the Gold&climidt Dynamo Equations, 789 



Appendix. 



More general solution of equations (1) and (2) (p. 763). 



It was pointed out by Mr. H. F. Thompson of Charterhouse, 

 to whom the equations were shown, that when the coefficient 

 ra=M/R was the same in both equations, the variables could be 

 separated by introducing £=u+ y, n=x — y ; for then 



dl 

 m -t*~~ Z (s®c_pt + nip tan pt) + b secp£=0, 



dtt 



m-rr — n (sec pt—mp tan pt)-\-b sec pt=0 ; 



whence ultimately 



where 0=p£, n=l/mp, and b=y ; 



with a similar expression for rj with the fractions inverted, and a 

 different arbitrary constant. A Fourier expansion can be made, and 

 coefficients calculated ; and though there is no guarantee that the 

 series will be convergent, this appears not to matter practically in 

 this case. 



It will be more convenient, however, to quote the not dissimilar 

 but completer solution with which Mr. C. A. Gaul of Marl- 

 borough has favoured me, for the case when the two coefficients 

 are different. His process, abbreviated, runs closely as follows : — 



The given equations are 



x=zm-r.(y cos pi) ; y=b + m ' -r(v coapt) ; 



or nx = cos d-j-(y cos 0) (i.) 



d 

 and n' y = ri b -{- cos 6 -r(x cos 6), .... (ii.) 



where n = l/mp, n' = l/m'p, 6=pt i s=sin0. 



d 

 Now since y (cos 0) = — tan 0, we get from (ii.) 



dx 

 n'y cos 0=n'& cos 0+ cos 3 0y — x sin cos ; 



substituting in (i.), we can soon find that 



c X x dv 



(1-s 2 ) 2 ^ _4*(1- S 2 )^ -v(l + nn'-2s>)-n>bs = 0. 



Phil. Mag. S. b\ Vol. 25. No. 150. June 1913. 3 H 



