68 PROCEEDINGS OP SECTION A. 



Let X and $ be the currents at any instant in A and P respectively, 

 and let the mutual inductance of A and F, when their axes are coinci- 

 dent, be m, and hence i7i cos oif at the time f. Also let r and I be the 

 total resistance and self inductance of the A circuit, and p, A. similar 

 quantities for the F circuit. 



Then when the armature is being driven at constant angular 

 velocity oj, and a; and 4- are flowing, the total number of lines linked 

 on A is — 



/.r -f m ^ cos wt, 

 and the number linked on F is — 



X ^ + mx cos (j)t. 

 Hence — 



rx + -77 ] i^v -\- m $ cos (uf > 

 J \ ^ I ™^ ««„ ..tf 



p$ + -7. * A^ + mx cos wt ^ = 7] 

 where rj is the applied steady e.m.f. in the F circuit. 



(I-) 



2. If we assume as the solution of these equations — 

 a;=xj2-{-x, sin (oit-\-c,) + x^ sin (2 od + c„) + x^ sin (3 W+c^) + &c. 

 ^=4/2+^. sin (w^+yj + 4\ sin (2 wt + y,) + i^ sin (3 w^+y,) + &c. 

 we can see at once on substitution that p^^ = 217, and that x^ = 0. and 

 it will be shown afterwards (section 15) that when x^ = 0, then t",, x^, 

 ii, ^4J fo) &c., vanish ; or, in words, when x„^= only odd harmonics 

 appear in x and only even ones in $. Let us therefore take — 

 x=x^ sin ((xit-\-G,) ~\-x^ sin (3 <iit-{-cJ + x^ sin (5 oj^+cj + &c. /jt , 

 ^ = 4/2 + L. sin (2 a)#+y,) + i, sin (4 «< + yj + &c. ^^'"^ 



Now any harmonic in either x or $, for instance .r,y sin (rjwt -j- c^), 

 being completely specified by x,j, c,,, and q, can, when its order q is 

 known, be represented by the vector drawn from the origin in any 

 reference piane to the point in that plane whose polar co-ordinates 

 are x^,, Cq, the constant term 4/2 in ^ being represented in the same 

 plane by the vector to the point 4i W^* 



The form of solution (II.) assumed may now be written — 



^ = 'ii + a3 + a, + a, 4-&C. nu \ 



4^ = aV2 + a, + a, + a, + &C. ^'^'^ 



where a,, a„ &c., a„, a„ &c., are vectors whose orders are indicated by 

 the subscribed numbers. Of these, one only, namely a^, is known, as 

 it is drawn to the point whose polar co-ordinates are 4, -/2, where 

 ^o := 2^/p. The others have to be determined. 



Note a. — In the sequel it will sometimes happen that a vector, 

 say av, originally assumed of order q will be used to represent a 

 harmonic of a different order, say q -\- \. In such a case it will be 

 written (a/J^y+i ; thus a^^ = x^ sin {qwt -\- Cq) but (a,/),, + i = x,, sin 



{(^+l)a.^ + C,}. 



Note h. — The length of a vector a will be written as d {I.e., with 

 the bar), thus ilj = x^., unless in cases where no ambiguity can arise, 

 when tt simply will be written for the length of the vector a. 



