On Current Curves. 259 



m K to m K +i t we mnst employ the general solution of (14), which for 

 the interval k, t K +i is 



* = 



in order that the current at the commencement of the interval 

 t K , t K +i may have the same value w v hich | it had at. the end of the 

 interval t K _\t K . The complementary function 



enables us to ensure this condition ; for, by taking the constant 

 A K +i of such a value that equation (15) is satisfied when i = i K and. 

 t = t K , there is no abrupt change in tjie current. The complementary 

 function, in fact, represents tjie, gradual; dying away of whatever 

 excess or defect of current there wpuld be in the circuit when m 

 changes. 



Equation (15) is true for alL values of i- between i* and i*+i ;. and, 

 therefore, enables us to find the time t K+1 at which the current 

 attains the known value H K + 1 /L. 



By changing K into K + 1 we obtain similarly the time t K + 2 at, 

 which the current has the value H K+2 /L, and so on. 



Thus the determination of k +1 is made to depend upon t Kl and in 

 order to make a start we must assume that the value of i- is known 

 for some 'definite value of t. When the number of alternations 

 per second is not great, it is not of much consequence what assump- 

 tion, within reason, is made, as, though the calculated curves will 

 vary with the assumption made, they will all eventually merge 

 into the true periodic current curve at some point which will be 

 exhibited when the first evanescence of Ae~ R '/ mL takes place. 



As this complementary fuiictipn is a continually decreasing quan- 

 tity, it becomes negligible w;hen it is allowed time enough. This 

 opportunity is affprded; when the straighter portions of the (B, H) 

 curve are. reached, and where the points on the curve can be taken 

 further apart. 



When, however, the period of alternatipn is short, in comparison 

 with the time-constant of the circuit, the evanescence of Ae- Ri / m1 ^ 

 4oes not so readily take plape,; and it will generally take several 

 qycles before the current shakes down into its tr,uly periodic form. 

 The preliminary assumption 913 ght therefore in such cases to be 

 ma,de with care, if it is desired to avoid the labour of calculating the 

 first cycles. But, if the per.ipdipity is quick enough, Ae~ R ' /7nL can be 

 taken as a constant, at any rate during the shorter intervals. 



Fig. 2 gives the plotted curve calculated for a circuit consisting of 

 500 turns surrounding an anchor ring, having, a coefficient of self- 

 induction (without the core) of 0'0004 henry, and a resistance of 



s 2 



