MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 29 



primed points are all in this plane, and we agree that A' or A occurs 

 in a term, according as P' or P is found in the subscript of the symbol 

 A or T used to designate the term, where 



Ap . p , = - A'a' + A'b' + B'a' - B'b' , (31) 



(A'b' + Fp'KB'a'+P'p') ,_ 



T r*- l °S (A'a'+P'p'KB'b'+Fpy ^ 



In these expressions every distance between points, such as A'b', 

 P'p', is a positive quantity. The formulas below are perfectly general, 

 but require the assignment of the capital letters JC' , A', B', P' to the 

 upper plane when both flows are above the earth and to the lower 

 plane when both flows are below the earth. They are most readily 

 checked by employing formulas (4a) and (24) in formulas (17), (18) 

 and (19). The results show that the mutual inductance is equal 

 to the Neumann integral between JC' and * augmented by terms 

 which depend only upon the arithmetical distances between the eight 

 points A', B', a', V, A, B, a, b. 



N(W-e)( u ,- e )=Njtr' J ,' + P'p'r P ' P ' + 2Pp'rp p -Ap- p * + Arp, 



where P'p>Pp', (33) 



= ^V + 2Zlo g [^,ifP<a„d,'are 



both at height Z, (34) 



= Njr'J+2Z log (1 + s 2 ), if A' B'b' a' is a 



horizontal rectangle and AB = s(Aa), (35) 



= iV^-V + 2Z[logi(l + Vl+^) + l-Vl+^ 2 + ^], 

 if A 'B'b' a' is a vertical rectangle with one 

 side on the earth and AB = s {A'a) = sZ, (36) 



N&-f){u-*) = NjV'J+P'p'Trp' - 2P'pYp> p - 2Pp'(T PP ' - T Pp ) 

 — Apy + 2Ap' P + 2Ap p > — 3Ap p , 



where P'p>Pp', (37) 



-Nt> - ~ zv , AAa){Bb){Ab'+Z)HBa'+Zy~ 



+4;(Ap> P — App), if P' and p' are both 

 at distance Z below the earth, (38) 



