218 PROFESSOR A. G. GREENHILL ON THE 



But a mere change of sign in ABEL'S results is sufficient to pass from the 

 logarithmic to the circular form of the third elliptic integral, and as it is the circular 

 form which is of almost invariable occurrence in dynamical problems, we shall adopt 

 it as our standard form. 



Applications of this third elliptic integral are introduced in the course of the 

 memoir to these problems, such as POINSOT'S herpolhode, and the spinning top or 

 gyroscope, the spherical catenary, the velarium, and the elastica under uniform 

 normal pressure. 



References to former articles on the same subject in the ' Proceedings of the London 

 Mathematical Society,' vols. 25 and 27, are given in the sequel in the abbreviated 

 form, L.M.S., 25 or 27 ; frequent reference is made also to KIEPERT'S articles on 

 the theory of elliptic functions in the ' Mathematische Annalen ' ('Math. Ann.,' 

 vols. 26, 32, 37). 



1. Working then with the third elliptic integral in the circular form, when the 

 elliptic parameter v is a fraction of the imaginary period w 3 , we change the variable 

 in the standard form of WEIERSTRASS, 



by putti 



= t , - ..- (1), 



(w-f r) 



n 



Pit V'r = .s cr, f"~U = b (2), 



*>V= V /-S J v=fw, (3), 



where / is a real fraction, so that the integral changes to 



i /_X ,/., ..(fr-'r) 



(where the elliptic arguments u and v may be supposed for a moment to be normalised 

 to the Jacobian form), and * is an elliptic function of u which we may denote by s (?/), 

 differing from WEIERSTRASS'S Vii by a constant, so that 



while 



cr = 



Considering that (HALPHKN, ' Fonctions Elliptiques,' 1, p. 222) 



is an algebraical function of x when v is an aliquot part of a period, we take 



__rV(v)(s-<r)-^/-Z ds 



- - 



