31 J SCREW CO-ORDINATES. 33 



30. The Intensity of the Resultant may be expressed in terms of the 

 intensities of its components on the six screws of reference. 



Let a be any screw of pitch p a , let p lt p. 2 , &c. p e be the pitches of the 

 six screws of reference w l} o&amp;gt; 2 , ... &&amp;gt; 6 ; then taking each of the screws of refer 

 ence in succession, for 77 in 29, and remembering that the virtual coefficient 

 of two coincident screws is simply equal to the pitch, we have the following 

 equations : 



ar 6 = ai + + cr-BT,,, + ap s . 

 But taking the screw p in place of 77 we have 



&amp;lt;*&quot;pa = l&quot;r ttl + tt/ ^ae. 



Substituting for ^ al ... -5r o6 from the former equations, we deduce 

 pj 1 * = $ OW) + 22 (/ V^ 12 ). 



This result may recall the well-known expression for the square of a force 

 acting at a point in terms of its components along three axes passing through 

 the point. This expression is of course greatly simplified when the three 

 axes are rectangular, and we shall now show how by a special disposition 

 of the screws of reference, a corresponding simplification can be made in the 

 formula just written. 



31. Co-Reciprocal Screws. 



We have hitherto chosen the six screws of reference quite arbitrarily ; 

 we now proceed in a different manner. Take for &) 1? any screw; for co 2 , any 

 screw reciprocal to a^; for o&amp;gt; 3 , any screw reciprocal to both &&amp;gt;! and &&amp;gt; 2 ; f r &&amp;gt; 4 , 

 any screw reciprocal to o) 1} &&amp;gt;.,, &&amp;gt; 3 ; for &&amp;gt; 5 , any screw reciprocal to a&amp;gt; l , &amp;lt;o 2 &amp;gt; MS, &&amp;gt; 4 ; 

 for &&amp;gt; 6 , the screw reciprocal to Wj, &&amp;gt;.,, &&amp;gt; 3 , &&amp;gt; 4 , o&amp;gt; 5 . 



A set constructed in this way possesses the property that each pair 

 of screws is reciprocal. Any set of screws not exceeding six, of which each 

 pair is reciprocal, may be called for brevity a set of co-reciprocals*. 



Thirty constants determine a set of six screws. If the set be co- 

 reciprocal, fifteen conditions must be fulfilled ; we have, therefore, fifteen 

 elements still disposable, so that we are always enabled to select a co- 

 reciprocal set with special appropriateness to the problem under con 

 sideration. 



* Klein Las discussed (Math. Ann. Band n. p. 204 (1869)) six linear complexes, of which each 

 pair is in involution. If the axes of these complexes be regarded as screws, of which the 

 &quot; Hauptparameter &quot; are the pitches, then these six screws will be co-reciprocal. 



B. 3 



