206 CELESTIAL MECHANICS. I. V. 19. 



square of D E (194) : consequently EL will be the space 

 described by the attractive force, while DE would have 

 been described by the velocity at D ; for the force may be 

 considered as uniform during the evanescent increments, 

 and the spaces described by such a force are as the 

 squares of the times : hence the joint result will be DL, 

 which is ultimately equal to DK, and the whole velocity 

 will be increased in the ratio of DK to DE, or DI to 

 DG, or BH to BF ; consequently, since H, I, and K are 

 ultimately equidistant from C, the velocities in AB and 

 AD, being- always equally increased at equal distances, 

 will therefore always remain equal at equal distances. 



Scholium 3. We may observe that every known force 

 in nature acts in conformity with this condition, and 

 operates always equally at equal distances from its origin : 

 as Laplace has himself remarked in this article, asserting 

 that F is always a function of /: and if the case were 

 otherwise, with respect to gravitation or magnetism, for 

 example, we might easily obtain a source of perpetual 

 motion, by causing a body to describe, in its descent, a 

 path in which the force is greater, and to ascend by one 

 in which it is smaller at the same distance. There is 

 indeed a supposed exception, in the hj^pothesis, which 

 Laplace has elsewhere adopted, respecting the extraordi- 

 nary refraction of crystallized bodies : but the exception is 

 by far too paradoxical, to be admitted by any person, not 

 previously determined to deduce the motions of light 

 from the laws of attractive and repulsive forces : for here 

 it is assumed that the force depends, not on the distance 

 of the attracting substance, but on the direction of the 

 motion, with which it varies perpetually. The New- 

 tonian demonstration of the laws of ordinary refraction 

 had the advantage, on the other hand, of simplifying their 



