312 THE THEORY OF SCREWS. [293- 



homography, and the correspondent 6 to any other screw &&amp;gt; is assigned by 



the condition that the anharmonic ratio of (a^^co is the same as that of 

 a a A A 



1/11/2(73(7. 



Reverting to our original screws a and rj, ft and , we now see that they 

 must fulfil the conditions 



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 p a 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 jB are the two 

 principal screws. 



Let 0, 6 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 



! cos 6 + & sin 6, . . . a cos 6 + /3 6 sin 6, 



