500 PHILOSOPHICAL TRANSACTIONS. [aNNO 1 768. 



It^. On the Connection between the Parallaxes of the Sun and Moon ; their 

 Densities ; and their Disturbing Forces on the Ocean. By Patrick Murdoch 

 D. D., F. B. S. p. 24. 



I. The length of a second-pendulum at the equator, and on a level with the 

 sea, being 36 inches 7-rV lines, Paris measure, according to the accurate obser- 

 vations of M. de la Condamine, that length, properly corrected, will, by the 

 reasoning in a former letter, (p. 87, &c. of this vol.) give the distance of a moon 

 circulating round an unmoveable earth, equal to 59.95792 semidiameters of the 

 equator. For the logarithm of this number, which is 1.7778438, write /. And 

 let L be the logarithm of some greater mean distance, inferred from observations 

 of the moon's parallax; and if r be the natural number of the logarithm 3 X 

 (l — /), and M be taken equal to ^r—., the mass of the earth will be to that 

 of the moon, as m to 1. Conversely, if m be any how determined, its equal 

 ^^-- J, and r, with its logarithm 3 X (l — /) are known; -J- of which is l — /, 

 to be added to /. For instance, if, with Sir Isaac Newton, we put m = 3g.788, 

 the distance will be 60.4557, the logarithm l being 1.7814372. 



II. If, for each of these three, the moon's mass, her accelerative force on the 

 earth, and her distance from the earth's centre, we write ip = I : the accele- 

 rative force of the earth on the moon will be represented by m, the mass just 

 now computed. And if f is the sun's accelerative force on the earth, x his dis- 

 tance in semidiameters of the lunar orbit, a the ratio of a sidereal year to a pe- 

 riodic month; we have (by cor. 2, prop. 4, Princip. 1) - = —; a given ratio in 

 given terms. 



III. The terms f, x, therefore, must involve a common factor; by which being 

 divided, the quotient may be — . And this might be obtained innumerable 



ways, were we to consider the ratio - merely as an abstract quantity, altoge- 

 ther unrestricted; it were only putting m' X a'' = f. And 



^—r- = .r, or m' ~ " «/" = p, and = x: so as the sum of the indices of m 



should be unity, and the difference of those of a should be 2. But though the 

 quantities f, x, are as yet unknown, they are not for that indeterminate and va- 

 riable, as such a liberty of substitution would import; and all substitutions 

 which imply the contrary, all indices which the theory disowns, or which are in- 

 consistent with observation, are to be rejected. In a word, the indices, n, p, 

 ought each of them to be unique, and determinate (sine compare),* as the 



• See Neut. Arith. Universal, in the schol. to prob. xxiv. The maxima and minima of variable 

 quantities; the coordinates belonging to a double point, or to a point of reflexion, or contrary 

 flexure, rays of curvature, limits of ratios, &c. All these are examples of the luiique; that is, of 

 quantities in a state that is distinguished from and exclusive of all others — Orig. 



