110 Prof. T. R. Lyle on the Variations of 



where x is the thickness, y the width of the strip, p the 

 specific resistance, and b the induction, provided that dbjdt 

 has the same value at all points of a cross section. 



In ring laminae when magnetized in the usual way neither 

 b nor dbjdt is constant across their section, but an upper limit 

 will be given to the rate of dissipation per cm. 3 at any point 

 where the induction is b by the equation 



dX_ x 2 /db\ 2 

 dt ~ 12p\dt)' 



and the rate of dissipation of energy by eddy-currents in the 

 whole ring will be 



x 2 /db\ 2 



: -j- x vol. of ring X space average of I -j \ throughout the ring. 



Now it can be shown that this average will not differ much 

 from (dB/dt) 2 where B has the meaning already assigned to 

 it, namely, the average induction across any section. The 

 latter statement is roughly indicated by the fact that though 

 the amplitude of b and hence of db/bt may vary considerably 

 from the inner to the outer radius of the ring, being greatest 

 at the inner radius as the amplitude of the magnetizing force 

 (which varies inversely as the distance from the centre of 

 the ring) is greatest there, still, since the inner circumference 

 is less than the outer one, there will, in making up the space 

 average of (db/dt) 2 , be a smaller relative volume of the iron 

 at the high induction than at the low induction. Hence, 

 finally, if E be the average eddy-current loss throughout the 

 ring per cm. 3 per cycle, we have, approximately, that 



if 



B = Bj [sin 0)^ + ^3 sin 3 (ft)^ — 3 ) 4-^sin b(cot- 5 ) + &c] 



j„(§ y-¥». 



then C/dB\ 2 7 , 2tt 2 , 



where 



$ 2 = B 1 2 {l + 9& 3 2 + 256 5 2 + &c.} 

 and 2 r 2 



It will be seen that $$ is a quantity of considerable 

 importance in this theory, and it will be called the effective 

 induction. 



It is the amplitude of the sinusoidal induction-wave that 



