72 PROCEBDmGS OP SECTION A. 



lience for the complete solution we have — 



a„ = — S.a^ = S,2,a„ = — HjSjSja, = &c., 



which gives a,, a,, a,^, &c., a^, a^, a^, &c., in terms of the known vector a^, 

 provided the continued fraction operators S^, 2,, S^, &c,, are deter- 

 minate. 



It is easily seen that they are determinate, for when q becomes a 

 large number — 



f,=2^ = A, + 2 

 m 



. T,/ + 1 = 2 — = Tf^ + y. (^See section 5.) 



m 



that is S„ 2,, Sj, 2^, &c., are recurring continued fraction operators, 

 and the recurring elements when reached are simple numbers. 



8. The form of solution obtained is one very easy of practical 

 application. In computing the contiuued fraction operators just so 

 many of their known /, t, elements (see section 5) need be taken 

 account of as are necessary to give the required degree of approxima- 

 tion. Moreover, in the case of a practical alternator, as the resistance of 

 the field coils is negligible relative to their reactance, all the t elements 

 are practically simple numbers independent of the Held resistance, 

 while for the t elements q is never a large odd number when /,/+2 differs 

 little from f,,. So that we could obtain S,, with considerable accuracy 

 by assuming the recurring stage to be reached, and, therefore — 



S,, = t,i — — 1 



which gives the quadratic in 8,/ 





■T/Z + l 



from which IS,^ can be obtained by ordinary algebra, 

 fin solving this quadratic the two operators that come under the 

 square root symbol will have to be reduced by the addition theorem in 

 section 3 to a single operator afi say, and the root of this ia\/ a i^/^.J 

 If .\ obtained in either of these ways be Sqi. ~ *<?, then as— 



8„ S,. 



2f/-i can be obtained by the addition theorem, and so on for S,/_2, 

 2^-3, &c. np to >Si. 



Let the results be written — 



S. = k,- *., 2, = <TJ- ^:' S, =.. S^-^, &C., 



and in general S^/= s (. '', ^p=(TpL ^'\ 



