1913]
TERMINATED RODS IN ELECTRICAL PROBLEMS
151
It is particularly to be noticed that although (29) is an overestimate, it vanishes when a tends to zero.
The next step in the approximation is the inclusion of Hl corresponding to the first root /^ of <£ (lib) = 0. For a given k, B has only one term, expressed by (13) when we write hl} H1 for h, H. In (16) when we expand (A + .B.)2, we .obtain .three series of which the first involving Az is that already dealt with. It does not depend upon H^. Constant factors being omitted, the second series depends upon
v J0*(ka)
and the third upon
kJa" (lea)
.(30)
•(31)
(hf - /c2)2 Jf (kb) '
the summations including all admissible values of k. In (24) we have under X merely the single term corresponding to JT3, /tj. The sum of (16) and (24) is a quadratic expression in Hl} and is to be made a minimum by variation of that quantity.
The application of this process to the case of a very small leads to a rather curious result. It is known (Theory of Sound, § 213 a) that k^ and hf are then nearly equal, so that the first terms of (30) and (31) are relatively large, and require a special evaluation. For this purpose we must revert to (10) in which, since ha is small,
Fo (ha) = log haJ0 (ha) + 2JZ (ha), so that nearly enough
J0 (Kb) = (h - k) bJ0' (Kb) =
' x ' v '
~- , log ha log ka
.(33)
O Y°\ ft IJ1 _«. A — —__
cUltl lii ---- /(< --- 7 Tr~77 7 \ i 7 .
&Ji(7a>)log7ca
Thus, when a is small enough, the first terms of (30) and (31) dominate the others, and we may take simply
(30) =
6 log &i a
.(35)
AI ±> n \ J- •/ /? 7\ -MW;
Also <f> (A^a) = — -.- -,-—-.—, 6 (7^6) = , \ •-.
/ '-- '—'-t,' r \ i / log&ja
Using these, we find from (16) and (24)
«V V~rt7""rt T" o / T 7 \ l" "
I
2 log 6/a 4 log2 ^a
.T0(k1b)J1(k1b) ' 4&1F0'(M)
ria(A?i6)-A?rah ......(3^)e surface, so that at all stages of the appro ination the calculated value of (14) exceeds the true value of (15). In t application to a condenser, whose armatures are at potentials 0 and