658 BELL SYSTEM TECHNICAL JOURNAL 



tances. In the third, fourth and fifth columns, 85 transient solutions 

 are given of which 39 are for the unit impulse, 30 for the unit step, and 

 16 for the suddenly applied cisoid. 



The causes producing the transients in Table II are but three in 

 number: the unit impulse, the unit step, and the suddenly applied 

 cisoid; and the mates for these causes are unity, p^^ and {p — po)-^ 

 as is shown by pairs (403*), (415*) and (440*). Multiplying these 

 three mates by the admittances and taking the mates of the products, 

 we have the effects, as is stated in the headings of the last three columns 

 of the table. 



To illustrate in detail the steps involved in finding a transient effect 

 with the aid of Table I, consider system No. 14 of Table II with the 

 cause equal to the unit step ^_i(0, X = |. The mate of the unit 

 step is p~^ by pair (415*). Multiplying this by F(/) as given in the 

 second column of Table II, we have up~^l + Vp/X)~^ for the cisoidal 

 coefficient. By pair (551) the mate of this is mVx exp(Xg) erfc VXg, 

 < g. Substituting for g the actual variable t, we have the transient 

 solution as given in the fourth column and fourteenth row of Table II. 



This simple example fully illustrates the three essential steps in 

 finding any transient effect when the admittance and pairs are known. 

 In this example the effect was considered to be the unknown. If 

 either the cause or the admittance were the unknown, the same pairs 

 would be involved but the two coefficients in a pair would be used in 

 the reversed sequence in all but one instance. 



There are still 32 squares of Table II left blank. It would be a 

 simple matter to place series solutions or integral solutions in each of 

 these squares. Thus if the impulse transient of column 3 is known, 

 the other two transients are given at once in integral form by pairs 

 (210) and (219); if the unit step transient of column 4 is known, the 

 suddenly applied cisoidal transient is written immediately in integral 

 form by the use of pair (220). The real problem is, however, either to 

 find closed form solutions in terms of known functions or to show that 

 this is impossible. When the failure of known functions has been 

 established, we should next consider the choice of new functions so 

 defined as to throw as much light as possible on the new solutions. 



Table II may be regarded as another table of coefficient pairs. 

 Column 2 contains cisoidal coefficients; column 3, the mates of these 

 coefficients; column 4, the mates of these coefficients when multiplied 

 by p~^', and column 5, the mates of these coefficients when multiplied 

 by {p — po)~^- The corresponding pair in Table I is referred to in 

 the lower left-hand corner of each square by its serial number. In a 

 few cases, two or three pairs are referred to and there it is necessary 



