120] 



FREEDOM OF THE SECOND ORDER. 



109 



Let X, JJL, v be three screws upon a cylindroid, and let A, B, C denote the 

 angles between /LI v, between v X, and between X //., respectively. If wrenches 

 of intensities X&quot;, // , v&quot;, on \, /*, v, respectively, are in equilibrium, we must 

 have ( 14): 



_X&quot; p&quot; v&quot; 



sin A sin B ~~ sin C 



But we have also as a necessary condition that if each wrench be resolved 

 into six component wrenches on six screws of reference, the sum of the 

 intensities of the three components on each screw of reference is zero ; 

 whence 



Xj sin A + /ij sin B + v l sin (7 = 0, 



X sin A + p, 6 sin B + v 6 sin G = 0. 

 From these equations we deduce the following corollaries : 



The screw of which the co-ordinates are proportional to aXj + bfa , ... 

 X 6 + bfj, G , lies on the cylindroid (X, //.), and makes angles with the screws 

 X, IJL, of which the sines are inversely proportional to a and 6. 



The two screws, of which the co-ordinates are proportional to 

 aXj 6/ij, ... aX 6 + &/A G , 



and the two screws X, /* are respectively parallel to the four rays of a plane 

 harmonic pencil. 



120. Screws on One Line. 



There is one case in which a body has freedom of the second order that 

 demands special attention. Suppose the two given screws 9, &amp;lt;f&amp;gt;, about which 

 the body can be twisted, happen to lie on the same straight line, then the 

 cylindroid becomes illusory. If the amplitudes of the two twists be 6 , $&amp;gt; , 

 then the body will have received a rotation 6 + &amp;lt;f&amp;gt; , accompanied by a trans 

 lation p e + &amp;lt;f&amp;gt; p&amp;lt;t&amp;gt;. This movement is really identical with a twist on a 

 screw of which the pitch is : 



O pe + 



& + &amp;lt;!&amp;gt; 



Since , &amp;lt;/&amp;gt; may have any ratio, we see that, under these circumstances, the 

 screw system which defines the freedom consists of all the screws with 

 pitches ranging from -co to +00, which lie along the given line. It 

 follows ( 47), that the co-ordinates of all the screws about which the 

 body can be twisted are to be found by giving x all the values from 

 QO to + GO in the expressions : 



a x dR * x dR 

 l + ~ &quot; K + 



in which 



