M» 



PHILOSOPHICAL TRANSACTIONS. 



[anno J 768. 



For, by the general law, a is to p, as the difference of the squares of z and 

 (|K -— 1) is to the square of (z — l) ; that is, as 2z — 1 to z^ — !2z + 1. And 

 the same way b is to ^, as 2t? — 1 to (f — 2^/ + 1 ; whence, halving the ante- 

 cedents, and retaining only z^, cP, in the consequents, we have the ratio of a to 

 b, as above. 



If the disturbing force of b is exerted at v, the opposite vertex of c, b will 

 now be to (p, as 2rf + 1 io d^ -{■ id ■\- 1 ; and, in strictness, we ought to take a 

 mean value of b: but this may be neglected as inconsiderable. 



Lastly, let the sun's distance be again expressed in semidiameters of the 

 lunar orbit; that is, if for z we write dx, and unity for <p, we have a : b :. 



F 1 



— : ^, or as M to a'*, as before. 



VII. In art. 5, it was found that, m denoting the density of the moon, and* 

 that of the sun, q^ being triple the ratio of the sun's mean semidiameter to the 

 moon's,* then will - = — ^. Whence it will easily follow, that the density of 



the earth is to that of the sun, as a^ X s^ to p^; p being the moon's horizontal 

 parallax, and s the sun's apparent semidiameter. 



VIII. Dr. M. has applied these rules, as in the following table, to the principal 

 hypothesis of the moon's mean distance. 



1° Supposing it of 6o.24 (from Dr. M. Stewart). 



2° 60.4 (cor. 7, prop. 37, Princip. 3). 



3° 60.455 (Neut. M. being 39.788). 



4° 60.493 (as by Mr. Short's calculations from the transit of $ .) 



The moon's mean distance, being in semidiameters of the equator. 



II. 

 III. 



IV. 

 V. 



VI. 



▼II. 



VIII. 

 IX. 



Parall. D 



? 

 Mass D 



O dist. to J) 

 Parall. 



D 

 Dens. O 



? 

 Dens. D 



? 

 Dens. O 



D 



Tides O 

 F. O on ? 

 F. ? on J) 



60.24 



57'4".I7 



70 4225 

 284.723 



2.77:1 



1.449:1 



4.0128:1 



2.5379:1 



1.593 :1 



60.4 



56'55' 



TT 



44.823 



356.885 



9"-58 



4.35:1 



0.9295:1 



4.045 :1 



3.9873:1 



1.9968:1 



60.45i 



56'52" 



39.788 



378.293 



9"J"' 



4.9 :1 



827 :1 



4.0559:1 



4 5544: 1 



2.1194:1 



60.493 



56'49".88 



36.99O8 



392.854 



8".69 



5.273 :1 



0.7707 :1 



4.06375:1 



4.8316 :1 



2.1981 :1 



• In the Principia, the semidiameters are l6'6", and 15'38j"; giving 9.0379755 for the loga- 

 rithm of j'. Others take a few secondi from cachj which does not much alter the value of y'. 

 — Orig. 



