366 Sir W. Thomson on the Motion of 



An isotropic helicoid may be made by attaching projecting 

 vanes to the surface of a globe in proper positions ; for instance, 

 cutting at 45° each, at the middles of the twelve quadrants of 

 any three great circles dividing the globe into eight quadrantal 

 triangles. By making the globe and the vanes of light paper, a 

 body is obtained rigid enough and light enough to illustrate by 

 its motions through air the motions of an isotropic helicoid 

 through an incompressible liquid. But curious phenomena, not 

 deducible from the present investigation, will, no doubt, on ac- 

 count of viscosity, be observed. 



Part II. 



Still considering only one moveable rigid body,, infinitely re- 

 mote from disturbance of other rigid bodies, fixed or moveable, 

 let there be an aperture or apertures through it, and let there be 

 irrotational circulation or circulations (§ 60, "Vortex Motion") 

 through them. Let £, r), f be the components of the "impulse " 

 at time t, parallel to three fixed axes, and X, fi, v its moments 

 round these axes, as above, with all notation the same, we still 

 have (§ 26, "Vortex Motion") 





(6) (repeated). 



But, instead of for T a quadratic function of the components of 

 velocity as before, we now have 



T=E + i{|>,w> 2 + ... +2[m, »]!«»+...}, . . (13) 



where E is the kinetic energy of the fluid motion when the solid 

 is at rest, and \ 5 \u, u]u 2 + ... j- is the same quadratic as before. 

 The coefficients \u, u\ , [u, v\ , &c. are determinable by a trans- 

 cendental analysis, of which the character is not at all influenced 

 by the circumstance of there being apertures in the solid. And 



dT 

 instead of f = -=—, &c, as above, we now have 



t dT iT/ rfT- y dT • 



f= ^ +I/ ' *=-&+**> s=m+in, 



dT 

 X= -z — \-Y{ny—mz) + (jl, /*, = &c, i/=&c, 



(14) 



where I denotes the resultant " impulse " of the cyclic motion 

 when the solid is at rest, /, m, n its direction-cosines, G its 

 "rotational moment" (§ 6, "Vortex Motion"), and w, y, zthe, 

 coordinates of any point in its "resultant axis." These, (14) 



