Lifting Power of the Electro-Magnet. 



506 



It dppears from the second column of this table that the power of a ring 

 decreases with an increase of its diameter, very rapidly at first, but more slowly 

 afterwards, so that its action continues sensible to a considerable distance. But 

 out of its plane the case is different, the total effect is much less ; but if z 

 have any considerable magnitude, it increases with the diameter of the ring. 



The case of a spiral is, however, not that of most ordinary occurrence, the 

 wire being generally disposed in a helix. To obtain its effect on the magnet's 

 element 4, we substitute for dc in (4) the differential of the helix. In this 

 curve if e = the slope of the wire. 



z = h6y.X.axie, dc = 



dz 



sm e 



as, however, the curve is inclined and its induction is in a plane perpendicular 

 to it, dc must be resolved in the direction of its base, and we have 



(ir- 1.3.5. h'r * ) 



•{2^+ 2.2.4.M' +'^''"J 



fiF.Ay.dz Ibr- 1.3.5. b h* 

 '^^- tane '^)2^+ 2.2.4. 



Putting 6- + r^ = <^, the integral consists of a series of terms, 



1 2n-4 



J ^, = b"-'yi^z{ 



{f + z')— 



2n- 



, + ; 



3.t-. u"'-'^2n -3.2n-5.t' . u'"-' 

 P.2n-2i 



2w-(2j + l)<^M^-(=""') 



which vanish with z, and therefore require no constant. 

 The series terminates when i = n, and its last term is 



b"-' { 2.4.6 .. . 2n-A 



Qn = 



: 21(1.3 .: 



H 



l.t' 



The term preceding this is obtained by multiplying it by ——^ ! ^^^^ next 



3 .f^ 2n 



by the additional factor ^ ' , , and so on till the last factor is 



■5.t' 



4 . It- 

 Having obtained Q„, the next term. 



2tt-4.M^" 



Q».2=Q-.x 



2m-2.2w r'b' 

 2n + l . 2n-l^T' 



