220 rHILOSOPHICAL TKANSACTIONS. x [aNNO 1738. 



the equipoise of the columns now calculated, art. 31 and 32. Let us first 

 suppose, that the earth is our fluid spheroid, composed of layers of different 

 densities; and that afterwards this fluid hardens into a solid, so that the difl^erent 

 strata, of which it is made up, are of no other use but to cause a gravity by 

 their attractions. Then let us suppose, that the seas and great waters about 

 the earth have a communication with each other, by means of some subterra- 

 neous canals. As the waters of the sea, which unite with each other, are pro- 

 bably homogeneous, the foregoing calculation, in which we have considered the 

 spheroid as a fluid, can no longer take place, because we have there supposed, 

 that the fluid contained in the canal bcn is of a density, that varies from the 

 centre to the circumference. From hence it seems to me, we must undertake 

 the computation of the equilibrium of the columns after another manner, thus: 

 We must examine whether two canals, as on and bc, which are filled with a 

 homogeneous fluid, will be in equilibrio, all the other parts of the spheroid 

 continuing as above. 



35. To do this, we will begin with finding the gravity of any column on, 

 fig. 15, arising from attraction alone. First then, we must resume the expres- 

 sion of the attraction in any point m, art. 7- Then we must multiply it by 

 f -\- \r, which will give 



2c/r' + r . sT 4.pcf>^r^ +^r , i cfrr +^r , 2cgr' + ?r ^ 



3+p 3 + P X 5 + p 3 +p X 5 +p 3+J 



And taking the fluent of this quantity, we shall have 



^e= + =^CL. + ==Jt^:i^-= + ^^li=, &c. for the 



Z + py.i+p 3 + p y. S + p 2+px3 + pxS+p 3 + ?x2 + 5 



gravity of the whole column on. 



36. If in this value we make x = 0, we shall have the gravity of the column 

 at the pole. 



37. And if we subtract the gravity of the column at the pole, from the 

 whole sum of the attractions of the column cn, we shall have 



*^'^' _. ._ + —^ ^— , which must be equal to the sum of the centri- 



3+px5+p 3+9x5+9 



fugal forces of the column cn, in order that the columns cb and on may be in 

 equilibrio. ( 



But we shall find this really to obtain, if we resume the quantity 



( — gSffl— — -I ^Ji ^Jl^) r ^ which,, by art. 31, expresses that part of 



3+px5+p 3+9x5+? e 



the centrifugal force in m, which acts according to cm. Then multiplying this 

 expression by r, and seeking the fluent, we shall have 



