CIRCULAR COILS OF RECTANGULAR SECTION. 435 



The term of the sixth order in the series for L is the value of the integral 



between the specified limits, and I have found it to be equal to, 

 2". *5. 7 a 5 [( 525y - 161 OfrV + 7706V + 108c) log ^ 



+ (^jjf- V-3220&V+2240&V 4 ) ti, 



OKQ /c 7 



Ijj. c r _ 2 " ( | ft" _ 46 V + 1 6V j IT, 



2161453 ,, 617423 M , 8329 M 4 . 4308631 

 " 2 3 .3.5.7 ~ 271^5 6 ~2T3~5 6 + 2'.3.5.7 j 



in which w, c, w, and ti have the significations assigned to them in 2. 



The method of integration indicated in this paper renders the determination of L in 

 series form comparatively easy for the special cases of a solenoid (c/b = o), and a flat 

 circular ring coil (b/c = 0), uniform current density being assumed. 



Thus COFFIN'S formula* for a solenoid can be easily obtained, and KAYLKwnt and 

 NIVEN'S formula for a coil whose axial dimension (b) is zero can be extended to the 

 sixth order, giving 



[7. c 3 lie 4 103* 8a 



L = 4^'a 11 + jtf + 2 . 3 .. 5 . + 2'.3.5.7. 



43cr c 4 429H57i><-' 1 



7 . 3*. a 3 2 s . 3 . V. a 4 2". 3'. 5'. 7'. "J ' 



which can also be obtained by putting b = in the general formula obtained above 

 for L, and remembering that when b/c = 0, o = 1, and w' = 1. 



* 'Bull. Bureau Standards,' 2, p. 113, 1906. 

 t RATLKIOH, ' Collected Papers,' 2, p. 15. 



3 K 2 



