306 THE THEORY OF SCREWS. [286- 



a screw a. But the initial velocity of the body in this case will not generally 

 be cos (a.rj) +PO.- It may be easily shown to be 



But we have also 



whence in all cases 



/3-&PT, = O-57 a f . 



This formula is therefore much more general besides being more concise 

 than that of 281. 



287. System with Two Degrees of Freedom. 



Let A, B, C, X, &c., and A , B , C , X , &c., be two homographic systems 

 of points on a circle. These correspond respectively to two homographic 

 systems of screws on the cylindroid according to the method of representation 

 in Chap. XII. Then it is known, from geometrical principles, that if any 

 two pairs, such as A, A and B, B , be taken, the lines AB , BA intersect 

 on a definite straight line, which is the axis of the homography. 



In general this axis may occupy any position whatever ; if, however, it 

 should pass through 0, the pole of the axis of pitch, then the homography 

 will assume a special type which it is our object to investigate. 



In the first place, we may notice that under these circumstances the 

 homography possesses the following characteristic : 



Let A, B be two screws, and A , B their two correspondents ; then, if A 

 be reciprocal to B , B must be reciprocal to A . 



For in this case AB must pass through 0, and therefore BA must pass 

 through also, i.e. B and A must be reciprocal. 



This cross relation suggests a name for the particular species of homo 

 graphy now before us. The form of the letter ^ indicates so naturally the 

 kind of relation, that I have found it convenient to refer to this type of 

 homography as Chiastic. No doubt, in the present illustration I am only 

 discussing the case of two degrees of freedom, but we shall presently see 

 that chiastic homography is significant throughout the whole theory. 



288. A Geometrical Proof. 



It is known that in the circular representation the virtual coefficient of 

 two screws is proportional to the perpendicular distance of their chord from 

 the pole of the axis of pitch ( 61). 



