PROCE EDINGS 



OF 



THE ROYAL SOCIETY. 



" The Third Elliptic Integral and the Ellipsotomic Problem." 

 By A. G. Greenhill, F.R.S. Eeceived December 22, 1903 — 

 Read January 21, 1904. 



(Abstract.) 



The elliptic integral of the third kind, which makes its appearance 

 in a dynamical problem, is of the circular form in Legendre's classifica- 

 tion, and thus the Jacobian parameter is a fraction of the imaginary 

 period, so that the expression by means of the theta function can no 

 longer be considered as reducing the variable elements from three 

 to two. 



Burkhardt* has given a series rapidly convergent for the numerical 

 calculation of such cases ; but the object of this memoir is to develop 

 the exact expression by means of an idea of Abel, given in the first 

 volume of ' Crelle's Journal,' 1826, " Ueber die Integration der 

 Differential-Formel 



(1) pdx/ JR, 



wenn R and p ganze Functionen sind." 



Abel proves practically that when the elliptic parameter is an 

 aliquot /^th part of a period, the third elliptic integral and the 

 associated theta functions depend on the /xth root of an algebraical 

 function. 



Thus, as shown in the paper, if we take the Jacobian elliptic 

 parameter 



(2) v = 2KV//x, /x = 27i+ 1, an odd integer, 

 the third elliptic integral in the form 



P (v) P - \xy dt 2 



* ' Elliptic Functions/ § 126. 



VOL. LXXIII. 



B 



