MATHEMATICAL APPENDICES. 



207 



where 



. 7T 



n sin - 

 n 



(2) 



I being the value of the continuous current between each pair of 

 brushes. 



The instantaneous current in the section of the armature is 

 therefore given by 



i = ii - / 



4 sin (pt - 0) 



- 1 



. 7T 



n sin 

 n 



(3) 



which has a mean square value given by 



16 sin 2 (pt - 0) 8 sin (pt - 6) 



dt 



-rrn sin 



This is a minimum when = 0, that is, in the section of the 

 armature situated at the middle of each phase. It is a maximum 



for 9 = "' 



n 



To obtain the mean square current over a complete phase, we 

 have therefore to take the mean value of (4) between the limits 



= -, and = : , and obtain the expression 



M m sin* - 

 The ratio of the heating of the armature of an ?i-phase rotary 



