156 THE THEORY OF SCREWS. [161, 



We can also determine the chord in a somewhat different manner, which 

 has the advantage of giving certain other expressions that may be of 

 service. 



Let U = be the cubic curve. 



Let V= be the equation of the two straight lines from the origin to the 

 points of intersection with the two equal pitch screws + 6. 



Let L = be the chord joining the two intersections of U and V, distinct 

 from the origin : this is, of course, the chord now sought for. Then we must 

 have an identity of the type 



cU=VX 



where c is some constant. For the conditions L = and V= imply U=0, and 

 L cuts U in three points, two of which lie on V, and the third point, called /, 

 must lie on X. The line X is otherwise arbitrary, and we may, for con 

 venience, take it to be the line 11 from the origin to /. The product VX 

 thus contains only terms of the third degree, and accordingly the terms of 

 the second degree in U must be sought in LY. 



Let U=u 3 +u 2 where u 3 and u 2 are of the third and second degrees 

 respectively, then cu. 2 must be the quadratic part of the product L Y. As L 

 does not pass through the origin, it must have an absolute term, conse 

 quently Y must not contain either an absolute term or a term of the first 

 degree. If, therefore, c be the absolute term in L, it is plain that Y must 

 be simply u 2 , and we have accordingly, 



c (u s + MS) = VX + (L + c) u 2 , 



where L denotes the value of L without the absolute term : we have con 

 sequently the identity 



cu 3 = VX + L u 2 . 



In this equation we know u z , u 3 , V, and the other quantities have to be found. 



If we substitute 



x = y cos /3 tan (a + 6), 



we make V vanish, and representing L by \x + py, we find 



X cos ft tan (a 6) p = c T~ 5 - ^ &amp;gt; 



h tan /9 + ra sin 20 



and after a few steps 



\ m cos 2a + tn cos 20 



c - h* tan /3 + ?/i 2 cot /3 sin 2 2 



p _ h sin /3 + m cos ft sin 2a 

 c ~~ h 2 tan /3 + ??i 2 cot ft sin 2 2 



