432 THE THEORY OF SCREWS. [393-397 



we obtain a cubic for X. The three roots of this cubic correspond to three 77 

 screws. Take the 6 corresponding to one of the 77 screws, then, of course, 

 6 will not, in general, belong to the four-system. We can, however, assign 

 to Q any pitch we like, and as it intersects 77 at right angles, it must cut two 

 other screws of equal pitch on the cylindroid ( 22). Give to 6 a pitch equal 

 and opposite to that of the two latter screws, then 6 is reciprocal to the 

 cylindroid, and therefore it belongs to the four-system. We thus have a 

 permanent screw of the system, and accordingly we obtain the following 

 result : 



In the case of a rigid body with freedom of the fourth order there are, in 

 general, three, and only three, permanent screws. 



394. Freedom of the Fifth and Sixth Orders. 



When a rigid body has freedom of the fifth order, the screws about which 

 the body can be twisted are all reciprocal to a single screw p. In general, p 

 does not lie on the system prescribed by the equation which the co-ordinates 

 of all possible 77 screws have to satisfy. It is therefore, in general, not 

 possible that the reaction of the constraints can provide an 77. There are, 

 however, three screws in any five-system which possess the property of 

 permanent screws without however making any demand on the reaction of 

 the constraints. The existence of these screws is thus demonstrated : 

 Through the centre of inertia of the body draw the three principal axes, 

 then, on each of these axes one screw can always be found which is reciprocal 

 to p. Each of these will belong to the five-system, and it is obvious from 

 the property of the principal axes, that if the body be set twisting about one 

 of these screws it will have no tendency to depart therefrom. 



A body which has freedom of the sixth order is perfectly free. Any screw 

 on one of the principal axes through the centre of inertia is a permanent 

 screw, and, consequently, there is in this case a triply infinite number of 

 permanent screws. 



395. Summary. 



The results obtained show that for a rigid body with the several degrees 

 of freedom the permanent screws are as follows : 



No. of Permanent Screws 



VI Triply infinite 



