THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 263 



This c is connected with FRIOKE'S T and t (' Diss.,' Leipzig, 1886) by the relation 





_ - C s + 4c 

 "~ 



The following table, analogous to the preceding one for /* = 6, gives the division- 

 values for /j. = 10, the six transformations being derived from the substitution 



and it may be compared with the corresponding results given by F. MULLER. 

 ' Archiv der Mathematik und Physik,' p. 161, 1884, ROHUE, ' Archiv,' 1886, p. 138, 

 and by W. GORING, 'Math. Ann.,' 7, p. 311, 1874, working on GAUSS'S unpublished 

 memoirs. 



For the application of this integral to a case of motion of the top, consult the 

 ' Annals of Mathematics,' vol. 5, 1903. 



The numerical value c = 2 will serve for verification ; this corresponds to the case 

 of ten cusps at intervals of JTT in the associated motion of the top (L.M.S., 27, 

 p. 602) ; and c 2 (sin 36 sin 18) will make K'/K = x /5 or 1, cases of Complex 

 Multiplication. 



