254 THE THEORY OF SCREWS. [235, 



11 11 



substitute p , = . = . 6 = x = -\ 



ft ?&amp;gt; ft y 



and suppose qi, q z , qa, q*, q*, q& to be in descending order of magnitude. 



&quot;X 2 -\ 2 &quot;\ 2 



Thus -JSL + _*_ + . ..+_*_=&amp;lt;). 



y - ?i y - & y - ft 



That is v(y - ft)(y - ?,)(y - ? 4 ) (y -?5)(y - &) + 



+ V (y - ?0 (y - 9-2) (y - &) (y - ? 4 ) (y - ? B ) = 0. 



In the left-hand member of this equation substitute the values 



qi, q*, q a , ?4&amp;gt; q*, q* 



successively for y ; five of the six terms vanish in each case, and the values of 

 the remaining term (and therefore of the whole member) are alternately 

 positive and negative. 



The five values of y must therefore lie in the intervals between the six 

 quantities q lt q 2 , ... q s , the roots are accordingly proved to be real and distinct 

 (unless one of the quantities \ 1} \ 2 , X 3 , \ 4 , \ 5 , X 6 = and a further condition 

 hold, or unless some of the quantities q 1} ... q e be equal). 



The values of p 1} ... p 6 are a, b, c; and we suppose a, b, c, positive 

 and a &amp;gt; b &amp;gt; c. 



The values of y lie in the successive intervals between 



1 1 1 _1 _1 _1 



c b a a b c 



and consequently of the roots of the equation in x. 



Two are positive and lie between a and b, and between b and c respectively. 



Two are negative and lie between a and b, and between b and c 

 respectively. 



The last is either positive and &amp;gt; a or negative and &amp;lt; a. 



236. The Pectenoid. 



A surface of some interest in connection with the freedom of the fifth 

 order may be investigated as follows. 



Let a be the pitch of the one screw &&amp;gt;, to which the five system is 

 reciprocal. 



Take any point on o&amp;gt; and draw through any two right lines OF, and 

 OZ which are at right angles and which lie in the plane perpendicular 

 to a). 



