418 THE THEORY OF SCREWS. [381- 



The equations to the axis of the screw are 



ap l + 3/0)3 zo) 2 _ bp 2 + za&amp;gt;i #a) 3 _ c/3 3 + xa) 2 yw^ 



0&amp;gt;1 &&amp;gt;2 W 3 



If x, y, z are all simultaneously zero, then 



ap 1= bp 1 = cp3 



ft)! ft&amp;gt; 2 0&amp;gt; 3 



and these are, accordingly, the conditions that the instantaneous screw passes 

 through the centre of gravity. 



With these substitutions the co-ordinates become 



p l7)l &quot; = (6 2 - c 2 ) ft&amp;gt; 2 ft&amp;gt; 3 ; Pals&quot; = (c 2 - a 2 ) 30i ; PM*&quot; = (a 2 - & 2 ) iw 2 , 

 PM 9 &quot; = (& - c 2 ) &&amp;gt; 2 a&amp;gt;3 ; PM&quot; = (c 2 - a 2 ) u,^ ; jp^&quot; = (a 2 - 6 2 ) &) 1 a) 2 ; 

 remembering that ^ = + a ; p 2 = - a, &c., we have 



% &quot; + V =0; 7 7 / + 7 74 // = 0; V+W-0; 



but these are the conditions that the pitch of 77 shall be infinite ; in other 

 words the restraining wrench is a couple, as should obviously be the case. 



From the equations already given, we can find the co-ordinates of the 

 instantaneous screw in terms of those of the restraining screw. 



We have 



__ 



~ 



2 (6 2 - c 2 ) (c 2 - a 2 ) (a 2 - 6 2 ) 



,-j-f O TT . JJ 



and O&amp;gt;i = tl - 77 -fj: ] f&amp;gt;2 = = ti T~, T, 77V j ^3 &quot; ~~7 77 /7\ 



a (r} 1 7/2 ) (773 ^4 j C ^7/5 f] 6 ) 



If we make 

 then we have 



3 

 &quot;* 



- . _ 



G &quot; 



2pT &quot;2o6c 



In these expressions, /&amp;gt; fl is the pitch of 6, and is, of course, an indeterminate 

 quantity. 



382. Remark on the General Case. 



If the freedom of a body be restricted, then any screw will be permanent, 

 provided its restraining screw belong to the reciprocal system. For the body 



