2 



Mr. A. G. Greenhill. The Third Elliptic [Dec. 22, 



subject to the condition 



<*) = 



is such that 



(5) 2^+* exp. (n + $)I(v)i 



= + + . . . ) X /T 1 



where 



(6) T l5 T 2 = 2/ 3 ± (1 + y) P + 2arf ± ajy, 



and x, y, y n are the functions employed by Halphen.* 



The calculation of the coefficients hi, h 2 , . . . can be carried out by 



the method of reduitesj avoiding the continued fractions employed 



by Abel, which make the order of the result higher than is necessary. 

 Here P (v) is of the nature of a zeta function, and equations are 



given for its calculation, as an algebraical function of a parameter. 

 But in most dynamical problems, such as Poinsot's herpolhode and 



the associated motion of the symmetrical top, the Jacobian parameter 



is of the form 



(7) v = K + 2K'i/fx, 



and this requires the resolution of Ti and T 2 , equivalent to a change 

 to an even value, in + 2, of /x. 



Then if p, nr denote polar co-ordinates in the invariable plane of a 

 Poinsot herpolhode, or of the angular momentum vector of a top, 

 and t' denotes the time, we can take 



(8) pf -■■«- = I (v), M P /Jc = t, 

 so that in such a curve 



(9) 2 (Mp/k) n+i exp. (n + J) (pf - w) i 



is an algebraical function ; and the curve is purely algebraical when 

 the constants are so chosen as to cancel the secular term pt'. 



The projection of the path of the centre of gravity of the top is the 

 hodograph of the herpolhode of angular momentum, and is thus 

 obtainable by a differentiation of the above ; some of these applica- 

 tions are developed in a memoir on the top, now appearing in the 

 ' Annals of Mathematics.' 



With p. = 8?i + 4 or 8n a further reduction is possible in degree. 

 We find for /x = 8n + 4, 



(10) (D - x 2 ) n+ * exp. (2n + 1)1 (y) i 

 = [R + R^+ • • • +(-l)^»] v /iX 1 

 ■\-i[R -K l x+ . . . +(-l)^«] x /iX 2 . 



* Halphen, ' Fonctions Elliptiques,' vol. 1, p. 102. 

 f ' Fonctions Elliptiqvies,' vol. 2, p. 576. 



