FREEDOM OF THE FOURTH ORDER. 233 



momental ellipsoid; every rigid body which fulfils nine conditions will 

 belong to this family. If an impulsive wrench applied to a member of 

 this family cause it to twist about a screw 0, then the same impulsive 

 wrench applied to any other member of the same family will cause it 

 likewise to twist about 0. If we added the further condition that the masses 

 of all the members of the family were equal, then it would be found that 

 the twist velocity, and the kinetic energy acquired in consequence of a 

 given impulse, would be the same to whatever member of the family the 

 impulse were applied ( 90, 91). 



223. Quadratic n-systems. 



We have always understood by a screw system of the nth order or briefly 

 an w-system, the collection of screws whose co-ordinates satisfy a certain 

 system of 6 n linear homogeneous equations. We have now to introduce 

 the conception of a screw system of the nth order and second degree or briefly 

 a quadratic n-system (n &amp;lt; 6). By this expression we are to understand a 

 collection of screws such that their co-ordinates satisfy 6 n homogeneous 

 equations ; of these equations 5 n, that is to say, all but one are linear ; 

 the remaining equation involves the co-ordinates in the second degree. 



Let #!,...,#&amp;lt;; be the co-ordinates of a screw belonging to a quadratic 

 w-system. We may suppose without any loss of generality that the 5 n 

 linear equations have been transformed into 



Q n+2 = ; n +3 = ; . . . K = 0. 



The remaining equation of the second degree is accordingly obtained by 

 equating to zero a homogeneous quadratic function of 



A A 



t/j ... t7 7 l+l 



We express this equation which characterizes the quadratic ?i-system as 



All the screws whose co-ordinates satisfy the 5 n linear equations must 

 themselves form a screw system of the 6 (5 n) = (n + l)th system. This 

 screw system may be regarded as an enclosing system from which the screws 

 are to be selected which further satisfy the equation of the second degree 

 [70 = 0. The enclosing system comprises the screws which can be formed by 

 giving all possible values to the co-ordinates 6 l , ...,0 n+l . 



Of course there may be as many different screw systems of the nth order 

 and second degree comprised within the same enclosing system as there can 

 be different quadratic forms obtained by annexing coefficients to the several 

 squares and products of n + 1 co-ordinates. If n = 5, the enclosing system 

 would consist of every screw in space. 



