VOL. XL.] » PHILOSOPHICAL TRANSACTIONS '2\Q 



, _JpgJg , gggg ^^ Q 



2+P + 9 X 3 + 9 X + 7 X 2 +;» \ + q y. 2 + q X 3 + q y. 5+ q ' 



where we have put e = 1 , for the greater simplicity of calculation. 

 Determination of such spheroids, as make the principle of the equilibrium of 



the columns, and that of gravity perpendicular to the surface, to coincide with 



each other. 



32. This equation informs us, that when out of all the infinite varieties, 

 which will be supplied by the equation of the densities D ^frt + gr^ + hr", 

 &c. we shall have taken at pleasure all the coefficients, and all the exponents, 

 one only excepted; if this last be such in respect of the others, that it may 

 fulfil the conditions of the foregoing equation, the spheroid, being supposed 

 in a state of fluidity, will be in equilibrio, because it will unite as well the prin- 

 ciple of a perpendicular tendency to the surface, as that of an equipoise of the 

 several columns. 



33. Before concluding this paper, I shall make a few reflections on the prin- 

 ciples we have now made use of, for determining the figure of a spheroid 

 revolving about its axe. 



The first principle which, after Mr. Huygens, we have had recourse to, and 

 which consists in making bodies gravitate perpendicularly to the surface, seems 

 to me of absolute necessity. For if there were never so little water on the 

 surface of the earth, it could not be at rest, if it had a tendency any how in- 

 clined to the surface. 



The second principle, made use of by Sir Isaac Newton, and which consists 

 in an equilibrium of the columns ce, cn, cf, could be thought necessary I 

 think, only for these two reasons ; the first is that which is usually assigned, 

 that at the first formation of the earth, it was probably in a state of perfect 

 fluidity; in which case it must acquire such a figure, as will result from the 

 equilibrium of the columns, and from the gravitation acting perpendicularly to 

 the surface. Indeed though this reason has a degree of plausibility, yet there 

 are many who think it to be of small force. Perhaps, say they, the earth has 

 never been in this fluid condition. 



ITie second reason, which I believe will have a greater weight with every 

 body, is this: considering the earth as it is at present, and without carrying 

 our thoughts so far back as to its formation, if the ocean, which is now on its 

 surface, has any considerable depth, and if its parts preserve a communication 

 with each other, from region to region, by subterraneous canals ; it can only 

 keep an equilibrium by this means, because its superficies is the same as it 

 would have, were the whole a fluid. 



34. This second reason has suggested a reflection to my mind, concerning 



p p 2 



