314 H. NAGAOKA: [VOL. W H. NAGAOKA.

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1 314 H. NAGAOKA: [VOL. W BY H. NAGAOKA. 1. The practical importance of having an accurate standard of inductance has lately been felt in electrical measurements relating to alternate currents and electric waves, and several formulas have been deduced for expressing the self-inductance of solenoid. They generally assume different forms according as it is short or long. Most of them are, however, complicated and not suitable for the use of experimental physicists and engineers. In the following, I propose to show that the self-inductance of a solenoid can be easily calculated by tabulating a certain coefficient _??_. The self-inductance of a very long solenoid is given by where n is the number of turns per unit of length; for solenoids of any length, it will be shown that where _??_ can be tabulated once for all as a function of angular aperture of the solenoid. When once the values of _??_ are known, the calcula tion of self-inductance is greatly facilitated, as the rest of the operation is a simple multiplication. 2. In a former paper,* I have shown that the mutual inductance of two coaxial solenoids is given by where n, n' are the numbers of turns per unit length of the solenoids; A, a the radii; 2l, 2l' the lengths; d the distance between the centres of solenoids; further, (2)

2 No. 16.] NOTE ON THE SELF-INDUCTNYGE OF SOLENOLDS. 315 where The case which deserves special, attention in the above formula is when the radii and the lengths of the solenoids coincide; i.e. when d=0, A=a, l=l'. In this case, M transforms into L. (3) Utilizing the relation we find that (4) Consequently (5)

3 which is identical with the formula obtained by Lorenz* aud Consequently the self-inductance of the solenoid is (6) Putting (I) we get L=4ƒÎn2. Area of Cross Section ~Length ~_??_ Area of Cross Section/ Length_??_ of the solenoid at the centre. Thus _??_ can be generally expressed as a function of the angular aperture. Since and 3. The coefficient _??_ can be expressed in various ways. (Ia) The expression (Ia) can be conveniently used when the length is very large compared with the diameter of the solenoid, while the second expression (Ib) will be found useful when the said ratio is very small. (Ib)

4 we find by (Ia) (I'a) For the convenience of calculation of the coefficients the following table is added The above series is only slowly convergent when the angular aperture is large, so that the limits of application are confined to small values The second operation (Ib) above indicated can be applied to the expression for E under the form where Thug (I'b)

5 4. On account of the simplicity of calculation in series proceed ing according to powers of ď or ď', the two series (I'a) and (I'b) above given will find special favour among practical scientific men, but the limits within which the series may be safely applied is too narrow that it is necessary to deduce other series which can be easily used within a long range in the value of a. For this purpose, expansions in q-series are specially to be recommended; q being given by where Since for k=sin45, it is gene rally sufficient to put for most practical purposes. Starting from the well-known expressions* we easily find by ( T) The limits of a, within which the above series can be conveniently used is wider than the expansions already given; for a=45 576q6= , so that the values of is right to six decimal * Jacobi, Fundamenta Nova, 105; Werke, 1, 161.

6 places for the above argument. By the way, it may be noted that for the same value of a, the term affected witn k'2 in (I'a) , and the convergency of (I'a) is slower than (III). 5. In order to arrive at expression which would give more exact values of it is necessary to transform formula (I) in ľ-functions. Since and we find whence Expressing this in terms of q' s we find for The above expression is applicable within, wide limits of a, and is rapidly convergent; although q12 is retained, it is generally sufficient for practical calculation from a=0 to a=45 to suppress terms beyond q4. For a=45, q4= and q6= , so that be tween the said limits

7 will give values accurate to six decimal places, which is superfluous in practice. Since a enters into the calculation before finding q, it would be more convenient to retain it in this form than changing it into a q-series. To find an expression of in terms of q1, by which the calcula tion of self-inductance for values of a from 45 to 90 may be easily effected, we have to express E and K-E by means of ƒö3 and (O, T1)'s. Since we find by ( T)

8 It is needless to remark that the convergence is extremely rapid. The slight inconvenience which is felt in the evaluation of the above expression is the presence of the term Even for small values of q1, it must be accurately known ; in fact, we shall have to push the calculations for q1 to several decimal places, which are quite unnecessary for finding the values of ' s, for the simple reason that the expression contains terms multiplied by It is con venient to calculate q1 from l1, where The two expressions ( Va) and ( Vb) for in terms of q and q1 resp. will enable us to calculate the self-inductance L by the formula ( U) with any desirable accuracy, while for practical purposes, the simpler expressions ( Va') and ( Vb') will generally suffice. 6. The application of quadric transformation to the elliptic integrals which enter in will lead to an expression, which is ex pedient for the evaluation of the coefficient, but for the use of physicists and engineers, those already given would be efficient for numerical calculation. It would however not be out of place to notice the different gates, which are open for expressing in a convenient manner. 7. The following numerical examples are given for the sake of comparison of formulas ( Ta') and ( V) For a=45, formula ( Ta') gives

9 The fourth figure is slightly in error as the formula (III) will show. For a=45, q= ; by formula (III) This coincides with the values. found from (IIIb) or from (I) by using Legeudre's tables of elliptic integrals. 8. Most of the formulas above deduced admits of easy calcula tion, but when the values of for different a's are once tabulated, they will have only theoretical interest. In the eyes of practical men, the result of the various calculations above given for finding will be of little value, when the table is constructed. The following tables of log10 and were calculated by Mr. T. Su gimoto from formula (I), by making use of Legendre's tables of elliptic integrals given in 'Exercises' vol. 3. It is given for every degree of a, but in practice it will be more convenient to tabulate _??_ for every tenth of a degree. The construction of such a table is not difficult, and I hope to publish it before not long.

10 TABLE OF LOG10_??_

11 TABLE OF _??_

12