304 ON THE THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 

 To connect up the results, //, = 20 and 40, 



s a> s <> /r , lX 



\ ' = a 3 in u. = 40 (21), 



s (w) s (2v) 



s (w) s (v) \e pi 

 are equal, so that 



%) + &20 / 9 0\ 



T- (Wh 



> e 



and thus, with a 30 = a, 6 20 = b, 



-~ 



, 



-20 



(26) 



'Math. Ann.,' 32, p. 119), and the preceding results are thus merely 

 bisection formulas for /A = 20. 



We arrive at the conclusion that it is the quotient of two theta functions, 

 6u and 6 (u 7;), with constant phase difference v, which is required in dynamical 

 application, the functions a, /8, y, S for instance employed by KLEIN in top-motion ; 

 but the separate theta function 6u has no mechanical interest. 



This quotient, qualified by the constant factors 00 and 6v, is an elliptic function 

 of u when v is a half-period, dn u for instance when v is the half-period K, and the 

 quotient is the /A*'' root of an algebraical function of the elliptic functions of u 

 when v is an aliquot /i th part of a period ; in this way we express the result of ABEL'S 

 pseudo -elliptic integral. 



The formation of this algebraical function for the simplest values of /A has been our 

 chief object, and in the course of the work the ellipsotomic problem has been carried 

 out of the determination of the Division- Values of the Elliptic Function. 



The Transformation problem may be considered solved at the same time by means 

 of symmetric functions of the division-values ; but as Transformation has no 

 dynamical utility, it has not been developed in this memoir. 



