290] DEVELOPMENTS OF THE DYNAMICAL THEORY. 307 



Let a, ft, 7 be three screws of one system, and let 77, f, be the three 

 corresponding screws, and, as usual, let r a f represent the virtual coefficient 

 of a and . Then whenever the homography is chiastic : 



This is geometrically demonstrated when the following theorem is 

 proved : 



If six points be inscribed on a circle, then the continued product of 

 the three perpendiculars let fall from any point in a Pascal line formed 

 from these six points upon three alternate sides of the corresponding 

 hexagon is equal to the continued product of the three perpendiculars let 

 fall from the same point on the other three sides. 



Let act , ft ft , 77 be the three pairs of sides, and write the equation 



afty = a ft y , 



then this represents a cubic curve through the nine points a a , aft , ..., and 

 this cubic can only be the circle and the Pascal line. 



289. Construction of Chiastic Homography on the Cylindroid. 



It is first obvious that, if two corresponding pairs of screws be arbitrarily 

 selected, it will always be possible to devise one chiastic homography of 

 which those two pairs are corresponding members. The circular construction 

 shows this at once for, join AB and A B, they intersect at T, then the line 

 TO is the homographic axis, and the correspondent to X is found by drawing 

 A X, and then AX through the intersection of A X and OT. 



290. Homographic Systems on Two Cylindroids. 



The fundamental theorem for the two cylindroids is thus expressed : 



Take any two screws, ct and ft, on one cylindroid, and any two screws, 

 77 and , on the other, it will then be possible to inscribe one, and in general 

 only one chiastic homography on the two surfaces, such that a and rj shall 

 be correspondents, and also ft and . 



For, write the general equation 



If, then, a, ft, 77, are known, and if 7 be chosen arbitrarily on the first 

 cylindroid, it will then be always possible to find one, but only one, screw 

 on the second cylindroid which satisfies the required condition. 



If a body had two degrees of freedom expressed by a cylindroid A, and 

 if an arbitrary cylindroid B were taken, then an impulsive wrench ad 

 ministered by any screw on B would make the body commence to twist 



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