TRANSACTIONS OF SECTION A. 529 



but not so in the shunt-dynamo. In each, the whole electric activity (that is, the 

 rate of doing work) ispc' ; or, by (3) — 



'H^^ (^) 



or, by (1) and (2)— 



KK' ^^^ 



Of this whole work, the proportions which go to waste ito heating the coils and to 

 work in the external circuit are — 



Rc- + R'c"^ . . waste . . . (6) 



h^^J^^^^l^-^Rc'^ + R'c'-^) . useful work . . (7) 



By making v sufficiently great, the ratio of (6) to (7) (waste to useful work) may 

 be made as small as we please. Our question is, how ought R and R' to be pro- 

 portioned to make the ratio of waste to work a minimum, with any given speed ? 

 or, which comes to the same thing, to make the speed required for a given ratio of 

 work to waste a minimum ? To answer it, let r be the ratio of the whole work to 

 the waste. We have, by (5) and (6) — 



^_ IV(RR')cc' V 



Rc' + R'c' KK' ' ' ' ■ ^^ 



For the single-circuit dynamo we have c = c', and (8) becomes — 



R + R' KK' ' -' 



SKK' ^^ 



where S = R + R' (H) 



Suppose now .S' to be given, and suppose for a moment / to be constant. The 

 problem of making r a maximum with v given, or v a minimum with r given, 

 requires simply that i?(^-i?) be a maximum ; which it is when R = ^S, that ie, 

 when the resistances in the working coil and the electro-mag-net are equal. But 

 in reality / is not constant ; it diminishes with increase of the magnetising force. 

 As it generally depends chiefly on the soft iron of the electro- magnet, and com- 

 paratively but little on the soft iron of the moving armature, or on iron mag- 

 netised by the current through the moving coils, it will generally be the case 

 that / will, cceteris paribm, be diminished by increasing R and diminishing R'. 

 Hence the maximum of r/y is shown by (10) to require R' to be somewhat greater 

 than |<S': how much greater we cannot find from the formula, without knowino- 

 the law of the variation of I. ° 



Experience and natural selection seem to have led in most of the ordinary 

 dynamos, as now made, to the resistance in the electro-magnet being somewhat 

 leas than the resistance in the working coil, which is in accordance with the pre- 

 ceding theory. 



"Whether the useful work of the dynamo be light-giving, or power, or heating, 

 or electro-metallurgy, we may, lor simplicity, reckon it in any possible case by 

 referring to the convenient standard case of a current through a conductor of given 

 resistance K connecting the working terminals of the dynamo. This conductor, 

 inaccordance with general usage, I call the ' external circuit,' which is an abbrevi- 

 ation for the part of the whole circuit which is external to the dynamo. In the 

 case of the single-circuit dynamo, the current in the external circle is equal to that 

 through the working coil and electro-magnet, or c of our notation. Hence, by 

 Ohm's law — 



1881. 



c = 



K+R + R' 

 M M 



(12) 



