A Treatise on the Theory of Screws

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 0., 04, 0,„ 06 in terms of 04 and 0.2. We get 0» = P0. + Q02; 04 = P'0, + Q'02; 0, = P"0X + $'02; 0, = P"'0X + Q'"02. 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„, 04, 0,. in terms of 0X, 0.2, 0a, when we have 02 = P0X + Q0S +P05, 0t = P'0X + Q03 + R'0S, 0S = P"0X + Q"03 + R"0a. I his symmetucal process is specially convenient when the screws of reference are six canonical co-reciprocals. The general theory may also be set down. An «-system of screws is defined by 6 - n linear equations. These contain 5(6 - n) = 30-5« 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 »-system. The original number in the equation 30 — 5?i may be written n (6 - n) + (6 - ») (5 - n). The redundancy of (6— ri) (5—n) expresses the number of constants necessary for specifying 6 - n screws in a system of the (6 - n)th order.