54 SIR G. GEEENHILL ON ELECTROMAGNETIC INTEGRALS. 



or, to the modulus K, 



p = joK = 4aKV P* 



- r, + r 8 v/WV,) a 



M = _2^(r,- 



Q = 27r(l-/)-4Kzn/K'. 



r,-r, _ OE OB BE 

 = 7-3 + r a OB OD == BD ' 



DPB = am/K',- DEP = am (l-/) K', 



The article in the 'Trans. American Math. Society,' October, 1907 (A.M.S.), may 

 be consulted for an elaborate and detailed discussion of the elliptic function analysis 

 and procedure of former writers, and a numerical calculation is given there for the 

 helix employed originally by VIRIAMU JONES. Measurement of fig. 2 gives K, \/<-, and 

 thence, -from LEGENDBE'S tables, K, E, Fi/r, and t /= Fi/r/K'. 



Another numerical application of these formulas can be chosen from the dimensions 

 of the current weigher at the N.P.L., Teddington, described in ' Phil. Trans.,' 1907. 



14. Integrate 12 witli respect to b to obtain the magnetic potential of the solenoidal 

 current sheet, or of the equivalent close helical winding in the ampere balance. 



In these integrations with respect to b the form 12 (MQ) of the III. E.I. comes in 

 most appropriate as not involving b in MQ, and then 



(]\ f o .77. .-> 7 f f A cos 6 + a 2 bdQdb 



7 i 'AacosO + a 2 



- ^TrO 



M 

 = 27T&- -P-60(MQ) 



This solenoidal magnetic potential is the same as that of the cylinder on which the 

 helix is wound, and so is the equivalent of the axial component of the gravitation 

 attraction of the solid cylinder, and this is the difference of the potentials of the 



