24 Mr. 0. Heaviside on the 



which show the expressions of Hi and IV, the second being 

 the coefficient of p, the first the rest. 



Similarly in other simple cases. And, in general, from the 

 detailed nature of the combination inserted at the end of the 

 line, write out the connections between the current and poten- 

 tial difference in each branch, and eliminate the intermediates 

 so as to arrive at Y=Z 1 C, the differential equation of the 

 combination, wherein Z is a function of p or d/dt. Put 

 2> 2 =—n 2 , and it takes the form Z 1 = R 1 , + L^p, wherein R/ 

 and L/ are functions of the electrical constants and of n 2 , and 

 are the required effective R/ and L/ of the combination, to 

 be used in (206), or rather in its z — l equivalent G x . 



As regards the z = end, it is to be remarked that, owing 

 to the current being reckoned positive the same way at both 

 ends, when we write V = Z C as the terminal equation, it is 

 — Z that corresponds to Z 1 . Thus — Z =E ' + L ^>, where, 

 in the simplest case, B ' and L ' are the resistance and induct- 

 ance of a coil. 



So far sufficiently describing how to develope the effective 

 resistance and inductance expressions to be used in the ter- 

 minal functions G and H, we may now notice some other 

 peculiarities in connection with the solution (19b). First 

 short-circuit the line at both ends, making the terminal func- 

 tions unity and = 0. The solution then differs from that 

 given in Part II., equation (82), in the presence of the quan- 

 tity K, the former Sw now becoming (K 2 + $> 2 n 2 )i, whilst P and 

 Q differ from the former P and Q of (78), Part II., by reason 

 of K, which, when it is made zero, makes them identical. If 

 we compare the old with the new P and Q, we find that 



U becomes L'-KR'/S^ 



R becomes R + KL'/S, 



in passing from the old to the new. Then the function 



R 2 + L' 2 n 2 (R-fKLVSj^CU -KRyS^^^R^ + L^ 

 » becomes K 2 +S 2 n 2 $W~ 



or is unaltered by the leakage. It follows that the equation 

 (85) Part II. is still true, with leakage, if in it we make the 

 changes (21 b) just mentioned, or put 



• ..(li-g), ..as+par, „ 



instead of using the v' and 7i expressions of Part II. 



At the particular speed given by % 2 = KRyL'S, we shall 

 have 

 P = Q = (i)*(R 2 + I/ 9 n 2 )*(K 2 + S 2 n 2 )* = (i)'(R'S + KL>, (236) 



} (*1« 



