Self-induction of Wires, 17 



When the rod is made a flat sheet, or a/b is very small, we 

 have Jj = ^7T/jb(a/b). 



Compare (29a) with Thomson and Tait's equation (46) 

 § 707, Part II. Turn the nab 2 outside the [ ] to nab z , and 

 multiply the 2 by 2. These corrections have been pointed 

 out by Ayrton and Perry. When made, the result is in 

 agreement with the above (29a), allowing, of course, for 

 changed multiplier. [I also observe that the — r in their 

 equation (44) should be + t, and the + t in (45), (the 

 second r) should be — r.] Such little errors will find their 

 way into mathematical treatises ; there is nothing astonishing 

 in that ; but a certain collateral circumstance renders the 

 errors in their equation (46) worthy of being long remem- 

 bered. For the distinguished authors pointedly called atten- 

 tion to the astonishing theorems in pure mathematics to be 

 got by the exchange of a and b, such as rarely fall to the lot 

 of pure mathematicians. They were miraculous. 



I now pass to a different problem, viz. the solution in the 

 case of a periodic impressed force situated at one end of a 

 homogeneous line, when subjected to any terminal conditions 

 of the kind arising from the attachment of apparatus. The 

 conditions that obtain in practice are very various, but 

 valuable information may be arrived at from the study of the 

 comparatively simple problem of a periodic impressed force, 

 of which the full solution may always be found. In Part II. 

 I gave the fully developed solution when the line has the three 

 electrical constants R, L, and S (resistance, inductance, and 

 electrostatic capacity), of which the first two may be functions 

 of the frequency, but without any allowance for the effect of 

 terminal apparatus. If we take L = we get the submarine- 

 cable formula of Sir W. Thomson's theory ; but although the 

 effect of L on the amplitude of the current at the distant 

 end becomes insignificant when the line is an Atlantic cable, 

 its omission would in general give quite misleading results. 



There are some a priori reasons against formulating the 

 effect of the terminal apparatus. They complicate the for- 

 mulae considerably in the first place ; next, they are various 

 in arrangement, so that it might seem impracticable to for- 

 mulate generally ; and, again, in the case of a very long sub- 

 marine cable, we may divide the expression of the current- 

 amplitude into factors, one for the line and two more for the 

 terminal apparatus, of which the first, for the line, is always 

 the same, whilst the apparatus-factors vary, and are less im- 

 portant than the line-factor. But in other cases the terminal 

 apparatus may be of far greater importance than the line, in 



Phil. Mag. S. 5. Vol. 23. No. 140. Jan. 1887. C 



