302 THE THEORY OF SCREWS. [283, 



multiplying the equations severally by pi,2h&amp;gt; an d adding, we get 

 dSpiXA = W Pi (Pi cos (&quot;ni ) + & cos (772 )+...+ &j cos (176)) 



= ^&quot;TSfa (since p^ is indefinitely large), 



whence we proceed as before and we see that the theorem (iii) remains true, 

 even if p^ or p$ or pg be infinite. 



If p a be zero, then in general cos aeo is zero. But in this case p a -f- cos aw 

 becomes d a the length of the perpendicular from the centre of gravity upon 

 a. Hence we have 



and the proof proceeds as before so that in this case also the theorem holds 

 good. 



Finally, let jj a be infinite, &amp;lt;u must then be of zero pitch and pass through 

 the centre of gravity and 



dp a = CD &quot;. 

 We have 



o&amp;gt; 1 = cosai, o) 3 = cos 

 so that the equations (i) become 



| a&amp;gt;&quot; cos (ai ) = 77&quot; rj l + p &quot;pi , \u&amp;gt;&quot; cos (ai ) = r) &quot;t] 2 + p &quot;p 2 , 

 | w&quot; cos (as ) = 7/&quot;7? 3 + p^Vs, 2 &amp;lt;u/ &quot; cos ( a 3 ) = 7y&quot; 74 + p &quot;pi, 

 \&amp;lt;a&amp;gt;&quot; COS (as) = r) &quot;&amp;lt;r)- a + p &quot;p 5 , \&amp;lt;o&quot; COS (05) = ^ &quot;^ + p&quot;p s . 



Multiplying these equations by +a^ lt a/3 2 , +b/3 3 , 6/3 3 , ... and adding, 

 we have 



| &amp;lt;o&quot; [a (fa - /8 a ) cos (ai ) + b (j3 3 - /3 4 ) cos (as ) + c (/8 5 - /S 6 ) cos (as )] = 17 &quot;^ 



Let a- be the screw belonging to the reciprocal system on which there is 

 an impulsive wrench of intensity a &quot; due to the reactions when an impulsive 

 wrench is administered on . Then we have 



/3a/3i = ^ &quot;^i + o-&quot; o-j ; - $afi t = %&quot; % z + o-&quot;V 2 , 

 whence 



/8a (fii y8 2 ) cos (ai ) = f &quot; cos (|i ) cos (ai ) + a&quot; cos (&amp;lt;ri ) cos (ai ), 

 with similar expressions for the two other pairs, whence by addition 



/8 {a (fr - yS 2 ) cos (ai )+ b (/3 3 - /9 4 ) cos (as) + c (J3 S - yS 6 ) cos (as)] = %&quot; cos (a|), 

 for since a and a- are reciprocal and p a = oo we must have cos (a&amp;lt;r) = 0. 



