656 BELL SYSTEM TECHNICAL JOURNAL 



The use of a table of pairs may also be stated in another and some- 

 what more general way as follows: 



For any system where the principle of superposition holds, any cause 

 C{t), its effect E{t) and the corresponding admittance F(/) are con- 

 nected by a relation which may be written in any one of three ways 

 which explicitly express each of the three quantities in terms of the 

 remaining two, as follows: 



E{g) = ^JclY{f)^lcC{gn 



C{g) = en 

 Y{f) = 



Y{f) 



diiEig) 



euc{g) ' 



where 5)7? is read "mate of." 



The use of coefficient pairs may be most simply illustrated by 

 reference to Figs. 3 and 4, in connection with the problem of finding 

 transient currents through a perfect condenser of unit capacity due to 

 impressed electromotive forces shown by each of the seven curves on 

 the right considered as functions of the time. Any curve on the right 

 being the cause, the next curve below it is the effect, considering Fig. 4 

 to be placed above Fig. 3. In the solution the first step is to find the 

 mate of the curve on the right. This is the curve on the left. This 

 mate is then to be multiplied by the admittance of the system which 

 is p for a unit condenser. Reference to the titles of the figures shows 

 that this product is given by the next lower curve on the left. To find 

 the mate of this last curve is the third step in the solution and for this 

 it is merely necessary to go to the curve on the right. The three steps 

 then take us from any curve on the right to the next curve below it. 

 Figs. 3 and 4, taken together, are a section of an infinite sequence of 

 pairs which illustrate an infinite number of possible transients in a 

 perfect condenser of unit capacity. 



If, on the other hand, the system consisted of a perfect reactance 

 coil of unit inductance and the impressed cause was again shown by 

 any curve on the right, the effect would be shown by the next higher 

 curve, assuming that the initial current at the beginning of time was 

 that shown by the extreme left of the upper curve. Thus, when the 

 cause is oscillating, there is one less half oscillation in the effect than in 

 the cause. This is for an inductance. For a condenser, conditions 

 are reversed; the effect has one more half oscillation than the cause. 



The scales of Figs. 3 and 4 may be changed to correspond to any 

 value of a, the parameter which appears in the coefficients of the pairs. 



