1 6 THE CYLINDROID. 



Each generating line of the surface is conceived to 

 be the residence of a screw, the pitch of which is deter- 

 mined by the expression p a cos */ + pp sin V. 



We shall now show that a cylindroid can be described 

 so as to contain any two screws. When a cylindroid is 

 said to contain a screw, it is not only meant that the 

 screw is one of the generators of the surface, but that 

 the pitch of the screw is identical with the pitch appro- 

 priate to the generator with which the screw coincides. 



Let the two given screws be 9 and 0, the length of 

 their common perpendicular be 7z, and the angle between 

 the two screws be A ; we shall show that by a proper 

 choice of the origin, the axes, and the constants p a and p^ 

 a cylindroid can be found which contains 9 and $. 



If /, m be the angles which two screws on a cylin- 

 droid make with the axis of x, and if z,, z z be the corre- 

 sponding values of z, we have the equations 



pe = pa cos *t + Pp sin % z i = (A - AO sin / cos /, 



A> - Pa cos *m +pp sin 2 ;;/. z 2 = (p a ~ Pi) si n m cos m - 

 A = / - m, h = Zi - z 2y 



which the axis makes an angle w with the axis of x is k = #> cos ! at -f sin* w. 

 The writer was informed by Dr. Felix Klein that Pliicker had also constructed 

 a model of this surface. 



Pliicker does not appear to have contemplated the mechanical and kinema- 

 tical properties of the cylindroid, with which alone we are concerned ; but it 

 is worthy of remark that the distribution of pitch which is presented by physi- 

 cal considerations is exactly the same as the distribution of parameter upon the 

 generators of the surface, which was fully discussed by Pliicker in connexion 

 with his theory of the linear complex. 



The name cylindroid was suggested by Professor Cayley in reply to a re- 

 quest of the writer. The word originated in the following construction for 

 the surface, which was also communicated by Professor Cayley. Cut the 

 cylinder cc* +y* = (pp p a }* in an ellipse by the plane z = x, and consider the 

 line x = o, y p$ p a . If any plane z = c cuts the ellipse in the points A, B 

 and the line in (7, then CA, CB are two generating lines of the surface. 



