11. ELECTRICAL RHEOMETRY. 15 



The value of r^ in this case, according to the notations already given, and the 

 positions of the assumed axes, becomes 



r' = [x — my + 2/" + '*^ 

 that is, making also here 



(5) P = VT? + W^, 



and substituting from (3) for x, y, 



(6) r =z V U^ + f — 2 am sin. ^ + (c^ — li^) sin? ^-j 

 therefore the expression for the force exerted by an element of an elliptical current 

 on a magnetic pole will be 



/.jrx kAds hd^. Vri? {a^ — {a? — If) sin? ^) + {ah — mh sin. ^f 



^ ' '*'~ ^^ ~ (62 +J2 _ 2 rtia sin. ^ + (a^ — h^) sin."" <j>) i 



In order to find the components, we shall take the equation of the surface of an 

 elliptic cone, wliich is 



(8) /ti = 6" {mz — nxf + c? (pz — «?/)- — a- W {z — nf = 0, 

 which, supposing ^ = for the case of simple obliquity, and after differentiation, 

 making z = 0, because all the points of the curve lie in the plane YX, will become 



^ = 2b'n'x = 2h''an^ sin. €>, 

 ax 



4- = 2 a^ n^ 7/ = 2 a" b n^ cos. A, 

 dy "^ ^' 



~ = — 21? mnx + 2 cr h- n = 2c? h- n — 2 6- amn sin. A, 

 az 



hh z=in^ a^ I? \j? (a? — (a^ — I?) sin? ^) + {ah — hm sin. ^f] ; 

 hence the components are 



/»2 n 



j^ _ r 1 5ff j hhn sin. q) d^ 



~'^ Ic^' dx~ I (f + b^ — 2 am sin. A + (d^ - 



Y= — 



Z = 



{I? + W — 2 am sin. ^ + {d^ — 6-j sin? ^) - 

 



k a n COS. ^ d^ 



{P + b^ — 2 am sin. ^ + {a^ — b-) sin? <^)- 







2 7t 



h {ab — bm sin. ^) d ^ 



{P + b^ — 2 am sin. ^ + {a- — h^) sin? ^) * 







in which the limits of the integrals are and 2 n. 



§ 5. Integration of Formula for the Ellipse. 

 The expression 



^^ ~ {P + b^ — 2 am sin. o + (a' — W) sin? a^ 



