108 THE THEORY OF SCREWS. [117- 



From the condition of reciprocity we must have : 



P\n\ (i cos I + & sin /)+...+ p ti rj 6 ( cos I + /3 6 sin I) = 0, 

 or, OT ar| cos I + zap,, sin I = 0. 



From this tan I is deduced, and therefore the screw becomes known 

 ( 26). 



In general if tn^ be the virtual coefficient of any screw 77 and a screw 

 on the cylindroid, we have 



V*r,0 = ^ar, COS I + CT^, SU1 I \ 



whence if on each screw a distance be set off from the nodal line equal to 

 the virtual coefficient between 77 and 6, the points thus found will lie on 

 a right circular cylinder, of which the equation is ; 



X? + y* = TS^X + TX^y. 



Thus the screw which has the greatest virtual coefficient with 77 is at 

 right angles to the screw reciprocal to 77, and in general two screws can be 

 found upon the cylindroid which have a given virtual coefficient with any 

 given external screw. 



118. Relation between Two Cylindroids. 



We may here notice a curious reciprocal relation between two cylindroids, 

 which is manifested when one condition is satisfied. If a screw can be found 

 on one cylindroid, which is reciprocal to a second cylindroid, then conversely 

 a screw can be found on the latter, which is reciprocal to the former. Let 

 the cylindroids be (a, /3), and ( X, fi). If a screw can be found on the former, 

 which is reciprocal to the latter, then we have : 



pAi (i cos I + /3j sin 1) -f . . . + p n ^n (oi cos I + /3 n sin I) = 0, 

 Pif^i (i cos I + ft sin 1) + . . . 4- pn^n ( a n cos I + ft n sin I) = 0. 

 Whence eliminating I, we find : 



As this relation is symmetrical with regard to the two cylindroids, the 

 theorem has been proved. 



119. Co-ordinates of Three Screws on a Cylindroid. 



The co-ordinates of three screws upon a cylindroid are connected by four 

 independent relations. In fact, two screws define the cylindroid, and the 

 third screw must then satisfy four equations of the form ( 20). These 

 relations can be expressed most symmetrically in the form of six equations, 

 which also involve three other quantities. 



