APPENDIX. 435 



centres, in the more difficult case of raoveable 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 -^ 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. AYe may 

 observe, in passing, that the vanishing of the expression 

 for the conjugate axis in some fundamental formula con 

 nected with the ellipse, for example, the subtangent, gives 

 rise to other curious properties of the curve similar to the 

 one noted in this theorem, which is itself related to that 

 peculiarity. (See a porism arising from this circum 

 stance; Life of Simsoji, p. 154.) The same theorem had 

 occurred to Lagrange himself, in examining the problem 

 of deflecting forces to two centres; it is indeed derivable 

 immediately from the case of that problem when one force 

 vanishes and the centre connected with it is in an arc of 

 the ellipse ; for then the radius vector belonging to that 

 centre becomes the chord. But Euler, long before either 

 of them, in 1744, had given the theorem for parabolic 

 arcs, which they only extended to elliptic arcs, and had 

 published it in the Berlin Mem. 1760. Yet when Lam 

 bert claimed it as his own in 1771, and Lagrange gave 

 him the honour of it in 1780, Euler, though he lived 

 three years after, never thought of reminding them of his 

 prior claims. It was thus, too, with the first of analysts, 

 respecting the extension of the Differential Calculus to 

 that of Partial Differences (Life of D Alembert, p. 466.}, 

 by far the greatest step in mathematical science which 



F r 2 



