A NEW CURRENT WEIGHER, ETC. 515 



SECTION 11. DIFFERENTIAL EFFECTS OF THE SEVERAL WINDINGS, AND THEIR 

 RELATION TO THE LINEAR DIMENSIONS OF THE COILS. 



On each fixed cylinder there are four helices, and on each suspended cylinder two 

 helices, and the diametral dimensions of those on the same cylinder are slightly 

 different. Let the upper helices on the left fixed cylinder be designated Ul and U2 

 respectively, the lower ones Ll and L2, and those on the coaxial suspended cylinder 

 a and b ; also let the helices on the cylinders to the right be represented by similar 

 letters characterised with a dash. Then the maximum force due to a current y k in the 

 left fixed helices and a current / per unit length in the current sheets equivalent to 

 the suspended helices may be written 



/y*(M in . + M OT .+M tta +M u .+M IJU +M, w +M lu +M u ,) = 2/y A M L = D L (say), 



where M U1(1 is the difference in mutual induction of the coil Ul and the circular ends 

 of a (i.e., Mj M, of formula (1), p. 507), and M L is the difference in mutual induction 

 of all the fixed left helices and the circular ends of the current sheet equivalent to a 

 and 6. For the system on the right there is a similar expression which may be 

 denoted by D R , and the sum D L + D R is conveniently written as D. 



In addition, there are secondary forces due to the mutual action of the fixed systems 

 and the opposite suspended ones. The maximum secondary effect due to the left 

 fixed system and the right suspended one may be written S L , and that due to the 

 other systems S H . Let S L + S K = S. 



The direct and secondary forces may aid one another, in which case the total force 

 measured by the balance is D + S, or they may oppose one another, the force thus 

 becoming D-S. The sum (D + S) + (D-S) gives 2D. If only one-half of the whole 

 system is used, D L or D H is obtained. In the determination of the E.M.F. of the 

 cadmium cell, the forces D + S and D S were measured in most cases. 



Estimation of Difference between Left-hand and Right-hand Systems of Coils. If 

 the two forces D L and D K act in opposition on the beam of the balance, the force 

 required to maintain equilibrium is (D L D R ) + (S L S R ) or (D L D B ) (Sj, S H ). By 

 reversing the current through all the coils on one side of the balance, one of these 

 states is obtained from the other. If both of the balancing forces are measured, the 

 mean is D L D R , which is equal to 2y'y A (M L M K ). Thus the mean balancing mass is 

 2y'y A (M L M H )/</, and is to be accompanied by a positive sign when the force acting 

 on it is in the same direction as D L , and by a negative sign when in opposition to I >, . 

 If mi is the balancing mass, M L M R = Wi^^y'y* ; a check is thus afforded on the 

 calculated difference M L M H . The calculated value of M L is 25962 '04 centims. (see 

 (22'), p. 514), and of M K 25960'43 centims., the difference being 1*61 centims. The 

 mass ?H, was determined on five different dates, and on each occasion the current was 

 reversed through all the fixed coils in order to reverse the direction of the force and 



3 u 2 



