200 ON FORCES OF ATTRACTION 



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 



771 TfL 7* 



forces only -^- and . When we add the other two - 

 r* q* m 



and the complication is not considered by Lagrange 

 m 



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 resultant 

 of r and g), and this will carry the ultimate diagonal repre- 

 senting the whole force applied to P beyond the axis S S f . 

 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 and . But this would 



r* (f 



cause these radii vectores to be produced, and make their 

 resultant also fall below the axis. It can hardly be doubted 

 that these considerations 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 centres, in the more difficult case of 

 moveable centres is certain. The justly celebrated LXVIth 

 proposition of the First Book affords ample proof of it ; and 

 indeed the LXIVth proposition comes so near the subject of 

 this note, that it may be correctly said to contain the grounds 

 both of Clairaut's and Legendre's more full investigation. 



11. In connection with this subject Lagrange expresses 

 great admiration of a theorem of Lambert, which no doubt 



is remarkable, that in ellipses ( the central force being as ^ ) 



\ r J 



the time taken to describe any arc depends only on the 

 transverse axis, the chord of the arc, and the sum of the radii 

 vectores at its extremities. We may observe, in passing, that 



