A Treatise on the Theory of Screws

 312 THE THEORY OF SCREWS. [293- homography, and the correspondent 0 to any other screw co is assigned by the condition that the anharmonic ratio of is the same as that of Reverting to our original screws a and y, ß and £ we now see that they must fulfil the conditions (iiWW) = (ØA^a), (<öi«2to2?) = (.010203/3) when the quantities in the brackets denote the anharmonic ratios. It can be shown that these equations lead to the formulae of § 281. 294. Exception to be noted. We have proved in the last article an instructive theorem which declares that when two cylindroids are given it is generally possible in one way, but in only one way, to correlate the several pairs of screws on the two surfaces, so that when a certain free rigid body received an impulse about the screw on one cylindroid, movement would commence by a twisting of the body about its correspondent on the other cylindroid. It is, however, easily seen that in one particular case the construction for correlation breaks down. The exception arises whenever the principal planes of the two cylindroids are at right angles. The two correspondents on P to the zero-pitch screws on A had been chosen from the property that when pa is zero the impulsive wrench must be perpendicular to a. We thus take the two screws on P which are respec- tively perpendicular to the two zero-pitch screws. But suppose there are not two screws on P which are perpendicular to the two zero-pitch screws on A. Suppose in fact that there is one screw on P which is parallel to the nodal axis of A, then the construction fails. We would thus have a single screw on P with two corresponding instantaneous screws for the same body. This is of course impossible, and accordingly in this particular case, which happens when the principal planes of P and A are rectangular, it is impos- sible to adjust the correspondence. 295. Impulsive and Instantaneous Cylindroids. Let X, X' be two screws on a cylindroid whereof a and ß are the two principal screws. Let 0, 0' be the angles which X and X' respectively make with a. We shall take the six absolute screws of inertia as the screws of reference and we have as the co-ordinates of X äj cos 0 + /3j sin 0,... as cos 0 + ß6 sin 0,