n. ELECTRICAL RHEOMETRY. 13 



' j^^ ihiiR sin, a j-^ ^i 2 52 ^1 ^2 ^ij . 



likewise 

 (15) (y 4Z;ni2j;os^p2£i_2i^7r> — c^i;n 



Avhereby we can deduce the resultant 



4) = n/ X- + Y^ + Z"-, 

 whenever we know B, m, n, p, that is, the radius R, the distance below the centre, 

 the length of the needle, and its deviation from the plane of the current. In the 

 particular case of the middle of the needle being in the centre of the circle for an 

 angle of deviation = 5, and a length I, we shall have ^^ = 0, m = Z cos. S, n = I 

 sin. S, a ^ 0, h^ = R I cos. 8, 



2 A Rl COS. S 



^ ^ R' + T' + 2Rl COS. 6' 

 and the three (15) are reduced to the two last components Fand Z, as is evident 

 on comparing the last formula of Savary, § 2. Further observations on this point 

 will be made hereafter, when the force of the other pole and terrestrial magnetism 

 will be taken into consideration. 



§ 4. Differential Formulae for Elliptical Currents. 



"When the current surrounding the needle runs in an elliptical conductor, the 

 conical surface (§ 1) will be that of an elliptical cone ; but as this may have either 

 double or simple obliquity, we shall suppose it to have only the simple, in order to 

 avoid a complication, which would lead to no advantage. We shall therefore 

 suppose the axis of the cone inclined to the transverse axis only, and lying in a 

 plane perpendicular to the plane of the curve. 



We must first find the value of the perpendicular line A, which measures the 

 height of the differential element of the conic surface. 



Following the same proceeding as for the circle : K 0, PI. I. fig. 4, be the centre 

 of the ellipse, A the pole, AB the height of the cone, tT a tangent line to the ellipse 

 at E, drawing from A, a line J.P perpendicular to this tangent and joining BP, we 

 shall have 



(1) AP~ z= AB'- + BP\ 



Letting fall from the centre a perpendicular OM to the tangent tT, it will be 

 parallel to BP, and the two right-angled triangles TOM, TBP, will give 



BP : OM :: BT : TO; 

 whence 



• BP:=9K-^ = :^{T0-B0) = 0M-B0:^. 

 TO TO ^ ' TO 



