64 SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 





and so on, with /,+/, = 2/. Because 



/ ^ ^ x/(a 3 + & a ) A 



(8) ysn/,sn/ 4 = ^ 2 + fc2 / + A . 



, 2A/> f A f 2A6 



(9) cn/ 8 dn/ s == , cn - 



(10) 8 n(/ 4 +/ 3 ) = - = s 



r 3 



= si " OBF 



Produce NP parallel to AB to cut the circle on ED again in P 4 , then 



A _NP BP 

 v/( a + ^) NB BP,' 



because P, P 4 are inverse points in the circle, centre N, through B ; so that 



(13) sn (/,-/;) = |^ sin BPP 4 = sin BP 4 P, 



(14) BP 4 P = am (/,-/ f ) G', OBP 4 = am (2-/,+/ ( ) G'. 



Thence a geometrical construction for am^G' and am^G', similar to that above 

 for/j and/j- 



The pole of the chord KB/ through B will lie on the line through A perpendicular 

 to AB, at A' suppose, and the tangent A'B/ will be parallel to AB. 



A whole chapter might be written of elliptic function theory, showing in this 

 manner the geometrical interpretation of the various formulas, especially of the 

 quadric transformation, in relation to co-axial circles. 



23. Our chief object was to employ a straightforward integration of MAXWELL'S 

 result as a direct road to the analytical results required in ampere-balance current 

 weighing. The check on the arithmetical calculations has been explained and carried 

 out in the 'Transactions of the American Mathematical Society' ('A. M.S.'), 1907, 

 56, p. 516. 



Considering that the chief analytical and numerical difficulties in these operations 

 arise in the III. E.I. expression of Q, and that this is cancelled by making 



(1) A = a, /=i i2 = w 



