ALTERNATE CURRENT GENERATOR. 69 



3. If we agree to indicate by /. the operation of rotating any 

 vector to which it is prefixed through an angle in the positive 



TV 6 - 



direction, then i a = — a or t ' = — 1, and t a = (cos 9 -\- i'^ sin 6)a 



- 



or i = cos 9 + 1-'^ sin 9. 



Also, if f = Dt ' , fa is the vector obtained by increasing a D times 

 in length and then rotating the increased vector through an angle/ in 

 the positive direction. Plane operators such as t are well known to 

 be subject to the same rules as ordinary algebraical symbols. Again, 



the sum of two operators «,t ', a, r-- can be expressed as a single 



operator A^'/' say, that is At'ra = a,r -a + «, r«a where a is any vector. 



Using the expression for r given above — 



77 77 TT 



A (cos (// + t- sin if/) a = aX^os 9^ + i- sin 9,) a -j- '•c,(cos 9, + t- sin 9,) a , 



so that — 



A cos «/'=«, cos 9^ + (7, cos 9. ; A sin i/' = «, sin 9^ + «, sin 9,. 



Thence — 



A' = f/j' -)-«/ + 2a, a, cos (9^ — 9,). 



a, sin 6, + a, sin 9, 



and tan li^ = ^ ' — . 



a^ cos Pj 4- «j cos a. 



Again, if ap = t sin (pojt + ■//,), — (a,, ) = pw i-^a,, , 



seeing that — (u^, ) = /?oj4- sin ipMt + y^, + ^ j; 

 hence for x and 4^ as expressed in section 2 — 



4. By means of the formula — 



2 sin a cos 6 ^ sin (a + i) + sin (a — i), 

 it is easy to show that 1x cos wt^ where x is the a series of odd order 

 vectors in section 2, is represented by the series of even order vectors 

 of which the one of the /)th order is the vector sum of a^, _ i and ay, + i, 

 or that — ■ 



Ix cos wt = (a,)„ + (a, + a,), + (a, + a,), + (a. + a,), + &c., 

 (a,)o being the resolved part uf a, along the y axis, that is along the 

 directiou of vectors of zero order. (See note, section 2.) 

 Similarly — 



2f cos wt = (a„ + a,), + (a, + aj, + (a^ + aj. + &C. 



Again, by means oi the formula — 



2 sm a sin b = cos {a — b) — cos {a -\- b), 

 it is easy to show that — • 



77 77 



2^ sin «)/ = I - (a„ — u,), + t ^ (a, + uj^ + &c. 



77 



= (. ^2 {(III - 1 — cq + i)r/ (where q is odd), 

 with a similar result for the product of x and sin o)t. 



