74 Mr. 0. Heaviside on the 



wherein them's are known from the connexions of the whole 

 system ; each normal system having its own p, and also a 

 constant A to fix its magnitude. The value of A is thus what 

 depends upon the initial state, and is to be found by an inte- 

 gration extending over every part of the system. In one 

 case, viz., when the initial state is what could be set up finally 

 by any distribution of steadily acting impressed force, we do 

 not need to perform this complex integration, since we may 

 obtain what we want by solving the inverse problem of the 

 setting up of the final state due to the impressed forces, as 

 done by one method in Part III., and by another in Fart IV. 

 If also the initial state of the apparatus be neutral, so that it 

 is the state of the line only that determines the subsequent 

 state, we can pretty easily represent matters, viz., by giving 

 to A the value 



A j t '(Bu«-Lw»)* :_ {2e) 



wherein U and W are the initial Y and C in the line, 

 whose capacity and inductance per unit length are S and L ; 

 so that the numerator of A is the excess of the mutual electric 

 over the mutual magnetic energy of the initial and a normal 

 state, whilst the denominator A is twice the excess of the 

 electric over the magnetic energy of the normal state itself, 

 which quantity may be either expressed in the form of an 

 integration extending over the whole system, or, more simply, 

 and without any of tbe labour this involves, in the form of a 

 differentiation with respect to p of the determinantal equation. 

 For instance, when we assume L = 0, and we make the line- 

 constants to be simply Rand S, its resistance and capacity per 

 unit length (constants), as we may approximately do in the 

 case of a submarine cable that is worked sufficiently slowly to 

 make the effects of inertia insensible, in which case we have 



dY dC • d' 2 Y 



-i= nc > -i= 8Y > &= RSy ; • <*) 



so that we may take 



771 



u = sin(mz + 6), w= — yrcos (mz + 6), . . (4<?) 

 if — m 2 =Bi8p; then equation (2e) becomes 

 Sjo'lJi* dz +-Yo + Ya 



/ ( 1 -cos 2 mZ . -^ F(W) }' 



(5.) 



■j jl— cos nu . — — T 

 where the undefined terms Y and Y 2 in the numerator depend 



