GRASSMANN'S FORMULA. 463 



478. (in.) Amplre's Formula. If we impose on the elementary 

 force the condition of being directed along the right line joining the 

 two elements, we get Ampere's formula ; this formula is the only one 

 which satisfies the general principle of action and reaction, and 

 consequently the essential conditions of a true elementary force. 

 For any other solution, the action of the element ds on the element 

 ds', will not be equal and directly opposite that of the element ds' 

 on the element ds. 



479. (iv.) Grassmanris Formula. Let us replace the arbitrary 

 functions X, Y, and Z, respectively by xf t yf, and zf, f denoting a 

 function of the distance r. The components of the elementary force 

 will then be 



This operation is the same as adding to the force given by 

 M. Reynard's formula, another force d 2 ^ the components of 

 which are 



The force itself is given by the equation 



[XViP = a*[f*ds* + r*(df)* + 2frd(fdr}}, 

 or, taking into account the relation dr = ds cos 0, 



~ 2 Wi) 2 = [flr + rdfj 



This force makes with the straight line r an angle, the cosine 

 of which is 



xd (xf) +yd (yf) + zd (zf) _ frdr + r*df_ d(rf) 

 ' 



