of a certain network of conductors 113 
This last result could have been obtained at once from (1). 
For &, is always greater than 2r though less than 2r+s when 
n>1, and &, diminishes as n increases. Hence R, has a limit 
_R., when n becomes infinite. . Putting R, = R, and R,,=R,, m 
(1), we have 
aie 
gab IR. 
‘Since &,, is essentially positive, this quadratic gives 
R,=r+vr + 2rs, 
which is identical with (138). 
§3. Mr J. M. Dodds, Fellow of Peterhouse, has kindly drawn 
my attention to the advantages of the use of hyperbolic functions 
in this problem. Since 
sinh (n + 1) 8+ sinh (n — 1) @= 2 cosh @ sinh n@ 
and sinh 29 = 2 sinh @ cosh 0, 
we have 
Reo = 2rt+ 
sinh(n+1)6@  sinh20  sinh(n—1)0 
sinhn@  sinhO sinhn@ ° 7" sy) 
Now, since A, = 2r +s, (1) can be written 
3? 2 
Ky, + s = 27 san imines ANS So araegr ae 
i tS S 8 
or er Re (15) 
Comparing (15) with (14) we see that, if 0 be suitably chosen, 
_ _sinh(n+1)@ 
tS snhnQ 7 es (16) 
When n=1, this gives, since R,=2r+s, 
cosh 0=(r+s)/s. 
Since es aaa 
we have = cosh @ + sinh @=*(r-+8 +1) = f 
u q 
and gis cosh @— sinh’ @ = —(r+s—t)=—. 
Hence 
SU DES ie Cue 
ies —s 
sinh n@ en) — ene p-g” ) 
which agrees with (11). 
