MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 27 



to Miss Frances Thorndike for the accuracy attained in the numerous 

 curves of Figs. 1-5 and 8, which should make them of practical value 

 in numerical calculation. 



Mathematical Appendix 



Additional mathematical results which have been employed in 

 connection with the figures and discussion of this paper are brought 

 together below for convenient reference. 



Formulas for Fig. 2 



2s= P +d + l,ti 0A = \, 0a = P , 



d i = l-\-p i -2pcos0, 

 sin 8 = dsln (0+<£), 



sin = p sin (0 + 0), — L ■£ ' j 



cos6=(2s-l)-2s(s-l)/p, Fig- 12 



3 =f(p,0)=pf(p- l ,6) 



= cos0(log-^+plog-^-) (20) 



s — P s — I 



(21) 



A . tan%(6 + <p) , . 1 



= COS \ Og , n ' + p log i-T i— 



{ B tan§0 H 6 tan|0tan|0 



= 2 cos log ^~ = 2 cos 6 log ( 1 + 1 /sin 1 0) , if p = 1 , 



i i 2 + p ■ i 9 , 2 + p .. 

 I — p p 



= cos0|^plog-^ -(l-p)log(l-p)J 



, p(l+2p) , 



+ 12(1=P) + '''' P< ' 



= cos/9[plogp-(H-p)log(l+p)]-j^p^/3« 



p(ll + 19p + llp 2 ) 



96(1 +p) 3 P "I" • • • i 



9 = 2 _ c -[Ar+(i-p)iog(i-p);/2p i (22) 



Vp 



