76 
MR. F. E. SMITH ON THE ABSOLUTE MEASUREMENTS OF A 
Calculation of M 2 , the Mutual Inductance of the Coils, and the farther Brush 
Contact Circle. 
The following formula, due to Rosa and Grover,* gives the mutual inductance 
of a single layer coil of length x and a co-axial circle of radius A in the plane of one 
end of the coil:— 
M = 
2t rWN 
d 
, , 3a 2 A 2 , 5a 4 A 4 ^ , 35a 6 A 6 v , 63a 8 A. 8 v 
I + ~r—— + A-2 + 712 -A-4 + - ■ - - Ag + , 
8 d i 64 d s 
512 d 12 
1024 dA 
where 
X 2 = 3 —4x 2 /A 2 , 
X 4 = 5/2-1 Ox 2 / A 2 + 4x 4 /A 4 , 
X 6 = 3 5/16 — 3 5ce 2 /2 A 2 + 21 tc 4 / A 4 — 4 cc 6 / A 6 , 
N = total number of turns in length x, 
a = radius of coil, 
A = radius of circle, 
x = axial length of coil, 
d — \/x 2 + A 2 . 
When the coil is long this formula is very exact and easy to use, and it was a 
simple matter to calculate M 2 with the precision necessary for our work. The results 
of 16 calculations are given in Table XIII. 
In Table XIV. the values of M 2 are given for coils, the mean diametral planes 
of which are distant from the brush contact circle by the amounts given in column 4. 
The results show the variation of M 2 with change (l) in the axial distance of the 
coils, (2) in the radius of the coils, and (3) in the radius of the contact circle of 
segments and discs. 
The summary on p. 78, Table No. XV., relates to the coils and brush contact 
circles of the Lorenz apparatus and sufficiently indicates how small corrections were 
made when the distance between the coils was varied. The contact circle of segments 
and brushes was practically in the midplane of the coils near to it; the method of 
ensuring this is described in Section 22. 
Correction for Conicality of Coils. 
The increment coefficient q (Table IX.) enables the change of mutual inductance to 
be calculated when the radius of a coil is changed by a small amount, but the change 
must be a uniform one. When the change is not uniform, the change in mutual 
inductance for an increase in radius of any part of a coil must be known, and the 
correction for conicality must be calculated in parts. In the past it has been 
customary to take the radius of a coil as absolutely uniform from end to end, but 
such a procedure invariably introduces errors into the calculation. 
* ‘Bur. of Standards Bull.,’ vol. 8, No. 1, p. 101. 
