S76 ■ M. Ampere's Memoir 



tween n and k, which I had before found by another process. 

 It is sufficient for this purpose to decompose the action which 

 each of the elements of the complete circuit exercises on the 

 element in consideration, into two forces; the one perpen- 

 dicular to this element, and the other which shall have the 

 same direction with itself, and which I shall call the elemen- 

 tary tangential force : then to sum up all the elementary tan- 

 gential forces produced by the complete circuit, and to equal 

 this sum with zero, which is the tangential force due to the 

 whole circuit. Thus then, if we represent by ds' the element on 

 which it acts, by ds an element of this same circuit, and other- 

 wise preserve the denominations of the memoir printed in the 

 Annalcs de Chimie et de Physique, tome xx., p. 398, et seq. 

 we shall have for the mutual action of the two elements, 



- i i' r' -"-*■ d (r* d' r (page 413) ; 



moreover „ ^'' , Ano\ 



cos/3 = - -^, (page 4.08), 



whence j, ^'' j j i ; e 



a' r = ^d s = — as' cos p, 



which changes the expression of this action into 



j/'d5'r^-"-*d(r*cos/3); 



for d s/ which represents the element on which the complete 

 circuit acts, is constant with respect to the characteristic d. 



In order to have the elementary tangential force, we must 

 multiply this value by cos /3, which gives 



i i' d s' r * -"-* cos ^ d (/■ cos /3), 

 which may be put under the foi m 



in'ds'r^-"-^* d(r*cosi3)% 

 Integrating by parts, we obtain for the total of the tangential 

 force 



in-'d5'{,.^-"-^^(/cos/3)'-(l-«-2/I-y7--"-'V*cos/3)'dr}i 



or itt'ds' \-—--(i-n-2k) dr (. 



( r J r ^ 



As the circuit is closed, r and /3 will take the same value at 

 the limits; thus the first part 



cos' /3 



will disappear. But it will not be the same with the second, 

 which cannot be calculated till we have replaced one of the 

 variables r and /3 by its value in the function of the other drawn 



from 



