622 Froceedings of the Royal Irish Academy. 



between a and fB ; and if ebe the angle between a and ;8, and be the 

 angle made by p with a parallel to the bisector of the angle e, then we 

 have from the above 



% -.h : : sin 2<^ : sin 2e. 



The equation of the surface S is now deduced for 



X 



tan 6 = - ; 



y 



whence we obtain the equation of the cylindroid iu the well-known 

 form 



^ "^ ^ sm2e -^ 



The law of the distribution of pitch upon the cylindroid can also be 

 deduced fi'om the same principles. If a and f3 are screws of zero pitch, 

 then any reciprocal transversal 6 will be also of zero pitch ; and as p 

 must be reciprocal to 6, it will follow that the pitch of p must be 

 equal to the product of the shortest perpendicular distance between p 

 and 0, and the tangent of the angle between the two lines. In short, 

 the pitch of p must simply be equal to what is sometimes called the 

 moment between p and 6. 



