78 'proceedings of section a. 



Assume — 



^ = -;^ + a, + a, + a, + &c. 



V = -^ + a, + a, + ttj 4- &C. 



wliere a„ is the vector to the point whose polar co-ordinates are /r, 

 '^/2, and a^ is, as before, the vector to the point ^^/p, '^/2. The other 

 vectors a^, a„, aj, a^, &c., a„ a„ a., &c., have to be determined. 



On substituting for x and ^ in the above equations it will be 

 found that the odd order vectors in x. and the even order ones in ^, 

 are determined by the same equations VI., section 6, as when « = 0, 

 and are completely independent of the even order vectors in x and the 

 odd order ones in ^, these latter depending only on e and vanishing 

 with e. 



This being so, a,, a„ a^, a^, &o., are given by the solution already 

 obtained, and a,, a,, Oj, a^, &c., will be given by a similar set of equa- 

 tions written down from symmetry. Thus the complete solution is 

 given bv — 



a,= - S,a. = S,Xa, = - ^X^^^, = &C. 



ao= — S,a, = 2,'*^,aj = — 2, SjSjttj = &c. 

 where a„ is the vector to 2rj/p, 7r/2, as before and 



a^ the vector to 2'-/r, — • 



Note. — In the former case (p = o) the t operators were all of odd. 

 and the r ones of even order. In this case the operators of either class 

 ai'e of both orders. 



The translation from the above vector solution to the ordinary 

 sine form follows as in section 10. 



16. In the preceding solutions the magnetic fluxes have been 

 assumed to bo in phase with the magnetizing current-turns, and so iron 

 loss due to hysteresis and eddy currents has been neglected. To take 

 account of the latter the interpretation of the well-known relation 

 B = /aH connecting steady magnetizing force and induction produced 

 has to be modified. The induction produced by H = H, sin (wt + c^) 

 is known to be of the form, 



B = «?, H, sin (ivf + c, — 8,) + higher harmonics, and attending 

 only to the fundamental harmonic m B, if H be represented as explained 

 in section 2 by the vector /?„ and B by the vector b^, then the above 

 trigonometrical relation may be written b^ = /jlJi^ where fn^ is the operator 

 Wj t " *^' . 



In a former paper* by me was shown how these 2J<^'>^'>'i^(fbiUfi/ 

 operaiors, as they may be called, can be determined. They depend on 

 the character of the iron and the thickness of the lamina?, on the 

 amplitude and period of the fundamental harmonic of the induction 

 oscillation they refer to, and to some extent on the wave foim of the 

 latter. 



* Variation of Magnetic Hysteresis with Frequencj\ Phil. Mag., Jan., 1905. 



