Dynamo-electric Machines. 295 



From 1 r 



and the above we get a quadratic in r, 



r2(p a -^i)-r« 2 (2-+l)-e 2 R = 0. 



Then from v m , 



,n + -r N ~ 



■-^-(^dA) 



we have 



e=r( TS )lw + -r) 



VR + W \ p / 



In the general case this is intractable ; but for particular values 

 of x something can be made of it. 



Let %=2, which corresponds to a wide range somewhat 

 below the point of saturation of the magnets. Then we get 

 another quadratic in r, 



r 2 L 2 e-*(l\v)A + r (2eRs-n (IXvf) + eR 2 = 0. 



The next step is to eliminate r between the two quadratics. I 

 have used the formula 



(a b 2 — a 2 b o y + {a x b Q — a b{) {a x b 2 — a 2 b x ) = 



for the eliminant of 



a Q r 2 + a x r + a 2 = 0, 



& r 2 + V + 6 2 = 0. 

 The computation is of great length, and I have not completed 

 it. It results in an equation of the 6th degree in p and v, of 

 which the terms of the 6th and 4th orders are as follows : — 



pXl\vyn 2 e 2 n+p*e 2 W> 



-p 2 (l\v) 2 l/— -(2v-3w) + neR 2 (l + ?)-4n e 3 R 2 l 



+ (MV„5{,( 1+ ?)-,( l+ f)} 

 + =0. 



These terms may be written shortly in the form 



ap 2 v* + bp*-cp 2 v 2 + dv* + =0. 



There can be no asymptotes except such as are parallel to the 

 axes. 



