125-127] MOTION OF A HELICOID. 191 



which will be consistent provided 



whence ulp=kBPI{AQ-WB(A-B}} ........................ (iii). 



Hence there are an infinite number of possible modes of steady motion, of the 

 kind above defined. In each of these the instantaneous axis of rotation and 

 the direction of translation of the origin are in one plane with the axis of the 

 solid. It is easily seen that the origin describes a helix about the resultant 

 axis of the impulse. 



These results are due to Kirchhoff. 



127. The only case of a body possessing helicoidal property, 

 where simple results can be obtained, is that of the isotropic 

 helicoid defined by Art. 123 (7). Let be the centre of the 

 body, and let us take as axes of coordinates at any instant, a line 

 Ox, parallel to the axis of the impulse, a line Oy drawn outwards 

 from this axis, and a line Oz perpendicular to the plane of the 

 two former. If / and G denote the force- and couple-constituents 

 of the impulse, we have 



Au + Lp = f = /, Pp + La = \=G, 



Aw + Lr = = 0, Pr + Lw = v = I 

 where OT denotes the distance of from the axis of the impulse. 



Since AP L 2 4= 0, the second and fifth of these equations 

 shew that = 0, q = 0. Hence w is constant throughout the 

 motion, and the remaining quantities are constant ; in particular 



u = (IF-GL)l(AP-L*\\ 

 w = -^ILI(AP-D) } 



The origin therefore describes a helix about the axis of the 

 impulse, of pitch 



G/I-P/L. 



This example is due to Lord Kelvin*. 



* I.e. ante p. 176. It is there pointed out that a solid of the kind here in 

 question may be constructed by attaching vanes to a sphere, at the middle points of 

 twelve quadrantal arcs drawn so as to divide the surface into octants. The vanes 

 are to be perpendicular to the surface, and are to be inclined at angles of 45 to the 

 respective arcs. 



For some further investigations in this field see a paper by Miss Fawcett, &quot; On 

 the Motion of Solids in a Liquid,&quot; Quart. Journ. Math., t. xxvi. (1893). 



