Lifting Power of the Electro- Magnet. 



501 



As the magnet has cylindric arms, BEG is a circle ; £', therefore, — rdrd6dl, 

 transporting the origin to the centre, we have 



fxFdcdl (b — r cos 6) . rdrdO 



dM = 



(1) 



{b- + r' + 2^ - 2br cos e)i ' 



Integrating this for 6 from to 27r, and for r from to r', we obtain the mag- 

 netic ibrce of a slice of the magnet whose thickness = d/, due to the action 

 of ^. 



This, however, assumes that each molecule is susceptible of magnetism up 

 to the full influence of the current on it, which can scarcely be the fact. Those 

 nearest the helix being most excited, must tend to induce polarities opposite to 

 their own on those next within, on which also the direct action is less ener- 

 getic ; and we may, therefore, expect to find a zone of intense magnetism suc- 

 ceeded by one weaker, null, or even reversed, followed by a series of similar 

 alternations. This does occur in compound magnets to a great extent ; and is 

 manifest in those experiments of Pliicker, which prove that a mass of iron is 

 less attracted than filings of the same metal, and tliese less than powder of iron, 

 more sparsely distributed by being diffused through lard. Of course the same 

 inductive interference occurs in the case before us ; but we know too little of 

 its laws to be able to introduce it into the calculation. 



The first integral belongs to a class which presents considerable difliculty 

 when its modulus is so near unity, as must be the case with the innermost 

 spires ; and among the methods of approximation which have been devised by 

 Euler and Legendre, none, on the whole, are as convenient for my purpose as 

 the common development by the Binomial theorem. Let ¥ -V r- -\- z^ — m', 



rdr = udu ; and expanding ( 1 2 — ) > ^"^^ omitting odd powers of cos 6, 



because the terms introduced by their integration vanish between the limits, 

 we obtain 



X —5- — 1 cos^ 

 vr 



dM= 



fxFdcdldude 



b-irA X 

 + Cx 

 ■vEx 



u- \4 



(2b)V' 



&c. 

 3t 2 



9 2P_ 



13 26- , 



12+1^-1 

 &c. 



cos^ 6 



cos^e 



