334 Mr. 0. Heaviside on the 



So far will define in the briefest manner, U, T, Q, and 

 activity. 



Now let the F's vanish, so that no energy can be communi- 

 cated to the system, whilst it can only leave it irreversibly, 

 through Q. Then letp l5 p 2 be any two values of p satisfying 

 (88) regarded as algebraic. Let Q 1? Ui, T± belong to the 

 system p x existing alone ; then, by (89) and (90), 



= Q 1 + U l + t 1 , or = Q 1 + 2 i ? 1 (U 1 + T 1 ); 



= Q 2 + U 2 + T 2 , „ 0=Q 2 + 2^ 2 (U 2 + T 2 ). 



But when existing simultaneously, so that 



Q=Qi + Q 2 + Qi 2 , U = U 1 +U 2 + U 12 , T*T 1 +T 1 +T MJ 



where U 12 , T 12 , Q 12 depend upon products from both systems, 

 thus : — 



Q 12 = 2 { Bntv?/ + B 22 v 2 v 2 ' + B 12 (v x v 2 ' + v 2 v x r ) + . . . \ , 



U 12 = A n %«/ + k 22 x 2 x 2 r + A 12 (^ 1 ^ 2 r + x 2 os-i) + . . . , 



Ti 2 = OnVjVi + G 22 v 2 v 2 + C 12 (vx Vi + v 2 v/) + . . . , 



the accents distinguishing one system from the other, we shall 

 find, by forming the equations of mutual activity 2EV = . . . , 

 and %Wv = . . . , that is, with the F's of one system, and the 

 w's of the other, in turn, 



= iQi 2 +p 2 U 12 + j p 1 T 12 , | 



= iQi 2 +p 1 U 12 +i? 2 T 12 ;J 



adding which, there results the equation of mutual activity, 



= Qi2+(pi+ j p 2 )(U 12 + T 12 ), or = Q 12 + U 12 + T 12 ; 



and, on subtraction, there results 



0=(Pi-K>(U 12 -T 12 ), .... (91) 



giving U 12 =T 12 , if the p's are unequal. But this property is 

 true whether the p's be equal or not ; that is, XJ n =T U when p, 

 is a repeated root. Various cases of the above are discussed 

 in 'The Electrician,' November 27 and December 11, 1885, 

 with special reference to the dynamical system expressed by 

 Maxwell's electromagnetic equations. 



The following applies to Maxwell's system, using the equa- 

 tions (4) to (10) of Part I. (Phil. Mag. August 1886). A 

 comparison with the above is instructive. Let E,, Hj and 

 E 2 , H 2 be any two systems satisfying these equations, with no 

 impressed forces, or e = 0, h = 0. Then the energy entering 

 the unit volume per second by the action of the first system 



