378] THE THEORY OF PERMANENT SCREWS. 415 



a (6, - 0.) + Y(0 r , + e G )-Z (0 3 + 4 ), 



6 (B, -0J + Z (0! + 2 ) - X (e r&amp;gt; + 6 ), 



c (A - 0) + X (A + 4 ) - I 7 (9, + 4). 



Before substitution for X, Y, Z it will be convenient to use certain abbre 

 viations, 



0i 2 = 1 ; &i 2 = pi ; 0i + 0. 2 = \i , BI + 2 = &)] , 



/]//}/ /} A Q i Q \ Q \ Q 



&quot;3 #4 = e 2 , 3 (7 4 = p 2 , &quot;3 + #4 = A-2- U 3 + (7 4 = &amp;lt;W 2 , 



$&amp;gt; # = e 3 ; 5 $ 6 = p 3 ; #5 + #/ = X 3 , 5 + fi = &amp;lt;w s - 



With these substitutions in v 2 the square of the velocity of the element we 

 readily obtain after integration and a few reductions and taking the total 

 mass as unity, 



2 c 2 ) 



. 2 (i)i bc^p^cos X 2 &amp;lt;w 3 &)j (c 2 a 2 ) 

 &amp;gt; 3 &)i XsO^a^ (a 2 6 2 ), 

 whence we easily find 



7/TT 



If T;/ , ... rj s &quot; be the co-ordinates of the restraining wrench, then, as shown 

 in 368, 



&quot;= 1^ T 



P*d0r 



whence we deduce the following fundamental expressions for the co 

 ordinates of the restraining wrench : 



pMi&quot; = acp 3 w 2 + abp 2 w s + (6 2 c 2 ) &) 2 &) 3 , 

 PM-i = + acp 3 (o. 2 abp 2 co 3 + (b&quot; c 2 ) &&amp;gt; 2 &&amp;gt; 3 , 

 PMs&quot; = abpico 3 + cbpsWi + (c 2 a 2 ) w^w^ 

 PPl&quot; = + abpitos cbpsW! + (c 2 a 2 ) o) s o) l , 

 PS^S&quot; = bcpzW! + acpiO) 2 + (a 2 6 2 ) w^w*, 

 Pets&quot; = + fop^ acpi(o 2 + (a 2 6 2 ) 



As usual, we here write for symmetry 

 p l= = + a; p., = -a; p 3 = + b; 



We verify at once that 



Pw&quot;0 1 + ... 



but this is of course known otherwise to be true, because the restraining 

 screw must be reciprocal to the instantaneous screw. 



