II. ELECTRICAL RHEOMETRY. 17 



(7) Cos.-'^ = -l^t^, 

 ^ ' 1 — q^ COS- C, 



and T will become 



f8^ T - 2{l-<fcos.^^r dj_ 



^ ' ' ~" (2 eaf f (f (1 — (i)l COS.' ^ ' A=" 



where c^ = ^ ) ^ J . 

 P (1 — (f) 



3. When both roots of the equation are imaginary, we shall make 

 Sin.' 4' = — u +^v y/ — 1 : 

 making also x/u' + v' = u, cos. B = —JL=, sin. 6 = —J=.-.., the two bino- 



mial factors will be reduced to the trinomial one. 



(9) Sin.^ ^+2^ sin? ^ cos. + [T, 

 in which cos. 6 is positive or negative, according as ij > or it <C 0. 

 Hence we shall have 



m 2d_± 



' (2 eaf [sin.^ ^ + 2 ^ sin:' ^ cos. 6 + (i") |' 



This quantity may be written 



2cZ^ 



rp__ COS.^ 4' _ 



^~ ,o \s , , /sin.*'^+2ixsin.-'^cos.e+ii^\ hin.^'^, + 2 u sin? ■^ cos. 6 + u\ 



(2 eaV COS.* 4^ \ — 4- , ) ^ ^ — j^ ~ 



^ ^ ^ COS.* ■^ '^ N COS.* -^ 



After convenient reductions by formuljE • = sec.^ =: 1 + tang?, making for 



cos? 

 the sake of brevity 



(10) 

 it will be reduced to 



A- = 1 + 2 ^cos.Q + i^- 



^ 2 „ cos.Q+a , . 2 T **"^i- 



Cos.- A = — L whence sm. a = — — 



A A 



rp 2d tang. -^ 



' ~ (2 eaf COS.* 4, {A' tang.* 4^ + 2 A ^ cos?\ tang? 4> +ft')|' 

 Let us now make 



Tang. 4> = 1-^ . tang, h a, 



Whence 



(11) {sin^^L= ^ ^ ^'"•' ^ ^ := f' (^ ~ ^"^- ^) , 



j ' A cos? i a + [I sin? ia A (1 + cos. a) + [i {I — cos. cr)' 



^21 ^ cos? ha J. (1 + cos. a) 



\ ' A cos? la + ^ sin? ha A (1 + cos. a) + ^i {1 — cos. a) 



These substitutions made in T^ will give 



^ "' '"" 4.{2aef{A^)% A*' 



The limits being (T ^ 0, c = 2 7t, having made as before A = "^1 — c^ sin? a, 

 in which (? =z h sin? X. 

 5 



