VOL. LVIII.] PHILOSOPHICAL TRANSACTIONS. 501 



quantities f and x are in nature. Dr. M. takes therefore /) = 1, and n = -J-; 

 that is, F = M"^ X a, X = —j. See the examples in the table subjoined, on 

 different suppositions of the moon's distance. 



IV. The accelerative forces of two spherical bodies, a, b, on 

 a third c, are directly as their masses, and inversely as the 

 squares of their central distances st, lt, or of a; and rfr 



F A rf' 



which may be thus expressed, - = - x - . But the masses, 

 A, B, being as their respective densities, which call s, m, 

 and the cubes of r, r, the semidiameters of the spheres, 

 conjunctly; if we write for 



- its equal, we have r = - X -r X -r. Let the mass c be 

 to B, as M to 1 ; andyi its force on B, will be ipM, or ip = 



/,.,. MF »X jF SX .^, 



- which gives — = - x -,, and -^ = — jt supposing the ap- 

 parent semidiameters of a and b to subtend the same angle at 

 the centre of c, and thence r to be to r, as xio d. 



V. But if SG, the semidiameter of a, be to the supposed 

 semidiameter sv, as ^ to 1, then, the density s remaining, the 

 accelerative force of a (proportional to its magnitude) will be 

 increased in the triplicate of that ratio, that is, we now have 



y. ^ — X ^, putting d= I. And the three bodies a, b, c, 

 representing the sun, moon, and earth ; and a being the ratio 

 of the periods of the earth and moon, it is 



-. = — , by the corol. quoted in art. 2. Whence - = —. ;. 



VI. This accelerative force of a remaining, imagine the semidiameter so to be 

 reduced to its former magnitude sv; and the density of a will, at the same 



time, be increased to *' =: ^^ x s, and - = — . In which case, namely, when 

 the apparent semidiameters of a and b (the sun and moon) are equal, their 

 powers to raise a tide at v, a vertex of c, will be as the densities s^, m:* that is, 

 as M the ratio of the earth's mass to the moon's, and q^ the duplicate ratio of 

 the year and month. 



Or thus: the distances of the bodies a, b, from the third c, being very great, 

 their powers to raise a tide at v, or their disturbing forces on the ocean, will be 

 directly as their accelerative forces at the centre of c, and inversely as their dis- 

 tances from it; that is, writing a, b, for the disturbing forces respectively; and 



for the sun's distance in semidiameters of c, the letter z, it will be a : 6 :: - : j. 



z d 



* See the last page of Or. Sanderson's Fluxions : or the late ingenious T. Simpson's Miscella- 

 neous Tracts, p. 13.--Orig. 



