258 PROFESSOR A. G. GREENHILL ON THE 



in the pseudo-elliptic case ; and cancelling the secular term pt by making p = 

 gives a purely algebraical herpolhode. 



With the degree halved by the use of the variable y, as in (8) 24, 



M P_ 



* 



? = /- (35). 



Ir y - 1 



In the associated motion of a top, the projection on a horizontal plane of a point 

 on the axis describes the hodograph of a herpolhode traced out by the axis of 

 resultant angular momentum of the top ; and thus EULER'S angles 6 and i/i for the 

 top are connected by a relation of the form 



1 M 2 sin ft> = - i d (pe') (36) 



(' Science,,' Dec. 20, 1901 ; 'Annals of Mathematics/ vol. 5, Series II., 1903). 



We may dispense with POINSOT'S rolling surface and consider the polhode as a 

 material line, or as the edge of a material cone, as in L)r. FR. SCHILLING'S model, 

 constructed by MARTIN SCHILLING, Halle a. S., and this is rolled on a fixed plane. 



According to M. DK LA GouRNElttE, the polhode is also a line of curvature, the 

 intersection of two confocal surfaces, an ellipsoid and hyperboloid of two sheets ; and 

 9 will now be the angle between the generating lines of the hyperboloid of one sheet, 

 one generator being perpendicular to the fixed plane, so that the other generator 

 moves parallel to the axis of the top in the associated motion (DARBOUX in 

 DESPEYROUS, ' Mecanique,' note 18). 



28. To utilise for ^ = -in -\- 2 the preceding results for odd order 2n + 1 of /^, 

 where 



ma. ,, x 



t = 11 (1), 



a -n J v '' 



put 



(a>) - s (4n + 2) v = co , D a , +1 - (3), 



and thence express , in, . . . in terms of a new parameter, and thereby express the 

 integral I (4r) of p = 4;i + 2. 



Returning then to the v of p. = 2 a + 1, which, when normalised to LEGENDRE'S 

 form, can be written 



and replacing y^ = by its equivalent D 2) , +1 = 0, the series of functions 



