888 Royal Astronomical Society. 



1st. A constant pulling force, equal to the mean value of the 

 force. 



2nd. A force pulling when Venus is in conjunction, pushing at 

 the time intermediate to the conjunctions, and pulling when Venus 

 is in conjunction again ; thus going completely through its changes 

 once between conjunction and conjunction. 



3rd. A force pulling when Venus is in conjunction, then pushing, 

 &c., going through its changes twice. 



4th. A force pulling when Venus is in conjunction, then pushing, 

 &c., going through its changes thrice. 



In this manner the forces go on, continually diminishing in magni- 

 tude. When we arrive at the 18th, the force is extremely small. 



The algebraical expression for the collection of these terms, put- 

 ting fl for the difference of mean longitude of the earth and Venus, is 



A-i-B . cos HC . cos 2 9 + D . cos 3 9 + &c, 



This is on the supposition that the orbits of the two planets are 

 circular and in the same plane. But, in consequence of their eccen- 

 tricities and inclinations, the forces of any one system alternately 

 pushing and pulling (Nos. 2, or 3, or 4, &c.) will not have the same 

 maximum magnitude throughout. But each can, in all cases, be 

 expressed by the combination of three such forces, in each of which 

 the maximum forces are equal throughout. Thus, if we combine a 

 large force, going through its changes twenty times in a certain 

 period, with a small force going through its changes nineteen times 

 in the same period, and another small force going through its 

 changes twenty-one times in the same period, then it will be found 

 that both the small forces increase the large force (whether in its 

 pulling or in its pushing state) near the beginning and the end of 

 the time; that both diminish the large force near the middle of the 

 time ; and that the two small ones destroy each other at a quarter 

 and three-quarters of the time. The elFect of this combination is 

 therefore precisely such as is spoken of above. 



Thus, then, for the complete expression of the force, we are 

 driven to an infinite number offerees following the law of alternately 

 pulling and pushing, but with very great variety of magnitudes of 

 force and of periodic time. The greatest portion of these produce 

 no sensible effect ; some because (though their magnitudes are large) 

 they act for so short time in one way, or their periods are so little 

 related to the periods of any movement of the moon, that their effects 

 never accumulate ; others because their magnitudes are small, and 

 there is no unusual circumstance favourable to their increase. 



But there is one of these forces which, in the algebraical expres- 

 sion, depends on 18 x mean longitude of Venus — 16x mean lon- 

 gitude of the earth, whose coefficient is exceedingly small, but which 

 goes through all its changes, from pulling to pulling again, in the 

 time, 



27'^ 13»» T'^SS^'G; 



or from pulling to pushing, in the time 



13d \^^ 33™ 47«-8. 



