E+ 2*3-^+^3^=0. 



Equation of Electrical Flow. 427 



equation may be written 



<m am 



dt +tlh M 



Tn this expression the two time-constants may be considered 

 to be independent. 



To obtain a geometrical representation of the changes : — 



(1) Suppose a line, whose length is r, to shrink logarithmi- 

 cally so that its change is represented by the equation 



dr _ r 



di~~T l J 

 where ti is a time -constant. 

 Then 



where a is the value of r at the beginning of the time, and t l 

 appears as the time taken for r to shrink to - of itself. 



(2) Secondly, suppose a straight line in a plane to con- 

 stantly change in direction at a uniform rate, in the same 

 sense. If is the angle measured from a fixed direction, 



de = 2ir 

 dt ~ t 2 ' 

 where t 2 is a time -constant. Hence 



Whence t 2 appears as the time required to describe 27T. 



(3) Suppose a line to undergo both the changes contem- 

 plated, which is possible, since one is a change of length, the 

 other a change in direction. Then, eliminating the time, we 

 have a 



2nty 



r — ae t , 



or r = atf"~tan0 9 where tan /3 = 1 . 



t 2 



This is the equation of the equiangular spiral, with the 

 characteristic angle /3, whose value merely depends upon the 

 two time-constants t x and t 2 . 



(4) Now imagine this length r constantly projected en 

 some fixed straight line, and for simplicity take this straight 

 line as at right angles to the direction in which = 0. 



Then the projection under consideration (E) has for its 

 expression & 



E = « . e tan £ sin 6 ; 



