338 Mr. 0. Heaviside on the 



the size of a particular normal system p 1} will be given by 



A _ UqI ~~ ™Q1 _ Uqi — T 01 _ Uqi~ T 01 naA\ 



'"Uu-Tn-aCUx-TO - p2 # • • < 104 ' 



'dpi 



by the previous, if U i be the mutual electric energy of the 

 given initial state and the normal system, and T 01 similarly 

 the mutual magnetic energy. 



And when we increase the number of variables infinitely, 

 and pass to partial differential equations and continuously 

 varying normal functions, it is, by continuity, equally clear 

 that the decomposition of the initial state into the now infinite 

 series of normal functions is not only possible, but necessary. 

 Provided always that we have the whole series of normal 

 functions at command. Therein lies the difficulty, when there 

 is any. 



In such a case as the system (71) of Part II., involving the 

 partial differential equation 



d 2 V dV d 2 V 



S= ES f +LS S • • • < io5 > 



wherein E, S, and L are constants, to hold good between the 

 limits 2 = and z=l, subject to 



V = Z Cat.£:=0, and Y=Z X C at x=l, 



there is no possible missing of the true normal functions 

 which arise by treating d/dt as a constant ; so that we can be 

 sure of the possibility of the expansions. Thus, denoting 

 RSp + LSp 2 by — m 2 , we may take the normal V function as 



u= sin (mz-\-0), (106) 



and the corresponding normal C function as 



w= + ^^ = + §£cos(m* + <9). . . (107) 

 mr dz m v 7 



Here 6 will be determined by the terminal conditions 



^=Z at*=0, -=Z ia t£=Z, , . . (108) 



and the complete V and C solutions are 



V = SAm6*", C = 2At0eP« .... (109) 

 at time t ; where any A is to be found from the initial state, 



