662 ELECTRICAL MEASUREMENTS 



bracket shows the number of the particular ordinate under 

 consideration. 



[Yp\i = Ak sin ka 



(Qfin . 



k ~- + ka) 



(QAn . 



2k - + fca) 



(j 

 (P- !)*--- + ka). 



Then the sum of the P ordinates is 



+ . + [Y P ]p = 



i f, /fc360\ _ /fc360\ 



m ka 1 1 + cos ^ p j -f cos 2 ^ p j + 



/\ ,/\l 



cos 3 ( p j + - . + cos (P - 1) (p) + 



/fc360\ , /A:360\ 



cos ka | sin ( ~p) + sin 2 ( p j + 



(21) 



k 

 Inspection shows that if -5- is a w?/ioZ6 number 



[ Y P ]i + [Fp], + [Y P ], + . . . + (Y P ]p = PA k sin ka = 



(22) 



That is, when p- is a whole number the sum of P equally spaced 



ordinates is equal to P times the first ordinate. This is the first 

 of the laws referred to above. 



k 

 If p is not a whole number the above series can be summed by 



aid of the following trigonometrical formulae. 



cos + cos 20 + cos 30 -f . . (+ cos (P - 1)0 = 



1X , cos(P- 1)0- cos P0 



2(1- cos 0r 

 sin + sin 20 -f sin 30 + . . . + sin (P - 1)0 = 



. (P - 1)0 . P0 



sin - ~ sin -=- 



(24) 



sin ^ 

 In this case when 



