Sine Galvanometer 'AW 



Thus in the first case assume the spirals displaced axially 2 mm., which is far in 

 excess of the displacement from the center which the magnet can ever have. No effect, 

 to 1 part in 427,000, will come from the displacement of those spirals which remain 

 between the planes distant (7.5 1) cm. from the center of the magnet. Of the 

 two remaining pairs of turns, the effect of one will be reduced 1 or 2 parts in 427,000, 

 that of the other 4 or 5 parts in 427,000. 



If we assume the Helmholtz dimensions, with a = 15 cm., none of the spirals but 

 the two pairs last considered will have their contributions at the magnet modified by 

 as much as 10 (or even 5) parts in 424,000. Of these two, the contribution of one will 

 be reduced by about 4 parts in 424,000; the contribution of the other will be reduced 

 by about 10 parts in 424,000. 



In the first case the total reduction of the intensity at the magnet is not more than 



1 part in 400,000; in the second, 1 part in 80,000. 



To calculate the fractional variation of the axial intensity in the central plane 

 normal to the axis for the pair of circles we have only to evaluate the second and third 



9 n 2 



terms within the braces of equation (6). If we write / = - - , give a and z the mean 



r 3 



values obtaining for our coil, and express y in mm., the equation becomes approximately 

 / = / (1 - 13 X 10- 8 y> - 8.7 X 10" V . . . ) (7a) 



The first correction-term in equation (6), however, which is strictly zero for a true 

 Helmholtz pair, varies greatly with the axial distance 2z. Hence, as our spirals are 



2 cm. wide, it is desirable to obtain a closer approximation. For this purpose we may 

 use the method of Lyle (I.e. ante) and consider each complete spiral replaced by two 

 circles of the same radius, a, symmetrical about the mean plane and distant /3 there- 

 from where p- is 1/3- (a"- p 2 ), 2a is the length of the spiral, viz, 2 cm., and 2 P is 

 the diameter of the wire, viz, 0.58 mm., and each circle carries the current of half the 

 turns of the actual spiral. Applying the formula (6) to the inner and outer pair of 

 these Lyle circles, and properly combining the results, we get in place of (7a) the much 

 more nearly true equation 



/ = / (1 + 10 X 10- 3 y - 8.5 X l(r 10 y* ) (76) 



No part of the magnet, when in proper adjustment, extends more than 10 mm. 

 from the center. At a distance of 11 mm. from the center (7a) gives (/ / )//o = 



2.8 X 10" 6 , while the much more nearly correct formula (76) gives (/ f <,)//<, = 



0.4 x 10~ 6 . On account of the construction no part of the magnet can ever be more 

 than 11.5 cm. from the center; and there is no occasion in practice for 11 mm. to be 

 reached. Thus (/ /o)//o will always be entirely negligible. 



14. The method of measuring the overall diameters of the spirals resembles, to 

 a considerable extent, the methods used in the National Physical Laboratory of England 

 by Ayrton, Mather, and F. E. Smith 1 . It is illustrated in Figure D of Plate 9. The 

 spool was mounted with its axis vertical and central on the adjustable leveling table in 

 the instrument testing room, and could thus be moved axially through any distance 

 required without rotation. The spool was so mounted, however, as to make rotation 

 about the axis quite easy. 



The measuring instrument was a U micrometer with two independent heads. The 

 frame which carried the heads was made of bronze and was carefully machined on 

 a Brown and Sharpe universal milling machine. The top surfaces to which the 

 micrometers were attached were made true and parallel and were so marked that when 



Phil. Trans. A. vol. 207, 190S, p. 463; and vol. 214, 1914, p. 27. 



