particle whose co-ordinates 

 are xy z resolved parallel to those axes are respectively 



i I vTT 

 ifr4 VT^ 



