346 Mr. 0. Heaviside on the 



butions per unit length is 



^ P^H, . 2irrdr+ ^ f "h/H," . 2irrdr 



which, by using the above expressions, becomes, provided 



L OA- ° l(E2 ~ F2) + C ^~^\ . . (126a) 



E and F being Y/k or the longitudinal electric forces at r—a x 

 or r = a 2 . But 



E-F = R"C, 



where R"= the R/' + R," of equation (56), Part II. ; and 



0=g+L oP + R-=g+^ + L>, 

 so (126) becomes 



The mutual electric energy is obviously 8V l V 2 per unit length. 

 By summation with respect to z from to I, subject to VC = 

 at both ends, we verify that the total mutual magnetic energy 

 equals the total mutual electric energy. The value of 2T in 

 a single normal system is, by (126a), and the next equation, 



LoC 2 + C^=^|(E' + LW . .(128a) 



per unit length ; and that of 2U is SV 2 . Hence, per unit 

 length, 



2(U-T) = SV 2 -C 2 ^(R' + L'p). . H29) 



In this use Y = u and C = w, equations (126), and we shall 

 obtain, for the complete energy-difference in the whole line, 



- {f Ji(W- • • }'| ^(jg +R' + lV)=M say, (130) 



which is the expanded form of 



r du __ dw~V r 2 d /it 7 YV 

 L dp dpJ L dp\w /J ' 



as may be verified by performing the differentiations, using 

 the expression for ujw in (127), remembering that m 2 in it is 



