120 CELESTIAL MECHANICS. I. ii. 8. 



'---^- . JNow at the beginning and end of the 



body's motion, the variations Sa', Sy, S2, must necessarily 

 vanish, because it is supposed to move from one fixed 

 point to another, the intermediate points only being sup- 

 posed to be subjected to an elementary variation : con- 

 sequently the value of ^fvds between these two points is 

 equal to nothing, and the quantity yids is a minimum, 

 since its variation vanishes. 



[Scholium. The nature of this fluent may be under- 

 stood by supposing the path to be changed towards the 

 middle by a slight variation ^x of x only ; the variation 

 ^fvds with respect to the whole portion of the path would 



uXcX 



then become -' , , which is equal to the product of tjie 



variation of x into the velocity of the body in its direction 

 at the given point.] 



Corollary 1. If the body is moving 

 freely on a given surface with a uniform velo- 

 city V, we hsive fvds=ti/'ds=vs, consequently s 

 is also a minimum, and the curve is the 

 shortest that can be described between the 

 two points. 



[Scholium. The curve in this case may always be 

 traced by bending a flattened wire over the surface, which 

 must necessarily determine the minimum, since the two 

 edges of the wire are of the same length, and the variation 

 of the length of the path contained within its limits va- 

 nishes: the same curve must also be that which a body 

 would describe spontaneously on the surface, because it 



