130 Prof. R. Clausius on the 



certain limit do we get negative values of i. This latter 

 interval we will for the present disregard, as we shall come 

 back to it. 



The previous form may also have the following form: — 



i = l;(pw — \vw — a — b) 

 + ±\Z(piv—\vw + a — b) 2 + 4z(q—pa+pb)w. . . (37) 



In this we will introduce two signs for the purpose of simpli- 

 fication; that is, 



v v 2 



w = 



' = *>(!- -A= t>_^ * 2 , - • • (38) 

 \ p ) H + pv + aif' v J 



c = q—pa+pb, (39) 



by which the equation acquires the following very compen- 

 dious form: — 



i= ±(jp W f — a —6) + \<s/{pw' + a— b) 2 + 4tcw. . (40) 



In order to gain some idea of the magnitude defined by this 

 equation, w 7 e may make the following considerations. The 

 constant c defined by (39) may be positive or negative accord- 

 ing to the construction of the machine, and may therefore 

 possibly have the value null. In this case, by the disappear- 

 ance of the sign of the root and eliminating the magnitude a, 

 there is a great simplification in the form of the equation, 

 since it passes into 



i=zpw'—b (41) 



If we represent the values of i graphically by taking w' as 

 abscissa and i as ordinate, we get a straight line, which cuts 

 the abscissa-axis in the point at which the abscissa is equal to 

 b/p, and rises from here with increasing abscissa under an 

 angle the tangent of which is equal to p. 



If c is not equal to null, then if the value w is referred to 

 iv', and we thereby consider i to be expressed as a function of 

 w' alone, we obtain a curve by graphical representation, which 

 differs the more from that straight line the greater the absolute 

 value of c ; the difference being such that its point of intersec- 

 tion with the abscissa-axis is in a different position, and that it 

 is curved in one direction or the other according to the sign of 

 c. The curvature, however, is not considerable, and decreases 

 with increasing value of w\ 



If the current-strength i is to be directly referred to the 

 number of turns v, we must substitute for w and w f their values 

 given in (38). It is particularly significant of these expres- 

 sions that they have the magnitude It in the denominator, from 



