THIED ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 219 



where IP (?>) = ^ / y> /7) % 



and denote this integral hy 1 (r), and work with it as our standard form of the 

 elliptic integral of the ITT. kind. 



The function employed hy HALPHEN (' F.E.,' I, p. 230) is now 



</> (", '") = x '(* ~ o-) exp { inP (/.) - /! (r)} (8), 



</> (", - r) = x /(x - o-) exp { - iuP (r) 4- i\ (,>)} (<)), 



and is a Lame function of the first order, satisfying his differential equation 



\ ( ^-29u-&v=0 (10); 



<f> du- 



thence the Lame functions of higher order may he derived hy differentiation, as shown 

 hy HERMIT.K, ' Comptes "Rendus,' 1877, and these can he employed in the prohlems 

 considered hy Professor G. H. DARWIN, in the ' Phil. Trans.,' 197, 1901, " Ellipsoidal 

 Harmonic Analysis." 



In the Hermite-Jacohi notation we may take 



, , 000 (u v) 00R (u i') 



$( u ' v >- U 0. /ex P (<*'')' or 0; /H/ , 'expfcznv) (11). 



'2. Next introduce the ./ and ?/ employed hy HALPHEX (' F.E.,' 1, p. lO-l), which 

 may be connected with the a, h, and /> of ABEL'S notation (' (Euvres/ II., p. 155) hv 



and put 



.s'- () = S = 4* (x + .r)- -{(!+?/) x + xyY (-2), 



*(u) = Vu+- x (3); 



then if we |>ut 



ft- o- = x (n) x (r) = x + x (4), 



-.(.)-* , w _(' + *+-4 (5) , 



'(,) = ;/(,)= v/-S=: 



The multiplicative values 



can now all he expressed rationally in terms of x and y, with the y functions of 

 HALPHEN ('F.E.,' 1, p. 102; also ABEL, '(Euvres,' II., p. 159; MONTARD-PONOELOT, 

 'Applications d'analyse et de geometric,' t. I., Paris, 1862), by means of the recurring 

 formulas 



s ( mr ) _ ., ( ,,,) = x * y- + :; y T", y (nr) = x y*i (9). 



ym~y,i~ y 



2 F 2 



