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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