22S Prof. E. Taylor Jones on most Effective Primary 



The angle <£ can be calculated from the ratio of the 

 frequencies, which is given by the equation 



n 2 2 14 u+ V(l— u) 2 + 4Jc 2 u _ m* 



ni ~ 1-tu— \\l — u) 2 + U 2 u 



In the present problem, in which the primary capacity Cx 

 alone is varied, we have to determine the conditions in which 

 U sin (f> is a maximum, u varying while k is constant. 



The function U has a maximum value of 1/k at u=l — k 2 y 

 and there is a series of values of h for which the maxima of 

 U and sin cj> occur at the same value of u. If h has one of 

 these values the optimum primary capacity is that which 

 makes u equal to 1 —k 2 . The necessary values of k may be 

 calculated from (11) by putting in this equation m = 1 — k 2 , 

 and n^h successively equal to 3, 7, 11, . . . The first four 

 values of the series were given in a recent paper* by the 

 writer, in which it was shown that this series of adjustments 

 has the further property that in any one of them the system 

 has unit efficiency, i. e. the maximum electrostatic energy in 

 the secondary circuit is equal to the initial electrokinetic 

 energy (JL^o 3 ) in the primary circuit, on the assumption of 

 negligible resistances and perfect suddenness of the inter- 

 ruption of the primary current. 



If k has not one of these special values the maximum 

 value of U sin <f> does not occur when 4> is exactly tt/2, i. e„ 

 when the maxima of the two waves occur simultaneously. 

 If, however, one of these coincidences occurs at a value of u 

 not far from 1 — k 2 , dXJ/du is small, and this coincidence 

 determines very approximately a maximum of V 2 . The 

 greatest maximum secondary potential for any value of k 2 is 

 then determined by that coincidence which comes nearest to 

 the maximum value of U. The matter may be illustrated 

 by the curves shown in figs. 1 and 2, in which the full line 

 represents values of U sin <j>, the broken line those of U, for 

 various values of u. The full-line curve therefore shows 

 how the principal maximum secondary potential changes 

 when the primary capacity alone is varied. The points of 

 contact of the two curves determine the coincidences, cor- 

 responding to (£ = 7r/2. 



Fig. 1 refers to the case in which k 2 , the square of the 

 coupling coefficient, is equal to 0'768, the value for an 18-inch 

 coil with which the writer has experimented. In this case 

 the maximum value of U occurs at u = 0*232, and the greatest 

 maximum of IT sin </> at u=0'll (?? 2 /?2 1 = 6*801), i. e. nearly 



* Phil. Mag. xxix. p. 3 (1915). 



