215] FREEDOM OF THE FOURTH ORDER. 



whence eliminating z and observing that X //, = 90 we obtain, 



p d = a cos 2 X + 6 sin 2 X, 



and eliminating p e , 



(b a) sin X cos X = z. 

 If we desire the equation of the surface we have 



y = x tan X, 

 and hence finally 



225 



Thus again we arrive at the well-known equation of the cylindroid. 



We can also prove in the following manner the fundamental theorem 

 that among the screws belonging to any two-system there are two which 

 intersect at right angles ( 13). 



Let 6 be any screw of the two-system, and accordingly the six co-ordinates 

 of 9 must satisfy four linear equations which may be written 



If be a screw which intersects 6 at right angles, then we must 

 also have 







inasmuch as these screws are reciprocal as well as rectangular. 



From these six equations O l ,...0 6 can be eliminated, and we have the 

 resulting equation in the co-ordinates of &amp;lt;f&amp;gt;, 



B. 



= o. 



15 



