﻿58 Prof. G. A. Schott on the 



lower limit, so that t becomes infinite although .r remains 

 finite. For this reason it is convenient to extend the in- 

 tegrals from a finite lower limit ^ to the upper limit ^, 

 the suffixes and x being used to indicate initial and final 

 values respectively. Using the notation of the exponential 

 integral we find from (39) 



,, _^ 0= P^%^ = ^EifeO-Ei&o) -Ei(- Xl ) + Ei(-Xo)} 



tl -t = C lC 2^d x = Um( Xl ) -Ei( Xo ) +Ei(- Xl )-Ei(- X o) } 

 Jx X 



'Rrf^XisinliXi- cosh %1 -xo sinll %o+ cosh ^o- • • 



\- (40) 



These expressions involve neither F nor 8, but only % and 

 ^ T . so that in this extreme case of the exponential motion 

 the result depends only on the initial and final velocities of 

 the electron, and not at all on the strength of the field or on 

 the precise value of 8. This fact of itself is sufficient to 

 prove that the exponential motion is not realisable experi- 

 mentally, at any rate not with the electric fields at our 

 command; a numerical example may make this clearer. 



Let us take the case of an electron which has its speed 

 increased by an electric field of 27,700 volt ,' cm. (giving F 

 equal to 10" 14 ) from /3 = 0'01 to A = 0-30, i. e. from %„ = O-01 

 tox 1 =0-31. 



With the help of tables of the exponential integral * and 

 of the hyperbolic functions we obtain the following results 

 for the two limiting motions : — 





Newtonian motion. 



Exponential motion. 





Units of § 3. 



C.G.S. units. 



Units of § 3. 



C.G.S. units 



v x — x .. 



. 4-8 . 1012 



0-88 



0-302 



5-5. 10 ~ u 



t,-t Q .. 



. 3-05 . 1013 



1-86.10-10 



3-458 



21 . 10 -23 



'tl udf - 



. 3-05.10-1-3 



2*4 .10-^1 



0040 



3-9. 10~ 3 



A comparison of the numbers in the last four columns of 

 this table shows conclusively the enormous difference between 

 the two limiting motions, and there can be no question that 

 the Newtonian motion is in far better agreement than the 

 exponential motion with what we know from experience. 

 Even if the hypothesis of § 6 be not exactly true, its deviation 



* Dale, 'Tables of Mathematical Functions,' p. 85 and p. 64. 



