148 COEFFICIENTS OF INDUCTION. 



DM 



hence the total flow of force 8M, or ds t due to the element ds, 

 and which traverses the circle of radius a, has the value 



sin<9 



f 2 f 

 M 



J01 JP 



dddp. 



The limit X is the angle of the tangent OT with the element ds; 

 we have evidently 



a 



cos^,= - , or sin 2 01 = 

 a + c 



The two limits p 1 and /o 2 , which correspond to the two segments 

 om l and om ti of the secant OP, are roots of the equation of the 

 second degree, 



sin <9 + <r20 + <r = 0. 



If the ratio - is very small, we may take 



and 



i 



and a first integration with respect to p gives 



IT 



SM = 2ds \ ( / . sin 2 ) sin 6 dO. 

 Jo, \ c / 



This expression, integrated with respect to 6, gives 



7T 



(2-1. sin 2 6>^ +2/.tan- | 2 ; 



as the angle is very small, the expression reduces to 



We' deduce from this, for the value of the entire flux, or the 

 coefficient of mutual induction, 



Sa 



