278 Mr. 0. Heaviside on the 



where R/ and R 2 ', L/ and L 2 ' are functions of p 2 . The utility 

 of this notation arises from R/ &c. becoming mere constants 

 in simple-harmonically vibrating systems. Let e m , V m , and 

 G m be the corresponding quantities for the particular m ; then, 

 by (56), 



e m - ^f=-L oP C m + (R' lm + L' lm p)O m +(R'z m + V, mP )C m . (62) 



Or 



e m -^- = (U' m +JJ m p)C m) (63) 



where 



R' m =R' lw + R' 2m ; L' m = L + L' lm + L' 2wr . (64) 



R' m and U m are functions of p 2 . Therefore, by (62), sum- 

 ming up, jv 



e-^=t(B! m + V m p)C m . . . . (65) 



Now, although R' m and U m are really different functions of 

 p 2 for every different value of m, since they contain m' 2 , yet 

 if, in changing from one m to another, through a great many 

 m's 9 from m = upward, they should not materially change, 

 we may regard R' m and U m as having the m = expressions, 

 as in the purely electromagnetic case, and denote them by R' 

 and L' simply. Then (65) becomes 



e-^=(R' + I/p)C (66) 



simply. The equation of V is now 



-i + ( S=^ + L '^; • • • (67) 



and that of C m being 



«»=(R'm + L , m ^ + m 2 /Sp)O m . . . (68) 

 in the m case, that of C becomes now simply 



S^ + ^ = (R' + L^)SjdC (69) 



The assumption above made is, in general, justifiable. 



Let us now compare these equations with the principal 

 ways that have been previously employed to express the con- 

 ditions of propagation of signals along wires. For simplicity, 

 leave out the impressed force e. First, we have Ohm's system, 

 which may be thus written : — 



-S= EC > -§= 8 ^ S= rs ^- • (70) 



Here the first equation expresses Ohm's law. C is the wire 



