Theory of Dynamo-electrica I Machines. 133 



to the value v\ of v, by iv 1 and w/. We thus obtain 



i 1 =z±(piv 1 / --a--b) + ^\/(pwi +a—b) 2 + 4cciv l7 

 from which follows 



(a + i 1 )piv 1 / + cw 1 = (a + i l )(b + i 1 ) (43) 



In this we have to determine for w 1 and w± their expressions 

 from (33) and (38), 



A* o 



w 1 = =r h ; w{— £ -, 



& + /*>! + avf' -R + p^ + v/ 



in order to determine v lt 



We will make this determination here only approximately. 

 For since the members containing the factors \ and a only 

 need, from their smallness, be considered with great velocities, 

 w r e may with the velocities in question reject them without 

 hesitation. We have then to put 



/ v i 



We thereby obtain first, 



R + p^i p{a + i 1 ) + c 



and from this we have further 



v = (a + h)(b + i i )'R ( . 



1 p{a + i 1 ) + e-{a^i 1 ){b + i x )p v ' 



For the ordinary purposes of practice we may simplify 

 still further. From the smallness of the magnetic moment 

 fju of the remanent magnetism, the strength of current % is 

 very insignificant compared with that which occurs when the 

 machine is in full work; and hence, as already mentioned, it 

 is usual to disregard the currents which occur with small velo- 

 cities of rotation, and which only slowly rise to the strength i, 

 and to consider that the production of the current only takes 

 place with a certain velocity of rotation. From this point of 

 view we have to assign a limiting value to the corresponding 

 velocity which it acquires when /-t, and therefore also i, ap- 

 proaches zero. If this limiting value of v x be called v , we 

 have, from the previous equation, 



tfo=- — : ;-, (45) 



pa + c—abp v ' 



