Equation of Electrical Flow. 



429 



It thus appears that the variables in the problem of elec- 

 trical discharge under consideration may be represented by 

 the projections of three sides of 

 a triangle, which is constantly 

 undergoing uniform rotation 

 and linear logarithmic shrink - 

 ing. Let the figure represent 

 a portion of the appropriate 

 curve whose characteristic angle 

 is /3, and let R be some radius 

 vector. Then the projection of 

 R on Y will be a maximum when the tangent at R is parallel 

 to X.- Let P be such a position, and let P M be the 

 tangent at P. Then MPO = POX = /3. 



Now suppose A the line representing (in its projection) 

 the effective electromotive force about to change sign through 

 the value zero. This means that the current is about to change 

 sign, and the condenser having been receiving current is 

 about to begin to be discharged, i. e. its charge and therefore 

 potential difference is a maximum. Then the line of P.D. 

 must make the angle P A with the line of effective E.M.F. 



Again, the E.M.F. of self-induction is zero when the current 

 is at a maximum, by the nature of the ordinary hypothesis. 



Therefore, when the line representing E makes an angle /3 

 with X, the line representing induced E.M.F. (F) must be 

 parallel with it. Hence it also makes an angle {3 with the 

 line of effective E.M.F. ; but in phase lags behind it, whereas 

 the P.D. is in advance by that angle. 



Thus, if on any line taken as base we construct an isosceles 

 triangle of appropriate base angles, the sides will represent 

 the P.D. of the condenser and the induced 

 E.M.F. of self-induction respectively, and 

 the base will represent the effective E.M.F. 

 It only remains to rotate the triangle with 

 appropriate speed and to allow it to shrink 

 at the due logarithmic rate. 



The properties of the triangle agree exactly 

 with the electrical properties. 9 



The angle ft is such that tan /3= ^h ; 



therefore 



cos' 2 & = 



■♦(?)' 



KR 



2 _KR 2 

 2L ~ 4L 



R 



Phil. Mag. S. 5. Vol. 35. No. 216. May 1893. 2 G 



