76$ Sir Oliver Lodge on 



Hence the total current in rotor is 



and the total current in stator is 



f # M * q « . . P 2 M 2 /2p 2 M 2 a « , , ,") j 

 Fyo+^+^ofj^ 3 cos 4pt+ ^(-^7-1)^2^ + 1) ( 



Thus y , the original exciting current, is a simple factor 

 throughout ; and the circular functions of even and odd 

 multiples of pt } which on general grounds were expected, are 

 obvious. 



A rather more general treatment. 



On showing this to my brother, Alfred Lodge, of Charter- 

 house, late of Coopers Hill, he pointed out that the neatest 

 substitution would be 



0=pt + i7r, 



so as to throw everything into the cosine form and get a 

 general method ol: formation ; subsequent re-interpretation 

 being as follows : — 



cos 6=— sin pt; cos 3# = sin 3p£; 



cos 20= — cos pt; cos 40 = cos Apt. 



The terms in the pt formulse will all be of the same sign, all 

 the x series being in sines, and all the y series in cosines of 

 multiples of pt ; while the 6 formulae are all in cosines, the 

 signs of successive terms being alternately + and — . 



If all the coefficients affected with the + sign are added 

 together, their sum will always exceed the sum of the — co- 

 efficients by unity ; a fact which furnishes a check on 

 arithmetical correctness. 



To calculate successive #'s and ?/'s proceed thus : — 

 Write A 



U =pt + J£7T. 



Then 



= 6' 1 7/ COS 6. 



pM. d e . a d Vi - nn\ 



• /2= R d&( ' 1 ' =c ^y^~j0 ( A sm 26) 



— c i c 2 'tjo cos 26. 



