Power by Dynamo-electrical Machines. 529 



equation of the second order, and could be solved as such ; 

 vet here also the fact that the square of v 2 only occurs with 

 the small factors <r and X may be used for simplifying the 

 calculation. For this the equation may be written in the 

 following form: — 



e+i 



r = (a+0(*+*r v fe+v , . (34) 



2 g+? 



(a + i)(6+i) P P 



In this we may omit the last term of the numerator, and from 

 the equation thus abbreviated calculate an approximate value 

 of v 2 . This value can then be introduced into that last 

 member, and thus the more accurate value of v 2 determined 

 by a second calculation. 



When this magnitude v 2 is known, the determination of the 

 last magnitude, with which we are specially concerned (that 

 is, the work done by the second machine T 2 ), follows directly, 

 as equation (20) can be used, which, after replacing 7 and 8 

 by their values, may be written as follows : — 



T 2 = T 1 -[E + P (r 1 + ^ + (l + ^)(, + A_.) W + i) ] i2 . ( 35) 



We see from this equation, in which the negative member 

 on the right hand has the factor ?, that T 2 is greater (that is, 

 is nearer the given value T x ) the smaller i is. But as, from 

 (33), i decreases simultaneously with z, and as, further, the 

 chief member of the expression for z in (29) is the fraction 

 Ti/pt'i, and therefore decreases when i\ increases, i also de- 

 creases when t\ increases ; and hence we get the result for T 2 , 

 that its value is greater as i\ is greater ; and it is in fact 

 known from practice that, in order to make T 2 as great as 

 possible in reference to T 1} the velocity of rotation of the first 

 machine must be increased to the utmost possible limit. 



According to (32), the velocity r 2 of the second machine is 

 greater the greater i\ and the smaller i becomes. But as, from 

 the preceding, with a given value of T 1? a decrease of i is con- 

 nected with an increase of i\, it follows from (34) that v 2 in- 

 creases with an increase of i\ in a still greater ratio than t^ itself. 



It will be clear from what has been said that in this case, 

 in which T x is considered as given, there can be no question 

 of a definite ratio between v 2 and i\ for which T 2 would be a 

 maximum. 



In this paragraph, from the magnitudes T x and i\ the three 

 other magnitudes i, v 2 , and T 2 were determined ; and in the 

 previous paragraph the magnitudes i and T 2 from the magni- 

 tudes Vi and i* 2 . In like manner a series of other calculations 



