144 CELESTIAL MECHANICS. J. ii. 12, 



sented by its relation to t and 5, and this function vanish- 

 ing, according to the conditions of the problem, when t 

 has a determinate value, independent of the arc described. 

 Supposing F, for example, represented by ST, S being 

 a function of s alone, and T of ^ alone, we shall have 



^=d(4^)=r^'?.^+ S^^;" which indeed might be 

 df dt ds dt dt' ^ 



written T j- + S — > since S can only vary with s, and 

 or Qf 



this expression could cause no ambiguity. " But, since 



tion of T, or of — j-, we may also suppose --— to be a func- 



tion of ii, and we may call it ^- ^^±^), and we 



dds_ ds« C dS , d« X ) „ „ . . 



shall have g^-^^^ ^j + ^-(jj^) j = -K. Such .s 



the expression for the resistance derived from the diffe- 



d* 

 rential equation — = ST; which comprehends the case of 



a resistance proportional to the two first powers of the 

 resistance, multiplied by constant coefficients : but other 



differential equations representing — • would give diffe- 

 rent forms to the expression of the resistance." 



[Scholium 3. Instead of attempting to show the 

 utility of this very general formula, which is certainly not 

 extremely obvious in its present state, it will probably be 

 more useful to insert here a more elementary view of the 

 properties of the pendulum, remarking first that this pro- 

 position i? only demonstrated with respect to the ascent of 

 a body in the curve to be investigated, and that the descent 

 will require some of the signs to be changed, the re- 

 sistance cooperating with gravitation in the one instance. 



