522 Prof. K. Clausius on the Transmission of 



the left hand, and introduce then the following abbreviated 



signs : — 





v\ + vl 



(6) 



E+/3(» 1 +» 2 ) + «r(^+rf) 



The equation then becomes 



1= / xv?,L^ M "yT' U ' ... (7) 

 (a + z)(64^) 6 + z v y 



If this equation be further multiplied by (a + i)(b + i), and 



then the members arranged according to powers of i, we 



arrive at the quadratic equation 



i 2 — (pu—\V--a-->b)i--epu + \a\J + ab=zO ; . (8) 



by solving which we obtain 

 iz=.\(jpu— XU— a — b) 



± 2 V(pu— \U — a— 6) 2 + 4^>w— 4\aU — 4a6. . (9) 



Of the two signs before the root we must choose the upper 

 one, in accordance with my previous remark in reference to 

 the corresponding equation ; and by using this sign we must 

 make the limitation that the expression only holds for such 

 values of u as are great enough to make it positive. For 

 small values of u, below a certain limit, which make this ex- 

 pression negative, the value null for i, given in (4), must be 

 taken to hold. If this is to be treated in all strictness we 

 should have, in this interval, not the exact value null, but a very 

 small value of i, depending on the remanent magnetism in the 

 fixed electromagnet of the first machine. But as this interval 

 only comprises those velocities of the first machine which 

 occur during the starting, and before the second machine is in 

 motion, the very small values of i within it are of small import- 

 ance, and we need only direct our attention to the values which 

 hold for greater velocities, and are defined by equation (9). 



This equation may be put in a somewhat simpler form. If 

 for the sake of shortness we put 



«/.*-. ^ET=_ — 7 * - . . (10) 



c = (e — a)p = q—pa+pb, (11) 



the equation changes into 



i=± (pu'-a-b) + W(P u ' + <*-*>)* + 4eu. . (12) 



