35] 



GEODESICS. PROBLEM OF AIM. 



103 



enables us to find the differential equations of such a line. Suppose 

 the surface is a sphere, then if the particle is started from a 

 point P (Fig. 24) with a given velocity v 

 in any direction, it may be made to arrive 

 at Q by the introduction of certain con- 

 straints, for instance, suppose it obliged 

 to move on a plane passing through P 

 and Q. The principle of least action says 

 that in the natural or unconstrained motion 

 it will go from P to Q along the shortest 

 path, that is, an arc of a great circle. 

 Of all possible paths there are two natural 

 ones by which the particle travels from 

 P to Q along a great circle, but leaving P 

 in opposite directions. It is only for the 



shorter of the two paths that the action is a minimum. This is an 

 example of a frequent occurrence in the calculus of variations, 

 namely, that an integral possesses the minimum or maximum property 

 only when its limits are sufficiently close together. 



We will illustrate this by a less simple example. Consider the 

 problem of shooting at a target, or the ideal case of a single particle 

 acted on only by gravity, which has been treated in 18. 



Suppose the particle projected from the point # , with the 

 velocity v ot so as to reach the point x 19 2 1 . If t be the time of 

 flight, we have by 18, 3) 



Fig. 24. 



from which 



= V x 2 + V? = 



or otherwise 

 13) \g*$ + 



X - * ) - V) t* + (! - *o) 2 + (! 



- 0, 



a quadratic in t 2 to determine the time of transit in terms of the 



given constants x 



, , 





Introducing the following letters for 



the range, its horizontal and vertical projections, 



r = 

 and solving the quadratic, 



14) t* = |rk) 2 -gh VW ~ 9^ ~ A' 2 }- 



If the radical is real - - which will be the case if the initial 

 velocity is great enough - - since the absolute value of the term 



