CIRCULAR COILS OF RECTANGULAR SECTION. 423 



2. It is well known that the self-inductance of a single circular conductor with 

 rectangular section for uniform current density is given by 



i ffc el* ri-Y ri*-x 



~ I I 



Mr J -jc J -a* J -S-Y J -ifc-x 



and for a coil of 71 turns is n" times this if MAXWELL'S correction* for space between 

 the wires be neglected, l> being the breadth and c the radial depth of the rectangle. 



The evaluation of this definite integral, even for the second order terms, has 

 presented considerable difficulties. The first correct result to this order was that 

 given by WKiNSTEiN.t So far as I am aware no one has published a determination of 

 it to the fourth order. By a method indicated in the appendix to this paper the 

 integration can be carried out to the fourth order without difficulty and to still higher 

 orders if desired. 



Thus the following expression for L has been obtained 



2 11 . 3 . 5 . T. 



23 6fr 4 7&V , 



or H . 2 . w 



28 21 



365906 4 -20356V-11442( 



2 3 .3.5.7 

 in which 



a = mean radius, 1> = breadth, c. = depth, d = jV + c 1 = diagonal of the rectangle. 



3. If r be the G.M.D. of the rectangle from itself, it is well known that 



log r = log d <{> 



where 



* -Elect, and Mag,' vol. II, 693. ROSA (see Bull. Bureau Standards/ vol. 3, p. 37, 1907) ha, 

 greatly improved on MAXWEr.L's correction. 

 ' t ' WiED. Ann,' 21, p. 329, 1884. 



