578 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779. 



urged, by forces perpendicular to ae in any small equal times, through the un- 

 equal spaces s and /; and let the magnitudes of the bodies be represented by a, 

 B, and c respectively. Then the space through which a is actually urged in that 

 time will easily appear, from mechanics, (see art. 13) to be represented by 

 AXAKx.+ BXBPx/ g^ J ^j^^ described by c is ^ x aex.+ bx bkx^ 



AX AE'+BX BE^+CX CE'' ' ■' A X AE'+BX BE'+CXCe" 



X CE. 



§ 3. The preceding article being well understood, whatever doubts may remain 

 concerning the motion of a ring of matter considered as detached from the earth, 

 we may be certain that the motion of the nodes of the equator can never be the 

 same, whether we suppose the ring at the equator to be fluid and to rest on the 

 surface of the earth, partaking of the diurnal motion, or whether we suppose it 

 hard and compact, and by its cohesion communicating a proportional degree of 

 motion to the different parts of the earth. In fact, the problem of the preces- 

 sion of the equinoxes, which has hitherto been considered as extremely difficult, 

 and in its solution drawn out by authors to a vast length, requires no principles 

 but the received doctrine of motion, and the application of the lever, which 

 have been used in the last article. In that article we supposed the bodies a and 

 B to be impelled by different forces in parallel lines, and we estimated the real 

 space, which either a or c in any small time would describe, in consequence of 

 those impulsive forces, and their mutual connection by an inflexible lever. Now 

 this is precisely what is required to be done in the case of the sun's unequal ac- 

 tion on the protuberant parts of the equator. The excesses or defects of that 

 unequal action are to be considered as forces applied to those parts, which would 

 move them according to the different circumstances through unequal spaces pro- 

 portional to the forces in equal times of action, provided the particles were at 

 liberty to move freely in the directions in which they are urged; and, lastly, the 

 real space must be computed through which a particle moves at some known 

 distance from the centre of the earth in consequence of these various forces. 

 This whole process will not differ from the easy example already described, except 

 in the length of the calculation, and the proper management of the doctrine of 

 fluxions; and it seems advisable in difficult subjects always to begin with simple 

 instances, before we proceed to those that are more complex, and to distinguish 

 the algebraical operations from the principles on which they are founded. 



§ 4. In order to determine how much any particle of the earth is affected by 

 the unequal action of the sun, let cadb (fig. 2) represent the earth, s the sun 

 at a great distance, and CD a plane perpendicular to the line sx joining the 

 centres of the sun and earth. If sk or st represent the accelerating force of 

 the sun on a particle at the earth's centre, and sl be taken to skl in the duplicate 

 ratio of st to sp; sl will represent the attraction on any particle p, and by the 



