CHAP. XIII] TORQUE AND TRACTIVE EFFORT 251 



d\\', is necessary because the stored energy decreases. From 

 eqs. (176) and (177) we get 



F= -W.d(R/ds ....... (178) 



In some cases it is more convenient to express F through M 

 and (P. We have 



F= -W*d((P- l )/ds=+l(<l> 2 /<P 2 )d<P/ds, 

 or 



(179) 



In the preceding formulae F is in joulecens, M is in ampere- 

 turns, is in webers, (P is in henrys, and (R in yrnehs. With 

 other units the formulae contain an additional numerical factor. 

 It is to be noted that the mechanical forces are in such a direc- 

 tion that they tend to increase the permeance and decrease the 

 reluctance of the circuit. This agrees with previous statements. 

 See Arts. 41 and 57. 



If the partial linkages are of importance, it is convenient 

 to express the stored energy in the form PT,= ii 2 L, because the 

 inductance L takes account of the partial linkages; see eqs. 

 (105) and (106), Art. 58. The energy equation, according to 

 eq. (177), is then 



Fds= -dW.= -ld(v*L) ..... (180) 



and the condition that there is no interchange of energy with 

 the line is 



d(iL) = ........ (181) 



The latter equation becomes clear by reference to eq. (106a), 

 because Li=n0 9q , where 0g is the equivalent flux under the 

 supposition of no partial linkages. The condition that then- 

 shall be no o.m.f. induced in tin- winding during the displace- 

 ment, is d(n0,)/<ft 0, whence eq. (181) follows directly. 



Performing the differentiations in eqs. (180) and (181), and 

 -iii-'ituting the value of Ldi from the second equation into the 

 first, we get that 



(182) 



When there are no partial linkages, L n 2 <P, and eq. (182) 

 becomes identical with (179). 



