FORMULAE EQUIVALENT TO THAT OF AMPERE. 461 



The intersection with the yz plane is 



Y (zdx - xdz) + Z (xdy -ydx) = ; 



from which follows 



Y_Z 



We have further 



r 2 ds 2 sin 2 = (ydz - zdyf + (zdx - xdz? + (xdy -ydx)* 



which gives 



_ a W sin 2 r (ydz - 

 r L 



Now, the expression - : - is the cosine of the angle which 

 rds sin 



the perpendicular to the plane rds makes with the x axis : hence the 

 quantity in brackets is the square of the sine of this angle, or the 

 square of the cosine of the angle /*' which the plane makes with 

 the x axis that is to say, with the element ds', and we have 



ads sin cos M' \Vdsds' . 

 / = - smtfcos/*. 



Thus the action of ds upon ds l is in the plane rds t perpendicular 

 to the element ds', proportional to the sine of the angle which the 

 element ds makes with the distance r, and to the cosine of the angle 

 which the element ds' makes with the plane rds, and lastly inversely 

 as the square of the distance. 



Let us take the plane rds as that of xz, and in this plane the line 

 OO', which joins the two elements, as the x axis. The force acting 

 on the element ds' placed at the origin of the co-ordinates is in the 

 xz plane and is perpendicular to ds'. To obtain its direction, we 

 must project trie element ds' on the xz plane ; a straight line in this 

 plane, perpendicular to the projection, will be the direction in ques- 

 tion ; it is perpendicular to the projecting plane, and therefore to the 

 element which passes through its foot in this plane. 



