DD) 
342 Dr Searle, The comparison of nearly 
Let A,, A, be the effective resistances of A when it is shunted 
by a), ds and By B, the effective resistances of B when shunted by 
b;, b;. Then 
1 = Lt ae i i = i at (1) 
a: = a A, mA oe ‘ 
1 SB a etal : 
Ses == i A ee 2): 
Bua Bayben, eb: Teoma (2) 
| oD Dee a 
Since A, = B, and aL, = B, meee} ofl fellate eertetetat fete (3), 
when the bridge is balanced in the two cases, we find, by elimi-— 
nating C and D, 
A. As = BB, wee etme ene teen ntsvenas 
Taking the square root of each side of (4), we have 
Jae O. 
The left side of (5) is the es mean of 1/A, and 1/A,. 
When A, and 4, are nearly equal, this is very nearly the same as 
the arithmetic mean. Thus, if 
See ey De ial 
Ai) Acces =A eee 
so that 1/A, is the arithmetic mean of 1/A, and 1/A,, we have 
fe fA 7 
ia, V acess, oe oe ae 
Hence, if A, and A, are so nearly equal that A,?/207 is negligible 
compared with unity, we may use the arithmetic mean 1/A, instead 
of the geometric mean. For example, if A,/a is 1/1000, A,?/20? is 
only 1/2,000,000, which is negligible in all but the most precise 
work. In that case, however, the resistances would, probably, — 
have been so well adjusted that A,/a is less than 1/1000. The 
same remarks, of course, apply to B, and B,. Replacing the ~ 
geometric means in (5) by the arithmetic means, we have 
(4, +a) ~3la ta) 
2 B, B 
“* Equation (4) is a quadratic for 1/A in terms of 1/B. _ If we solve it, we find 
SA) oe WN ON ae 
Vi TA 0h) \B Oh Fane — 2 Naar 
week can be used in any case where a,, d., b,, b, are not very ge compared 
with B. 
