June 3, 1910] 



SCIENCE 



867 



/ 



SPECIAL ARTICLES 



VARIATIONS GRAPHICALLY 



The usual developments by which the cal- 

 culus of variations is rigorously established, 

 however cumbersome, are nevertheless satis- 

 factory in so far as the reader knows what the 

 aim is. But with a student, as a rule, they 

 remain hazy. He acquiesces, of course, but 

 he loses faith and the cloud may not be lifted 

 during the whole of his subsequent course in 



the motion along it. Any two points, a and 

 a', h and h', may therefore be regarded co- 

 temporaneous at pleasure. We may express 

 this by putting Si = 0, as in the figure. Any 

 variation is possible, but the motion along s' 

 must nevertheless be regarded as continuous; 

 i. e., the experimental motion is conceived as 

 taking place, any assistance from without 

 being admitted. The figure then shows at 

 once, if we pass from a to h' in the two ways. 



dynamics. I may therefore ask for indul- 

 gence if I publish the following simple 

 treatment, because it has borne fruit and is 

 intelligible to anybody who understands the 

 equation s^vt. 



Let s be the curve along which the motion 

 of a particle actually takes place. Suppose it 

 is to our advantage to consider what would 

 happen if the motion proceeded along any 

 other infinitely near curve s', selected at ran- 

 dom but with the object stated. The nota- 

 tion would be less cumbersome without the 

 differential coefficients x, etc., but it is more 

 direct to use them. 



1. St = 0. There are two cases. In the 

 first, the curve s' is quite arbitrary, and so is 



Sx+ {x + ex)di = xcU + 6x + -Sx-di, 



(1) 



the obvious meaning of the last equation. 



2. St not zero. In the second case the path 

 s' is still arbitrary, but it may be regarded as 

 a smooth wire along which a bead of the 

 given mass slips by the same forces that move 

 it naturally and without the wire along s. 

 The two motions here are necessarily continu- 

 ous and both are prescribed. Hence cotem- 

 poraneous points a and a', h and V, are pre- 

 scribed, and an interval of time St must elapse 



