76 Mr. 0. Heaviside on the 



primary and the secondary currents, their expansions must be 



Also, the excess of the mutual electric over the mutual mag- 

 netic energy of the initial state V , C 1? C 2 , and the normal 

 state represented by (ICte) is 



Y «=-4w>- (LA + M(y + (LA + MCl) i^rfc> I {12e) 



and this is what must be added to the numerator in {be) to 

 obtain the complete value of A, if we also add the corre- 

 sponding expression Yj for the apparatus at the other end, if 

 it be not initially neutral. Using this value of A in (Xe) and 

 in (He) with the time-factor e pt attached, and in the corre- 

 sponding expansions for the other end, we thus by (\e) and 

 (11^) express the state of the whole system at any time. 



Since, initially, V is U, and independent of the state of the 

 terminal apparatus, it follows that in the expansion 



U = 2A^, 



the parts of A depending upon the apparatus contribute 

 nothing to U, so that, b}^ (be) and {12 e), we have the iden- 

 tities 



= 2A^^-, 0=$Aw u, = SA /"7 ,. . (13.) 



for all the values of z from to I. 



It may have been observed in the above that the use of (9e) 

 was quite unnecessary, owing to the forms of the normal 

 functions in (10*?) being independently obtainable from our 

 a-priori knowledge of the terminal apparatus in detail, from 

 which knowledge the form of Z in (8e) was deduced ; so 

 that, without using (9e), we could form (lltf) and (12^), I 

 have, however, introduced (j)e) in order to illustrate how we 

 can find the complete solution, without knowing the detailed 

 terminal connexions, from a given form of Z. We must 

 either decompose dZ Q /dp into the sum of squares of admissible 

 functions of p } multiplied by constants, say, 



^ =«i/i 2 + « 2 / 2 2 + «3./? + -.., • • • {Ue) 



where a x , a 2 , &c. are the constants, and f\,f% the functions 

 of p : or else into the form of the sum of squares and pro- 

 ducts, thus 



^=a,/i ! + « 1 / ! ! + «,/ ! , + Vi/ ! + W J + (15«) 



