A Treatise on the Theory of Screws

378 THE THEORY OF SCREWS. [346- In the choice of a screw-chain about which a mass-chain with four degrees of freedom can twist there are three arbitrary elements. We may choose as the first screw of the chain any screw from a given four-system. If one screw of the chain be moved over a two-system, or a three-system included in the given four-system, then every other screw of the chain will also describe a corresponding two-system or three-system. 347. Freedom of the fifth order. In discussing the movements of a system which has freedom of the fifth order, the analogies which have hitherto guided us appear to fail. Homo- graphic pencils, planes, and spaces have exhibited graphically the relations of the lower degrees of freedom; but for freedom of the fifth degree these illustrations are inadequate. No real difficulty can, however, attend the extension of the principles we have been considering to the freedom of the fifth order. We can conceive that two five-systems are homographically related, such that to each screw on the one corresponds one screw on the other, and conversely. To establish the homography of the two systems it will be necessary to know the six screws on one system which correspond to six given screws on the other: the screw in either system corresponding to any seventh screw in the other is then completely determined. In place of the methods peculiar to the lower degrees of freedom, we shall here state the general analytical process which is of course available in the lower degrees of freedom as well, A screw 9 in a five-system is to be specified by five co-ordinates dlt d2, da, di, d6. These co-ordinates are homogeneous; but their ratios only are con- cerned, so they are equivalent to four data. The five screws of reference may be any five screws of the system. Let </> be the screw of the second system which is to correspond to d in the first system. The co-ordinates of may be referred to any five screws chosen in the second system. It will thus be seen that the five screws of reference for are quite different from those of d. The geometrical conditions expressing the connection between </> and d will give certain equations of the type cf>2= U2-, ^>5=US, where Ult U5 are homogeneous functions of d2, 0S. These equations express that one d determines one <p. As however one <[> is to determine one d we must have also equations of the type di=U'- 02=U2’-, ds=U5, where Ux', Us' are functions of </>,, From the nature of the problem these functions are algebraical and as they must be one valued they must be rational functions. We have therefore