416 THE THEORY OF SCREWS. [378- 



These equations enable us to study the correspondence between each 

 instantaneous screw 6 and the corresponding restraining screw 77. It is to 

 be noted that this correspondence is not of the homographic, or one-to-one 

 type, such as we meet with in the study of the Principal Screws of Inertia, 

 and in other parts of the Theory of Screws. The correspondence now to 

 be considered has a different character. 



379. Limitation to the position of the Restraining Screw. 



If a particular screw 6 be given, then no doubt, a corresponding screw r) 

 is given definitely, but the converse is not true. If 77 be selected arbitrarily 

 there will not in general be any possible 6. If, however, there be any one 0, 

 then every screw on the same axis as 6 will also correspond to the same 77. 



From the equations in the last article we can eliminate the six quantities, 

 6 l , . . . (i ; we can also write ??/ = i)&quot;^, . . . rj n &quot; = tj&quot;r) n where 77&quot; is the intensity 

 of the restraining wrench and ^,,..1)^ the co-ordinates of the screw on 

 which it acts. 



We have a (%&quot; + V ) = 2abp 2 a) 3 2ac/&amp;gt; 3 &amp;lt;w 2 , 



2 c 2 r) 1 + 7; 2 , p. 2 p 3 

 whence = 6 c , 



a 7?! 7/2 C0 2 W 3 



and from the two similar equations we obtain, by addition, 

 b- j^ 77! + 7/ 2 c 2 a 2 773 + 774 a 2 6 2 % + 77,; _ 



a % ^2 b *73 - *?4 c 775-776 



It might at first have been supposed that any screw might be the possible 

 residence of a restraining wrench, provided the corresponding instantaneous 

 screw were fitly chosen. It should however be remembered that to each 

 restraining screw corresponds a singly infinite number of possible instan 

 taneous screws. As the choice of an instantaneous screw has five degrees 

 of infinity, it was to be presumed that the restraining screws could only 

 have four degrees of infinity, i.e. that the co-ordinates of a restraining screw 

 must satisfy some equation, or, in other words, that they must belong to a 

 screw system of the fifth order, as we have now shown them to do. 



380. A verification. 



We confirm the expression for the co-ordinates of 77 in the following manner. 

 It has been shown ( 376) that so long as retains the same direction and 

 situation, its pitch is immaterial so far as 77 is concerned. This might have 

 been inferred from the consideration that a rigid body twisting about a 

 screw has no tendency to depart from the screw in so far as its velocity of 

 translation is concerned. It is the rotation which necessitates the restrain 

 ing wrench if the motion is to be preserved about the same instantaneous 



