406 PHILOSOPHICAL TRANSACTIONS. [aNNO I767. 



reciprocally as the rectangle clb. For, drawing cd perpendicular to la, the simple 

 force towards c will be to the compound force towards a, as cl to 2ld, that is, 

 as bl to 2CL. Hence, since the simple force is as 



— -, the compound force will be as ——^ — , or as . 



CL*' ^ CL» X lb' cl X LB 



Schol. 2. If the two bodies were equal, the limit of attraction would be an 

 infinite plane, bisecting perpendicularly the distance of the two bodies. — Schol. 3. 

 f'utting the distance sc = d, the semidiameter of the greater body s = b, and 

 that of the less c = k; if then it were as the distance d : k :: \^ s -\- »/ c\ V c, the 

 point A would touch the surface of the body c. And the same would happen 

 to the point o if, diminishing the distance a little, it be d : k v. is/ s — »/ c: >/ c. 

 But if the distance d were less, the problem would be impossible. 



Prob. 1. The same things being supposed; to find the locus in which the 

 forces of the bodies may be to each other in a given ratio. 



Let the given ratio be that of h to c, viz. of the force of the greater body to 

 that of the less. Cut sc produced in E and p, fig. 3, so that se be to ec, and sp 

 to PC, in the subduplicate ratio of s io h: then the required locus will be the 

 superficies of the sphere pfe, described on the diameter pe. Which is easily 

 demonstrated as prob. 1 . 



Corol. 1. If cs becut in G, soas that cobeto gs, as - X ce to es; then the 

 point G will be the centre to which is directed the gravitation compounded in the 

 superficies pfe. This is easily proved by joining fs, fg, fc, and drawing gk 

 parallel to sp. — Corol. 1. And if in the diagonal fg there be taken fh = fc, 

 and KM be drawn parallel to sf, the compound force in the point f will be 

 reciprocally as the rectangle cfm. This is proved as cor. 4, prob. 1 . And the 

 same things are to be understood of the interior surface pfe, and the points 

 £-, h, k, m, in fig. 4. — Corol. 3. Where h is less than c, the centre g will be within 

 the surface pfe, as in fig. 4. Where greater, the centre g will be without the 

 surface pfe; and that the farther from the body c, the greater the given ratio is. 



Schol. If the given ratio be the same as s to c, the spherical surface pfe will 

 change into a plane, as in schol. 2, prob. 1, the point p going off infinitely. If 

 the ratio were greater, the point p would fall on the contrary side of the centre s ; 

 and the surface would again be spherical, and excentric of the greater body; and 

 its diameter would be found as above; but if the ratio should be less than (b -\- dy 

 X s to b\ or greater than h^s to {k + d)^ X c, the problem would be impossible. 



Prob. 3. To describe the motion generated by the conjunct forces of the 

 corpuscles of the attracting bodies s and c. 



If the bodies f and c be included in a fluid medium, in which there are 

 immerged corpuscles specifically lighter or heavier than the medium, the corpus- 



