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Full text of "Scientific Papers - Vi"

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