464 ELEMENTARY ACTIONS. 



Finally, the angle 8 which it makes with the element ds is 



or 



If we impose on this force the condition of being perpendicular 

 to the right line which joins the two elements we have then 



'(</)- Oj 



from which is deduced 



/-A, /-; 



hence the value of the added force is 



i = afds sin = ds sin 9. 



It will moreover be seen that this force d 2 ^ makes with the 





element ds an angle equal to + -. 



2 



When the two elements are directed along the same straight line, 

 as the force d 2 ^ * s nu ll> an< ^ tne force given by M. Reynard's 

 formula is also null, the action of the two elements is null. 



On this hypothesis, which is that of Grassmann, the true force 

 d^ will be the resultant of a force which is inversely as the square of 

 the distance, and of a force ^Vi which is inversely as the distance, 

 is perpendicular to the right line joining the elements, and whose 



direction makes with the element ds an angle equal to - + 6. 



Several other conditions might be imagined equally compatible 

 with experiment; but these few examples will suffice to show the 

 indeterminateness of the problem, and to point out the principal 

 solutions. 



