Resistance and Inductance of a Helical Coil. 83 



The equation (13) has a solution of the form 



^=A O + A 1 cos0 + A 2 cos2(^+ (16) 



where the functions A are independent of <f>. By substitution, 

 the accent denoting d/df>, 



(g- P ^~ + k 2 p)(A <) + A x cos <f> + A 2 cos 2</> + . . .) (A l cos </> + 2 2 A 2 cos 20 ) 



~^cos 2 a{2cos</>Ao / + (l + cos2(^)A 1 ' + (cos</) + cos3^)Ao / + } 



h j& cos2 ^ 2 "" 3 cos2 a ^ A °' + cos * Ai ' + cos 2( ^ A2 ' + ^ 



2 



+ £- 2 cos 4 a{2cos2</>A '+(cos^ + cos3</>)A 1 ' + (l + cos4^>)A 2 / + } 



_ c ^{(l_cos20)A 1 + 2(cos^~cos3^)A 2 + 3(cos2^-cos4<^)A 3 + } 



+ _£_ cos 4 «{ (cos <f> — cos 30) Ai + 2(1 —cos 40) A 2 4- 3(cos 0— cos 50) A a + }, 



4a 2 



we deduce the following system of equations, where 



D - ( |^ + ^ < 17 > 



DA - -£- cos^A/ - |~ cos 2 * (2 - 3 cos 2 «)A ' + -£^ cos 4 *A s ' 

 _A*cos 2 « + ^ 2 cos 4 « = 0, 



DAi--^ 1 - ^cos 8 «(2A '+A 2 ')-^co8 2 «(2-3cos 8 «)Ai' 



p la ^ a 



+ £ 2 cosXAi' + A 3 0- ^ cos**+ ^ 2 cos 4 a(A 1 + 3A 3 ) = 0, . (18) 



and so on. But it is evident that A , A„ A 2 form a 



sequence, each member of which is of a higher order in a~ l 

 than those before. 



For the purpose of this investigation, A is required to 

 order /3 2 /a 2 , and therefore, as an intermediary, Ax to order p/a 

 only, and A 2 not at all. Therefore the equations may be 

 written more simply in the forms 



o 2 o /« o o \^A cos 2 a d , A v -n 

 DA - 4 2 cos 2 «(2-3 cos-a)-^ = -j^W), ] 



A! cos 2 « dA , 



DA X = p — -. — . • • • J 



p a r dp J 



and Ai must be eliminated. 



G2 



