II. ELECTRICAL RHEOMETRY. 19 



Z = 



2 ^ /{ah — hm) g^ — 2g [ah — hmg) dn? ^ 



1 ( 



U 



+ {ah — {(f + gr) hni) sin} ^\d^ 



/~A' 



Making, for the sake of brevity, in the value of X, 

 3£=q^ + g 



N=2(fc^—{q^ + g), 

 p=2g^c^-{q^ + g)-c^g-; 

 and in the value of Z, 



M^ = ab— {q^ + g) bm, 



N' = {ab — bmg) {I — 2 c" g) — bmq\ 



P'=^{l — <fg) {{ab — bm) {I — c' g) — 2 h)nq'\ ; 

 and in both 



{2eayfq-{q^ + l)l ~ ^' 

 we shall have 



hn 



(1/ 



Q 



/ 3P sin} <p , N^d<p _P^d^\ 







and taking the complete functions, as in § 3, 



X=-2Q(^{F^-E^)-^r+-^^E\ 



bn\ c' ^ >^ c' c* b} r 



in which b^ is the complement of the modulus in order to distinguish it from the 

 semiarc, b. 



II. In the same way we shall find X, Z for the case of both roots being positive, 

 but, as it offers no difficulty, we shall only proceed to enlarge a little on the third case. 



III. In the case of imaginary roots, the supposition w = 90° — 2^' and (11) give 



Sin. co = cos. 2^ = 1 — 2 sin} ^ = 1 — 2 ^i {I— cos, a) 



J. (1 + COS. (t) + (U (1 — COS. a) 

 _ J. (1 + COS. a) — jtt (1 — COS. a) 

 J. (1 + COS. cr) + |ti (1 — COS. ff)' 



j~, • o 1 o 1 • ) 2 "^ A u ■v/ (1 — COS. a) (1 + COS. a) 



Cos. CO = sin. 2'^ = 2 cos. 4- sm. ■4' = — -~- ^^ ^ -/^ ^ 



J. (1 4- COS. (t) + ju (1 — COS. a) 



2 "^A [I . sin. a 



A{1 + COS. a) + ^ {1 — COS. a) 



