30jS Mr. Challis on the Theory of the 



Hence as=-{F{r-at)-f{r + at)\, 



and -, may be neglected whenever r is not small. The inte- 

 gral {B) was long ago obtained by Euler, and has more recently 

 been arrived at in a different manner by M. Poisson, {Mem. 

 Acad. Scien. 1818) who considers it a particular case of the 

 general integral. But nothing presents itself in the above solu- 

 tion to forbid our concluding that it gives the proper general 

 integral of the differential equation. It may be no objection 

 to our conclusion, that this able analyst has expressed the 

 general integral under a less simple form, by the method of 

 Definite Integrals, because that method, as he says, is not. 

 unique and determinate, and may not therefore be very proper 

 for ascertaining the simplest form of the general integral. The 

 use that will be made of the integral will elucidate this point. 

 The equations x- + y- = r-, and x" +f+ z-^r", shew that in both 

 cases r is the distance of the point under consideration from 

 the origin of co-ordinates. Equations (//) and (B), shew that 

 in both ^ is a function involving r and t alone. Therefore 



since 



d(t> , d(i> 



adt dr 



V and as are functions of r and t alone. Hence may be inferred 

 that primarilij the motion of every particle is at a given instant 

 some function of its distance from a point in a line drawn through 

 it in the direction of its motion. The generality of this inference 

 is legitimate, both because nothing was said about the manner , 



