434 Dr. Glazebrook on the Practical Apjjlication of the 

 Hence expanding as far as {/JLa/2} 4 , we find 



[x^a 2 



1- 



_5 /*V 

 12 4 



+ 



.,.). 



or neglecting ^\/x 2 a 2 in comparison with unity, 



9 O 



and 





! I a 3 



W 



(II.) 



3/ V'6^ + 4^2 



It is clear that if a is 2 miles, the value of k is about 

 20 per cent, too great. 



To compare these two formulae I. and II. we may, since 

 ft/2?* is less than unity, expand the logarithm in I. ; we thus 

 obtain 



F-^-il- 1 -^ \ 



In the Stockton experiments a = 2, y = '4, r = 204 ; and 

 the term neglected is about '02. 



A result of considerable importance may be deduced from 

 the above formulae. 



It is clear that for similar circuits similarly placed the 

 vertical force varies as the linear dimensions of the circuit 

 multiplied by the current. Now if the number of cars per 

 mile of track is to remain constant, the current must vary as 

 the length of the track. Hence the force is proportional to 

 the square of the linear dimensions. 



Fin-. 3. 



B' 



Thus if we have two parallel lines AB, A'B' (fig. 3), of 

 which A'B' is double the length of AB and at double the 

 distance from the source, the effect of A'B' is four times that 

 of AB. This is readily seen from first principles. 



