138 PHILOSOPHICAL TRANSACTIONS. [aNNO 1751. 



XL A Letter from the Rev. Patrick Murdocke, F. R. S. concerning the Mean 

 Motion of the Moon's Apogee, to the Rev. Dr. Robert Smith, Master of Trin. 

 Coll. Camb. p. 62. 



A warm dispute arose lately at Paris between M. de BufFon and M. Clairaut ; 

 the latter pretending that the Newtonian law of attraction is inconsistent, with 

 the motion of the moon's apogee ; and that its quantity ought not to be expressed 



by - of the distance, but by two, or perhaps more, terms of a series, as \ •\- — ; 

 which new doctrine M. Clairaut had got inserted in the memoirs of the Aca- 

 demy, and M. de BufFon had followed him close with another memoir, confuting 

 it. At first it was impossible to judge of the validity of M. Clairaut's reasoning, 

 because he kept his calculus a profound secret. But an absurd consequence of 

 his new law of attraction occurred as soon as M. de BuiFon mentioned the thing, 

 that, " if we should put the attraction, expressed by his two terms, of an assumed 

 quantity g, and resolve the equation, there would necessarily arise 1. different 

 values of the distance x, for the same attractive force." 



Suspecting therefore, that some error must have slipt into M. Clairaut's 

 reasonings (as he himself afterwards found there had), Mr. M. tried whether, 

 by an arithmetical calculation from Sir Isaac Newton's propositions only, the 

 motion in question might not be accounted for. By Mr. Walmesley's ingenious 

 treatise on the same subject, it appears that however M. Clairaut's hypothesis is 

 given up, yet a notion still prevails as if Sir Isaac Newton's propositions, con- 

 cerning the motion of the apsides were mere mathematical fictions, not applicable 

 to nature. The following calculation however of Mr. M. shows the contrary. 



Of the mean Motion of the Moons Apogee, according to Sir Isaac Newton. 



The rule given by Sir Isaac Newton, in the Qth section of his first book, is to 

 this purpose : 



1. That, supposing the common law of attraction, and that a central body t 

 attracts the body p, fig. 8, pi. 3, revolving round it in an orbit nearly circular, 

 with a force as unity ; if to this be added a constant force, whose ratio to the 

 former is expressed by c ; then the angular velocity of the body p, in an immove- 

 able plane, will be to its angular velocity, reckoned from the apsis of its orbit, 



1 -I- c 

 in the subdublicate ratio of 1 -f- c to 1 -j- 4 c, or as \/ :^—-r ^'^ unity. And there- 

 fore, if a represent any arc described by the revolving body in an immoveable 

 plane, then a X V' will be the corresponding arc in its orbit, reckoned 



from the apsis. And their difference a x {s/ £ — - — 1), willbethe regress of the 

 apsis. But if the force of the central body t be diminished by some constant 



