382 Special Reports 



As will be seen below, the Helmholtz relation diameter = twice axial distance is 

 not exactly satisfied for the spirals of the instrument described here. For our case, 

 if we assume the spirals uniform and exactly alike, with diameter and axial distance 

 equal to the mean diameter and axial distance, it happens that the axial intensity at 

 the center may be calculated as n times that of the central pair with an error less than 



1 part in 800,000. These conditions are, of course, not exactly realized, but in any 

 case the error is quite negligible. 



Furthermore, it has been shown by Lyle, 1 Rosa, 2 Searle, 3 and others that a circular 

 or helical turn of round wire whose diameter is small in comparison with that of the 

 turn, produces very nearly the same magnetic intensity at points remote from it as if 

 the current flowed through a linear turn coincident with the axis of the wire, and this 

 whether the current density is uniform over the cross-section of the wire, or inversely 

 proportional to the distance from the axis of the turn. For the central part of the 

 field of a pair of coils such as we are concerned with here the approximation is exact 

 to about 1 part in 2 X 10 6 . 



13. In order to calculate the constant G of the spiral coils used in this work, there- 

 fore, it is necessary only to know the mean diameter d = 2 a of the spirals and the mean 

 axial distance x 2z between the corresponding parts of the two groups of spirals, 

 to apply the standard formula for the axial intensity at the center of a system of two 

 equal coaxial circles, traversed by unit current, with the Helmholtz relation very nearly 

 satisfied, and to multiply by the number N/2 of turns in each spiral. Thus we have 



2ira*N 4ird*N 



G= (a'+z>) 3/2= (d-f-.T') 3/2 (5) 



In order to show that the field within which the magnetometer magnet is capable 

 of being placed while the instrument is being used is nearly enough uniform, we shall 

 assume at first a system of two coaxial circles. 



The axial intensity per unit current due to a single circle of radius a at a point 

 distant y from the axis and z from the plane, or r = (a* + z*)' A from the circle, is known 

 to be 4 



/ = ?^j l +3y_ 2 ( fl2 _4 22 ) + 45^( + 8z<- 12 a*2>) . . . \ (6) 



r 3 | 4r 64 r 8 ^ 



First assume a true Helmholtz pair with diameter 30 cm. To find the fractional 

 diminution of the intensity in moving 1 cm. along the axis (y = 0) from the center, 

 we may calculate/ for z = 7.5 -f- 1 and for z = 7.5 1, add, subtract the sum from 



2 / calculated for z = 7.5, and divide the result by the last quantity. We thus obtain 

 about 10 parts in 424,000. Similarly, for distances of 0.5 cm., 1.1 cm., 1.2 cm., and 

 1.5 cm., we obtain 2, 14, 20, and 48 parts, respectively, in 424,000. 



Next assume the circles have the mean radius and axial distance for the actual 

 spirals, viz, 14.9518 cm. and 14.9966 cm., as obtained from Tables I and II below. 

 Proceeding as above, we find for distances 0.5 cm. and 1.0 cm. fractional diminutions 

 not greater than about 1 part in 427,000. For distances 1.1 cm., 1.2 cm., and 1.5 cm., 

 the fractional diminutions are 2, 5, and 27 parts in 427,000. 



From these data it is easy to see that the axial variation of the axial intensity in 

 the case of our spirals is entirely negligible, whether we assume the actual mean linear 

 dimensions or the approximately correct Helmholtz dimensions given above. 



' Phil. Mag. (6), vol. 3, 1902, p. 310. 



' Bulletin of the Bureau of Standards, vol. 2, 1906, p. 71; and vol. 3, 1907, p. 209. 



With Ayrton, Mather, and Smith, Phil. Trans. A. vol. 207, 1908, p. 541. 



See Gray 'a Absolute Measurements, vol. II, p. 248. 



