104 DYNAMICS OF A RIGID BODY. 



The pitch of each reciprocal screw is equal in magni- 

 tude, and opposite in sign, to the pitches of the two 

 screws of equal pitch, in which it intersects the cylin- 

 droid ( 24). Now, the greatest and least pitches of the 

 screws on the cylindroid are/ and pp ( 20). For the 

 quantity p a cos 2 l + p$ sin *l is always intermediate be- 

 tween / a cos 2 /+/ a sin 2 / and pp cos z l + p sin 2 /. Hence 

 it follows that the generators of the cone which meet the 

 cylindroid in three real points must have pitches inter- 

 mediate between p a and pp. It is also to be observed that, 

 as only one line can be drawn through the vertex of the 

 cone to intersect any two given screws on the cylin- 

 droid, so only one screw of any given pitch can be found 

 on the reciprocal cone. 



One screw can be found upon the reciprocal cone 

 of every pitch from - oo to + oo . The line drawn through 

 the vertex parallel to the nodal line is a generator of 

 the cone to which infinite pitch must be assigned. Set- 

 ting out from* this line around the cone the pitch gra- 

 dually decreases to zero, then becomes negative, and 

 increases to negative infinity, when we reach the line 

 from which we started. We may here notice that when 

 a screw has infinite pitch, we may regard the infinity as 

 either + or - indifferently. If we conceive distances 

 marked upon each generator of the cone from the 

 vertex, equal to the pitch of that generator, then the 

 parallel to the nodal line drawn from the vertex forms 

 an asymptote to the curve so traced upon the cone. It 

 is manifest that we must admit the cylindroid to possess 

 imaginary screws, whose pitch is nevertheless real. 



The reciprocal cone drawn from a point to a cylin- 

 droid, is decomposed into two planes, when the point 

 lies upon the cylindroid. The first plane is normal 

 to the generator passing through the point. Every line 

 in this plane must, when it receives the proper pitch, be 



