IN CYLINDRICAL CONDUCTORS, ETC. 
89 
With wire of a given diameter, the value of R'/L depends upon the values of D, a, b. 
If the length of wire (l) is also fixed, a may be expressed in terms of l, D and b, since 
l = 2tt ab/JJ ..(75) 
Writing D = gd, b — we have from (72), (73), (75) 
The minimum value of R'/L is required for variations of g and tj ; the former gives 
the best spacing of the wires with a coil of given shape, and the latter gives the best 
shape. 
Best Spacing .— (76) is minimum with regard to variation of £ when 
f = 3«„G/(1 + F) > . . . ..(77) 
and then 
R'/L = fE(g-J{3G(l + F}»^.(78) 
Condition (77) shows that at the best spacing the proximity losses are one-third the 
skin losses. If the best spacing is not employed, then, writing R'/L = r, and letting 
t oj be the values of r, f when the spacing is best, 
T/r« = i(f/fo) i {S+te/f) 2 }.(79) 
from which the following values are found:— 
£/£ 0 (= D/D 0 ) = 0-6 0-7 0-8 0-9 1-0 1-1 1-2 1*3 
t/t 0 = 1-120 1-053 1-019 1-004 1-000 1-003 1-012 1-023 
i/io — 1-4 1-5 
t/ To = 1-037 1 • 055 
Best Shape. —Keeping the best spacing, the best shape i§ that value of ajb which will 
make (ajbf uJ/X. a minimum. The following values are obtained from the table given 
above for u n and X :— 
a/b = 1-0 1-2 1-4 1-6 1-8 2-0 2-2 2-4 
X/w„* {a/bf = 1-147 1-165 1-172 1-174 1-172 1-167 1-161 1-152 
so that if condition (77) is possible the best shape is ajb = 1-6, and then 
R'/L = 1‘872R (d/l 3 )* {G (l +F) 3 p.(80) 
When z is very large, F = 2, G = \/2z/4:, so that at very high frequencies 
R'/L = 0'557Rz (d/l 3 )% 
or with 
R - ipllird 2 , 2 2 = 2t r 2 fd 3 /p, 
R'/L = 3T 5{fp/ld)K 
VOL. CCXXII.-A. 
o 
(81) 
