ALONG THE CIRCUMFERENCE OF A CIRCLE. 67 



18. In this way we have obtained two pair of conjugate motions, the periodic 

 time of the second pair being double of that of the first, and the intensities of 

 gravitation being 



G = NZ . NZ 



a 



G^ = NZ . NA 



G = NZ . NE 

 y 



G s = NC . NE 



such that any one of the four motions being known the other three may be found. 



19. Here it is to be observed, that the periodic time of 7 and 8 is double that 

 of a and /3 ; in order to bring it to be the same, we must quadruple the intensities 

 of the gravitation acting on these bodies, so that for all the 



Periodic Times alike 

 G a = NZ . NZ , G^ = NZ . NA 



G = 4NZ . NE , G. = 4NC . NE . 



20. And if the intensities of gravitation be supposed the same for all the four 

 bodies, their periodic times will then be proportional to the square roots of the 

 preceding intensities : so that if we put T a , T^, T , T s for the periodic times on 

 the supposition of one gravitation, we have 







Gravitations 



alike 





T 



a 



T 



y 



= NZ ; 

 = 2 V(NZ 



• NE) ; 



T = 

 T = 



NE 

 ■2V 



21. These two pairs of conjugate motions are so connected, that from one of 

 them the other can be found, the law of connection being contained in the pro- 

 portion 



NI: IZ: : NE + NZ : : NE-NZ 

 or in 



NI + IZ: NI-IZ: : NE : KZ . 



If then from the pair a, (3, we deduce the pair 7, 8 ; we may again from this latter 

 deduce another pair of conjugate motions which we may mark 7^ 8 l ; and from 

 this again another pair 7 2 , 8 2 , and so on without end. Or if we regard 7, 8, as the 

 original pair, and deduce a, /3, from it, we may thence deduce a new pair a x , /3 V 

 and from that again, another a 2 , /3 2 ; and so on, so that we have a progression of 

 conjugate motions extending indefinitely each way, and such that any one of the 

 series being known, all the others can be thence deduced. The latter branch of 

 the progression, viz., that from 7, 8, to a, /?, and thence onwards is that which 

 is available in our research. 



VOL. XXIV. PART I. T 



