460 ELEMENTARY ACTIONS. 



475. FORMULA EQUIVALENT TO THAT OF AMPERE. We have 

 seen (349) that the action of two elements of the contour of two 

 magnetic shells, which is equivalent to the elementary electro- 

 dynamic action, may be expressed in an infinity of different ways, 

 with this condition that the resultant of the actions of a closed 

 circuit on an element has a determinate value. 



476. (i.) Formula of M. Reynard. The first form which we 

 have met (348) for the action of ds upon ds' is, by supposing the 

 element ds' at the origin of co-ordinates and directed along the 

 x axis, a force whose components are 



x r* y 



The factor a in these equations represents the product Il'^y', and 

 x, y, z are the co-ordinates of the element ds. 

 The force itself is expressed by the formula 



ll'ds'ds . 



/= sin 6 cos IM, 



in which is the angle of the element ds with the right line ?*, and 

 p! the angle of the element ds' with the plane rds. 



If d$ is the angle under which the element ds is seen from the 



element ds', an angle which is equal to - - , this formula may still 

 be written 



/= 



which is the formula of M. Reynard. 



In order to determine the direction of this elementary force, we 

 observe, in the first place, that it is perpendicular to the element ds' 

 since/,. = 0. 



It is in the plane rds. The equation of this plane, of which 

 X, Y, and Z are the co-ordinates, is 



X (ydz - zdy) + Y (zdx -ydz) + Z (xdy -ydx) = 0. 



