68 THE THEORY OF SCREWS. [77, 78 



But it will be observed that there is here a mass of not fewer than 

 20 independent constants, while the cylindroid is itself completely defined by 

 eight constants ( 75). The reason is that these four equations really each specify 

 one screw, i.e. four screws in all, and as each screw needs five constants 

 the presence of 20 constants is accounted for. 



But when it is the cylindroid alone that we desire to specify there is no 

 occasion to know these four particular screws. All we want is the system of 

 the fourth order which contains those screws. For the specification of the 

 position of a screw in a four-system three constants are required. Thus the 

 selection of four screws in a given four-system requires 12 constants. These 

 subtracted from 20 leave just so many as are required for the cylindroid. 



This is of course the interpretation of the process of solving for 3 ,0 4 , r&amp;gt; , 6 

 in terms of 9 and 2 . We get 



3 = P0, + Q0 2 6 4 = P 6, + Q 2 5 = P&quot;0, + Q&quot;0 2 ; 8 6 = P&quot; 0, + Q &quot;0 2 . 



Thus we find that the constants are now reduced to eight, which just serve 

 to specify the cylindroid. 



An instructive case is presented in the case of the three-system. The 

 three linear equations of the most general type contain 15 constants. But 

 a three-system is defined by 9 constants ( 75). This is illustrated by solving 

 the equations for 0. 2 , 4 , K in terms of 1; 3 , r ,, when we have 



0, = P0, + Q0 3 + RB S , 

 4 = P B, + Q S + R0 5 , 

 6 = P&quot;0, + Q&quot;d, + R&quot;0 5 . 



This symmetrical process is specially convenient when the screws of reference 

 are six canonical co-reciprocals. 



The general theory may also be set down. An ?i-system of screws is 

 defined by 6 n linear equations. These contain 5(6 n) = 30 5n constants. 

 We can, however, solve for 6 n of the variables in terms of the remaining n. 

 Thus we get 6 n equations, each of which has n constants, i.e. n (6 n) in 

 all. This is just the number of constants necessary to specify an ?i-system. 

 The original number in the equation 30 - 5n may be written 



n (6 - n) + (6 - n) (5 - n). 



The redundancy of (6 n) (5 n) expresses the number of constants necessary 

 for specifying 6 n screws in a system of the (6 n)th order. 



