NOTE ON THE SKEW SURFACES 
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But we have 
and, writing 
we find 
«! 2 + ßi + 7i 2 = 1, 
a u ßi, 7i = a + a'ds, ß+ß'ds, 7 + 7'ds, 
aiXj + ßßj + 77j = a 2 + ß 2 + <y 2 + (aa' + ßß' + 77') ds = 1 ; 
the equation thus is 
that is, 
the required equation. 
(«!«' + ß,y' + y x z r ) - (a«' + /%' + 7/) = 0 ; 
o!x + ß'y' + 7 ' z = 0, 
Calling the inclination of the generating line through the point (x, y, z) to the 
tangent of the line of striction the “obliquity,” and denoting it by co, we have 
ax' + ¡3y' + 7z = cos co. 
Calling the inclination of the two consecutive generating lines divided by the 
shortest distance between these lines the “torsion,” and denoting it by t“ 1 , we have 
a' 2 + ß' 2 + 7 2 = 
Sin- ft) 
In proof hereof, if for a moment <f> is the inclination of the lines L and L l to 
each other, then 
cos <j> = aa 1 + /3/3i + 77 x ; 
and therefore 
sin 2 <f> = (/3yj - &7) 2 + (7a! - 7! a) 2 + (a- a^) 2 , 
viz. writing 
«1, fti, 71 = a + o-'ds, /3 + /3'ds, 7 + yds, 
this is 
sin 2 </> = ds 2 {(/37'- /3V) 2 + (7a' — 7 / a) 2 + (a/3' — a'/3) 2 } 
= ds 2 {(a 2 + /3 2 + 7 2 ) (a' 2 + /3' 2 4- 7 /2 ) — (aa' + /3/3' + 77O 2 }, 
= ds 2 (a' 2 + /3' 2 + 7 /2 ); 
whence, if for a moment the shortest distance between the two lines is called S, 
then we have 
sin (f> _ _j _ sin 
S ds sin &) 5 
that is, 
sin 2 û) sin 2 , Q , , 
the required equation.