434 APPENDIX. 



+ 2 y ^ ; w = 2, and consequently m -f 2 = seems 

 necessary to his process. 



There seems reason to suppose that the kind of reasoning 

 on which we have relied as to the identity of the trajectories 

 had influenced Legendre in confining his investigation to 

 the case of curves which have not infinite branches. He 

 expressly says (Ex. de Calc. Int. 11. 372.), that he 

 confines himself to curves where the orbit is restricted to 

 a definite space. Certain it is, that the reasons applied 

 to the identity in the case of curves returning into them 

 selves is wholly inapplicable to curves having infinite 

 branches. 



10. The extreme complication of the problem arising 

 from the resultants passing through innumerable points in 

 the axis has been above noted, as regards the case of two 



forces only ~ and . When we add the other two 

 f 9 f i 



~ and the complication is not considered by Lagrange 



to be increased (p. 99.), and probably it is not as regards 

 the analytical investigation. But it certainly is increased 

 as regards the geometrical construction ; for we then have 

 to take the resultant of P c with P C (which is the re 

 sultant of r and q), and this will carry the ultimate dia 

 gonal representing the whole force applied to P beyond 

 the axis S S . Lagrange indeed does not take P C into 

 his analysis, because he supposes the forces r and q to act 

 in the same line of the radii vectores with the forces 



-s- and -r-. But this would cause these radii vectores 

 r 2 g 2 



to be produced, and make their resultant also fall below 

 the axis. It can hardly be doubted that these considera 

 tions weighed with Sir Isaac Newton, in disinclining him to 

 the investigation of a problem which could afford no hope 

 of a geometrical, or of any synthetical solution. That he 

 had deeply considered the subject of attraction to various 



