528 Prof. R. Clausius on the Transmission of 



instead of (26) we construct the equation 



(«+,)(&+,")->, U+i7V + b+t)p m * W 



In this we will introduce a simplifying expresion z with the 

 meaning T . ft yy x x r 



-^-fc+?)(' + 5+?)? • * * (29) 



by which the equation changes into 



> & -z (30) 



/ 



(a+i)(b+i) 



The two equations (27) and (30) are of the same form, so 

 that the solution of one at once gives that of the other. 

 Selecting equation (30) for treatment, we bring it by multi- 

 plication with (a+t)(b + f) into the form of an ordinary cubical 

 equation, that is, 



?-(z-ey-(a+b)zi-abz=0, . . . (31) 

 and apply to this the well-known method of solving cubical 

 equations. Only one of the three roots thereby obtained can 

 be used on account of the sign, and this is determined as 

 follows: — Let <£ be the angle between and it which satisfies 

 the equation 



- f __ (*-«)'+ |(.+»X«->-HF*» . . (32) 



i will be represented by the following expression: — 



i=l(z-e) + 2 K /(z- e f + -d{a + b)z.cost. . . (33) 



o 



In reference to the cosine which occurs as a factor of the 

 last member, it follows from equation (32) that while z in- 

 creases from to x , the angle cf> decreases from it to 0, from 

 which it follows that 0/3 decreases from w/3 to 0, and accord- 

 ingly cos ((/>/ 3) increases from \ to 1. Here it is to be observed 

 that the first increase is pretty rapid, so that in the values of 

 z which occur in practice, cos (0/3) cannot be very different 

 from 1. 



Equations (32) and (33) may also be used in a twofold 

 manner. If z is replaced in them by T 2 f pv 1 , thev form a solu- 

 tion of equation (27), and give that approximate value of i 

 which has been designated above by \ f . By replacing it in 

 (29) we obtain the value r, and, knowing this, we can by 

 another application of equations (32) and (33) calculate the 

 stricter value of i. 



After determining in this way the strength i, the first of 

 the above-named three magnitudes, there is no difficulty with 

 the two others. 



To determine v 2 equation (5) can be used. This is an 



