11. ELECTRICAL RHEOMETRY. H 



Sin. cj = 2 sin. ^ cos. (p cos. a — sin. a + 2 sin.^ <^ sin. a 

 Cos. CJ = — 2 sin. (|) COS. ^ sin. a — cos. a + 2 sin.^ ^ cos. a 



R {R — p sin. a — m cos. a) = R (R + (1 — 2 sin."^ ^) (j) sin. a + m cos. a) ) 



= R'^ + h^ — 2hUin.^^; 

 therefore the expression of the three components will be 



7t 



X ^ I 2 k7i R {2 sin. ^ cos. ^ cos. a — sin. a + 2 sin.~ ^ sin. a) d^ 



(5) (Y = I 2 h n R ( — 2 sin. ^ cos. ^ sin. a — cos. a + 2 sin.^ ^ cos. a) d <p 

 _ __ , 



7t 



2h{R^ + h^ — 2 h^ sin.^ ^) d^ 

 where for brevity we have made 



(6) A = v/ 1 — c~ sin,:^ ^. 



To integrate these formulae further transformations must be made, only in order 



to reduce them to more simple forms. We have 



1 _ c^ sin. ^ COS. ^d^ ihr^rf^fav^ f sin.^c o s.^d^ _ 1 

 '^ ^ A ~ A'' , tueieiorej ^3 — ^2"^- 



We have likewise 



/OS r — c^sin.^^d(p r(l — c^sin.^^ — l)d^ 



^ ^1 — c^sin.^(p 



= f-l^£jt=ld^-f-^Jl= = fAd^-fii. 



^ ^l—d'sin.'^ ^ ^1—c'sin.^^ -^ ^ ^ A 



Hence 



(9) S"^^ — S\i.i^^s% =?(^-^'' 



making with Legendre 



Again differentiating the quantity — ^^-- — '-^, and reducing afterwards the terms 



to the common denominator A'', and taking away the common factors in the last 

 fraction, we shall have 



^ ^ A ""W~~A^~~ A / *^' 



which by the algebraical division and convenient reductions will give 



(11) <« "^^ = [s(i-?) +,4 (1 -.'.«.•*)] <^*. 



Making 



(12) b'' = l — c% 



