Lifting Power of the Electro-Magnet. 



which destroy each other, except the term 



Fb"*' . 2n+l ■/■""- 



which belongs to the exponent n + 2. If the term of ?i = T„, 



Pb"-' r" 



503 



T = 



-* n — 



and the next. 



-' nt2 — ^ n X 



2n-l.M-"-»' 

 2n + l.2n-l.b\r^ 



Vi) 



n . n + 2 . u* ' 



from which the successive terms of the integral C which all vanish when r — 0) 

 are easily formed. We thus find, calling the section of the magnet ■nr'^dl — A , 



{ b 1.3.5. bV 1 . 3 . 5 . 7 . 9 . 6= . r* 



dM=ixFA 



2u^'^ 



2 . 2 . 4 . m' "*" 2 . 2 . 4 . 4 . 6 . m" 



1.3. 5 .7. 9. 11. 13.6'.?-° 

 + 2. 2. 4. 4. 6. 6. 8. m'^ "'' 



' X dc. 



(4) 



This converges sufficiently, unless b is nearly =r ; then, notwithstanding the 

 simplicity of the law of continuation, the computation is tedious. But as soon 



as 71 is so large that -, — ^— may be neglected, it can be nnich simplified : for, 

 4^y. ^ n' + 2?i ^ ° 1 . ' 



calling — j— = p, (3) becomes 



n + 2 

 and ,c being the number of steps, 



« + 4 



n + 2x 



(•5) 



Even this is too slow ; but it enables us to compute x terms per saltum. The 

 sum of them is (m = ^n), 



"Im + l 



p'ni p^in 



"'■«i + 2''"m + 3' 



m 



~xV 



or developing and arranging according to the powers of — , and putting 



m 





p-^2p'+. . . .+Xp^: 



(i-py 



|l-(.i-+l)p-^ + .rp-*' 



■B, 



