IN CYLINDRICAL CONDUCTORS, ETC. 
97 
Thus, if accented letters refer to the stranded wire coils, the relative values for equal 
inductance are 
(I'/l) 3 = (a'/aj 1 * = (V2UYn/2 
and 
t'/t=V2/s u {(l’2U) p n/2}\ 
in the extreme case, when s = 81, p = 4. n — 3-3, V = 1 -58 1, a' = 2 -5a, t fr = 0*32. 
(25) Modification of Formula when Strands are few in number— The value of the 
field Hj was calculated on the assumption of a large number of strands, giving 
Hj 2 = 8l 2 /d 0 2 . If there are two touching strands each of diameter 8, the field acting 
on one strand due to a current 1/2 in the other is I/<5, and d 0 (the diameter of the 
circumscribing circle) is 28. Hence, for this case, H/ 2 = 4l 2 /d 0 2 . The factor 2/df in 
the formula (84) must therefore be replaced by 1 /df. With three strands whose 
centres form an equilateral triangle of side 8, H x 2 = 4l 2 /3 S 2 and d 0 = 2*155(5, so that 
2 jdf is replaced by 1*55 jdf. 
With four strands in square order, the centres form a square of side 8, H x 2 = 9l 2 /8<5 2 
and d 0 = 2*414(5; 2/df is replaced by 1 -65/df. These new values react on the 
conditions (87), (88), (89) ; the “ shape ” condition (87) is practically unaltered. The 
“ spacing ” condition (88) gives slightly too high a value for D, viz.:— 
S = 2 3 4 
D/D 0 — 0*90 0*96 0*97 
where D 0 is the value calculated by (88) and D the true value. 
The factor 1*111 in (89) must be replaced by 1 *054, 1 *078, 1 *084 when s = 2, 3, 4. 
These differences are small, so that the theory may be safely applied even when the 
strands are few. 
(F). Many-layered Coils. 
(26) Coils of Many Layers .—Let the winding section of the coil be b x c. Let c/b 
be small, and let b be small compared with the radius of the coil. Let there be m layers 
in the depth c and n turns per layer. 
The field at any point in the section will have two components H 4 and H c parallel 
to b and c respectively, which will act independently in producing eddy-current losses. 
As regards H c , the field acting on a single wire is the same as that for a single layer 
coil for which D = b/mn. Thus the added resistance due to the action of H c is 
^tt 2 HG (z) ( mnd/b ) 2 , 
d being the diameter of the wire and R the direct current resistance of the coib 
As regards H 4 , each layer behaves as a current sheet having current density nl/b. 
VOL. CCXXII.-A. 
P 
