112 =Dr Searle, Calculation of the electrical resistance 
The method employed in obtaining the value of R, as a function 
of m is indicated by Bromwich (Infinite Series, p. 21, Example 26). 
We have, from (1), | 
Ryna(2r+s—2)+s(2r—2) 
R,-“2#= 
s+ Rr 
Mie (Ryra— 2) (2r+8—2) — (a —2re— 2rs) Ae 
s+ hy. 
The second term in the numerator vanishes if 
oe —(2re — 2rs =, . 2.0 eekeen eee (3) 
or if g=rtVP+%esaertt. «002 ol Oeeee eee EE (4) 
Since there are two values of « satisfying the condition (3), we 
obtain the two equations 
Cie Ger 
Rn-1r-t= ae fa) 
Roi-—r+t)\r+s+t 
Ryn ae ds ne 
Putting perts+t, g=r+s—t, eee (7) 
so that r+Lti=p—s, r—t = — Ss) soe (8) 
we obtain from (5) and (6) 
eee (9) 
R,ts—¢ Re he.9 0) 
This is the required formula of reduction. Since R,=2r+s, we 
obtain by successive applications of (9) 
Rnrts Be ie 2r 4+ 2s —p (ay a (2) (10) 
Rnt+ts—q 2r+2s—q\p p 
Solving (10) for R,,, we have 
+1 _ _nt+l1 
Ria! 2 3, (11) 
El 
which is the complete solution of the problem. 
§ 2. We can write (11) in the form 
JB es 8. os setlyeeee Sereeeeee 
1— (g/p) 
Since p>q, it follows that (q/p)” vanishes when n becomes 
infinite. Hence, if R. be the resistance between A, and B, when 
the number of connecting wires between the cables is infinite, 
Ro =p—sS=rtt=r4Vr4 Irs. ic... (13) 
