66 



Bowditcli on the elements of the orhit of a comet. 



i + nq 



I, wlience we get m 



if n 



Oj and as the true orbit 



is supposed to be obtained in this case, we must have f=S,^=C. 

 By substituting these values of n, t, g, in the equations (3), they 



become T 



S 



m, (T 



S),G 



C 



:m(G- 



C), and by re- 



jecting the commou factors T 



S. G 



C, we get from both of 

 them 7K = 1, as it ought to be. But if we make the same substi- 

 tution in Newton's formulas (3), they become 2(T — S) = 



?«(T 



S), 2 (G 



C) 



»2(G 



C). both of which give m 



% 



which is double of its real value just found, agreeably to the re- 



L 



mark abovementioned. Many other cases equally simple might 

 be shown in which the mistake is very apparent. 



The commentators Le Seur and Jacquler, iu their edition of 

 the Principia, Gregory in his Astronomy^ and Emerson in the work 

 abovementioned, have attempted to prove the correctness of New- 

 ton's rulesj by the following method. By comparing (as we Itavc 



I L 



done) the first and second operations, they find that the increment 

 P in the longitude of the node, increases the time T by the incre* 

 raentf — T, whence they find by the rule of three, that the incre- 



ment 



T 



S 



T 



t 



■ p 



P in the longitude of the node would prod 



crement S — T in the time T, by which 



would bee 



'!> 



I 



equal to the observed value S, they then put ^, 



S 



1 



of the node m P 



t 



S 



m, and call 



By proceeding 

 nne. manner with the quantities G,^, C, they find that the 

 ut P in the longitude of the node increases the quantity G 

 G, and then by the rule of three they find that the incre- 



ment 



G 



G 



C 



P in the 





of the node would prod 



an m- 



crement of C 



i 



G in the quantity G, by which means it would be 



I, 



qual to the value C deduced immediately from the oh 



