APPENDIX. 427 



can read that paper without feeling that the acknowledg 

 ment was too coldly made, after he had gone so far as to 

 suppose that the whole Xewtonian doctrine was over 

 thrown, and to propose a new law of - 2 + ^-, the whole 



of this arising from his own error. It is to be remarked, 

 however, that the investigation of 1745 was in all respects 

 most accurately conducted, and must have led to the same 

 result as in 1748 but for the supposition that certain quan 

 tities might, safely be neglected. Even in 1745 Clairaut, 

 upon Newton s assumption of the excentricity of M being 

 nothing, comes to his conclusion that the proportion of the 

 axes is as 69 to 70. 



4. Legendre treats the subject very fully, as far as re 

 gards two centres, and also confining himself to the forces 

 being inversely as the square of the distance (Exercises 

 de Calcul Integral part iv. sect. 2.). He deduces from 

 his analysis several theorems, two of which he regards as 

 very remarkable. The first apparently strikes him in this 

 light, because it shows the same orbit to be produced by 

 the combined action of the two forces directed towards 

 two foci, as either force would produce acting on the body, 

 and directed to one of the foci. If V is the velocity at the 

 vertex of the ellipse which would make the body de 

 scribe that curve when acted upon by the force directed 

 to one focus, v the velocity at the same point which 

 would make the body describe the ellipse when acted 

 upon by the other force directed to the other focus ; then 

 if the two forces act together upon the body, and I is the 

 initial velocity, or velocity of projection, it will describe 

 the same ellipse, provided I 2 = V 2 + v 2 . 



5. The other theorem follows from his integration which 

 gives the expression for the time. It is that if two equal 

 forces act upon the body directed to the two foci, and the 

 masses of the attracting bodies consequently are equal, the 

 revolving body will describe the ellipse in a shorter peri- 



