THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 223 



The ellipsotomic problem of the determination of the division-values (Thettu'erihe) 

 of the elliptic functions resolves itself thus into a consideration of the curve in x and 

 y given by this rational equation (4), which may thus be called the alUpsotomic 

 equation, by analogy with the ci/clotomic equation for the circular functions; and the 

 problem may be considered solved when ;c and // can be expressed, rationally or 

 irrationally, in terms of a parameter; according to POINCAUK ( Bulletin de la Societe 

 Mathematique de France,' 1883, 11, p. 112) this can always be effected by uniform 

 functions of an independent variable. 



The details have been carried out in the ' Proceedings of the London Mathematical 

 Society' (L.M.S.), 25, for values of /A up to 22 inclusive, omitting 19, which still 

 remains awaiting solution. 



When fj. is an even number, 4// -4- 2 or 4n-, x (i/x/') < (I'M.,) is a root of S = 0, 

 so that the cubic S can be resolved into factors, and we can employ the functions of the 

 Second Stage of LEGKNDHK and JACOBI, as required in most dynamical applications. 



But when /u, is odd, this resolution cannot be effected algebraically, and WKIKU- 

 STRASS'S functions of the First Stage must bf employed. 



The ellipsotomic relation (L.M.S., 27, p. 405) 



are equivalent, so that 



/.even, *?='*"* "I V = r*f ; , ... (7). 



y ^ (p. 2) y ', ((*. - 



Writing r for " 3 , and a> for w :i , 

 /* 



^, ;/ (r) = x* { "' '"y,, (8) 

 (HALPHEN, ' F.E.,' I, pp. 102, 198) ; and changing n into ^ u, and dividing, 



"--' -- r^-m.'.X^ K---M /<A 



LO-CX^A j (J). 



But 



a- (^ _ .) v = a- ('2<a - nv) = 6 1 """ '" <rnv ( I U), 



so that 



_i'' -' 



c. M rrv = .r~'' X '' 



and generally 



' 



and this is KIEPERT'S function T( ), defined in ' Math. Ann.,' 32, p. G. 



V / 



