Notices respecting New Books. 311 



p is now supposed to be ni, so that the operator q must be 

 expressed in the form P + Qi. This is done by the ordinary 

 algebraical process of expressing q as 



{RK-LS» 2 + i (HS + LK>}*, 



and then expressing this as P + Q^, which is easily done. We 

 have then 



, r -x(-p+%) . . 



=e e~ F z e- — p sin 1lt, 



and, from the nature of the operator e n , this becomes 



V = e e -p - c (sin nt — Q.v), 



the value of the current strength being then easily deducible by 

 another (and simpler) operation. 



There is no doubt that this operational method very greatly 

 curtails the mathematical work in a large number of cases, 

 although it occasionally happens that the process of " algebrising " 

 the operational form of solution involves some troublesome 

 mathematical work of the ordinary (or humdrum) sort. Mr. 

 Heaviside is, however, always at home with this method, while 

 beginners will have to mind their steps ; but they will be amply 

 rewarded for any time and trouble that they take in studying the 

 process. 



A particularly interesting illustration of Mr. Heaviside's opera- 

 tional mode of treatment is the case of a " distortionless " circuit — 

 i.e., oue in which K/L = K/S — at one end of which is applied a 

 voltage varying in any assigned way with the time. It is to be 

 observed that when the distortionless relation E/L=K/S is 

 fulfilled, the general operator, q, becomes rational in p, and we 

 have 



trTTr , P 



1 



q— \/EK + L » where v= -j==* 



Equation (/3) gives in this case Y = e~ x (' / * R+ ;)Vo, where V is the 

 voltage impressed at the origin. If now V =/(«)= any function 

 of the time, this gives 



Y=e /(*~> (>) 



for the potential at distance x at any time t. The current strength 

 at this point is deduced just as simply ; for, if C is its value, we 



know that— + (R + L/>) C = 0, while, since V=e-^V , we have 

 doc 



_= — qV ; so that, in all cases, 



°-(wS> v (f) 



