APPENDIX I. 487 



But n - 2 linear equations in an -re-system determine a cylindroid and hence we 

 see that all the screws on this cylindroid will be principal Screws of Inertia. 



In like manner if there be k repeated roots, i.e. if 



Pi Pa &amp;gt;* 



then a 1} ... a k are arbitrary but a fc+1 , ... a n must be each zero. We have thus n k 

 linear equations in the co-ordinations. They must also satisfy 6 - n equations 

 because they belong to the %-system and therefore they satisfy in all 



Q-n + n k=6k equations, 

 whence we deduce that 



If there be k repeated roots in the determinantal equation of 86 then to those 

 roots corresponds a k-system of screws each one of tvhich is a principal screw of 

 inertia and there are besides n -k additional principal Screws of Inertia. 



So far as the cases of n = 2 and n = 3 are concerned the plane representations 

 of Chaps. XII. and XV. render a complete account of the matter. 



Let (Fig. 10) be the pole of the axis of pitch, 58, then may lie either 

 inside or outside the circle whose points represent the screws on the cylindroid. 



Let (Fig. 22) be the pole of the axis of inertia, 140, then must lie inside 

 the circle, for otherwise the polar of would meet the circle, i.e. there would 

 be one or two real screws about which the body could twist with a finite velocity 

 but with zero kinetic energy. 



We have seen that the two Principal Screws of Inertia are the points in which 

 the chord 00 cuts the circle. If could be on the circle or outside the 

 circle then we might have the two principal Screws of Inertia coalescing, or we 

 might have them both imaginary. As however must be within the circle it is 

 generally necessary that the two principal Screws of Inertia shall be both real and 

 distinct. 



But the points and might have coincided. In this case every chord through 

 would have principal Screws of Inertia at its extremities. Thus every point on 

 the circle is in this case a principal Screw of Inertia. 



We thus see that with Freedom of the second order there are only two possible 

 cases. Either every screw on the cylindroid is a principal Screw of Inertia or 

 there are neither more nor fewer than two such screws, and both real. 



If a and /3 be any two screws on the cylindroid then the conditions that all 

 the screws are Principal Screws of Inertia are 



With any rigid body in any position we can arrange any number of cylindroids 



