416 PROCEEDINGS OF THE AMERICAN ACADEMY. 



A complete series of stationary vector- diagrams might be presented 

 following those of Figure 17, and corresponding to those of Figures 7 

 and 12 ; but in. view of the relative simplicity of formulas (78) to (88), 

 such vector-diagrams have more theoretical than practical interest. 



Apeeiodic Case. 



When p, the semi -resistance of the circuit, is just equal to the surge 

 impedance c,, = Vis of the same ; then the circuit is aperiodic, and 

 there is neither a circular angular velocity w, nor a hyperbolic angular 

 velocity O. The aperiodic case may also be represented by a special 





P 



1 <*> 



^ * p ^ at Q 



Figure 19. Aperiodic case as Figure 20. Aperiodic case as 



limiting condition of ultra- limiting condition of o. c. cir- 



periodic circuit, when wq = ^. cuit, when j- = ojq. 



i2 stationary vector-diagram. w stationary vector-diagram. 



rotating vector-diagram as has been suggested by Macfarlane ; but it is 

 easier, for practical purposes, to treat it as a limiting case of the ultra- 

 periodic circuit. We have by (78) 



u = Uo cot i/' €-^i sinh (nt + gd-^4i). volts (89) 



Now let n become very small, as in Figure 19. Consequently, \p be- 

 comes very small; so that coti/' may be replaced by \/\p, and 

 sinh(0^ -h gd-^^) by {m + gd'^ij^). 



Hence U4,=o = ^ r'' (^t + gd'^ip). volts (90) 



But Q = -Li}/ and r/rf~V — ^ when i// approaches zero ; thus 



M^=0 = —^ e--t (x^pt + i//) == U^^ {-It + 1), volts (91) 



and q^^Q=i Q^r'-i'^-it -\- \), coulombs (92) 



i^=Q = — -^ = Q^^-W'-K amperes (93) 



