152 



THE THEORY OF SCREWS. 



[160 



160. Equation to plane section of Cylindroid. 



Each generator of the cylindroid is the abode of a certain screw, and 

 accordingly each point in a plane section will lie on one screw, and generally 

 on only one. We may, accordingly, regard the several points of the cubic as 

 in correspondence with the several screws on the cylindroid. It will often be 

 convenient to speak of the points on the section as synonymous with the 

 screws themselves which pass through those points. 



We must first investigate the equation* to the cubic curve produced by 

 cutting the cylindroid by a plane situated in any arbitrary position. 



Fig. 35. 



Let OX and OF (Fig. 35) be the two principal screws of the cylindroid of 

 which OH is the nodal line. Let XYl be the arbitrary plane of section. 

 The position of this plane is defined by the magnitudes h, a, (3, whereof h 

 is the length of the perpendicular from on XY, a. is the angle between OR 

 and OX, and ft is the angle ORl, or the inclination of the plane of section 

 to the principal plane of the cylindroid. 



Draw through H the line 1$ parallel to XY; then we shall adopt IN 

 as the new axis of x and SIR as the new axis of y, so that if P be any point 

 on the surface, we have PN = y and IN = x. The dotted letters, x , y , z 

 refer to the original axes of the cylindroid. Let fall PT perpendicular on 

 the plane of OXY, and TM perpendicular to XY. Then we have MN= 



whence 



y + z cosec ft = h sec /3 (i), 



* Transactions of the Eoyal Irish Academy, Vol. xxix. p. 1 (1887). 



