﻿Electrodijnamic Phenomena suggested by Gauss. 455 

 is necessary for the propagation of the action. To determine 

 this value, Betti developes the function in powers of a --, and 

 limits himself to the first powers by putting 



The limitation to the first two powers he justifies on the ground 

 that or and - are small magnitudes. But, according to his own 



assumption the duration of a period of the function represented 

 by <f>(t) is also very small, and in the course of his deduction 

 there is even the following passage :— " Now let the duration p 

 ot a period be very small as compared with the time in which 

 the electrical action propagates itself through the unit of length, 

 so that a (the magnitude lying between and p) may be neg- 

 lected in comparison with 



r a 



c 



If the function c/>(/) has so short a period, and its value there- 

 lore changes so rapidly as is here presupposed, it must have 

 very great differential coefficients. If such a function is to be 



developed in reference to a magnitude which contains -, and in 



comparison with which the duration of a period is ver/smaU ic 

 is impossible to neglect in this development all higher powers 

 than the second. To see this, since Betti has named no special 

 conditions as regards the nature of the periodical functions, it is 

 sumcient if we consider any given function whose period has the 

 duration p. Let it be the following • 



. 2tt 

 sin — t. 



P 



Putting in this t+V in the place of t, and developing with re- 

 spect to powers of P, we have 



sin — 

 P 



P P P l.2\pj Sln p l 



JP* /2tt\ 3 2tt 



_ In this series we see at once that when the magnitude p in 

 the denominator is small as compared with the magnitude P in 

 the numerators, we must not think of restricting ourselves to 

 the first three members. 



Since in Betti's paper the entire further calculation depends 

 upon the development of that series in which all members higher 



2H2 ° 



