60 SIE G- GEEENHILL ON ELECTKOMAGNETIC INTEGEALS. 



In making these verifications use must be made, of the differentiation formulas 

 given in the 'American Journal of Mathematics' ('A.J.M.'), 1910, p. 392, where 

 D denoting rfrf = (A 2 + a 2 +& 2 ) 2 -4AV, 



f 



W 



<> 



with the check formulas 



(8) ;n-- A ;rA ^TT = < 



dP A '*Q 7. 



a -jr A -yf o -57- = 0, 



a6 a& ao 



_ 



: -- -- r^ 



da da da 

 Pa_QA-i2& = W, 



dW dW dW 



~ Q ' : " 



Reviewing these calculations it will be noticed that the S.F. again shows generally 

 a marked superiority over the P.F. in its analytical simplicity. 



This N in (l), 18, is the expression which gives the potential energy (P.E.) of 

 the two co-axial helical currents, or their equivalent current sheet solenoids, investi- 

 gated by V. JONES, 'Roy. Soc. Proc.,' 1897, or the mutual P.E. of the two pairs of 

 equivalent end plates (' A.M.S.,' 1907, 59). 



But as it is the force only between the two currents which is required, and this is 

 given by d~N/db = L, we need only calculate the end values of L for measurement 

 in the current weigher. 



20. As another illustration of the extra analytical simplicity of the S.F., take the 

 calculations of BROMWICH (' L.M.S.,' 191-2), where the results are expressed in a series 

 for the attraction and P.F. of a circular disc, the circle on AB, where the surface 

 density is a- = ky", varying as the n th power of the distance y from the centre 0. 



