CHAPTEK V. 



THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE*. 



50. A Plane Representation. 



The essence of the present chapter lies in the geometrical representation 

 of a screw by a point. The series of screws which constitute the cylindroid 

 correspond to, or are represented by, a series of points in a plane. By 

 choosing a particular type of correspondence we can represent the screws 

 of the cylindroid by the points of a circle &quot;f. Various problems on the 

 cylindroid can then be studied by the aid of the corresponding circle. We 

 commence with a very simple process for the discovery of the circle. It will 

 in due course appear how this circular representation is suggested by the 

 geometry of the cylindroid itself ( 68). 



It has been shown ( 13) that the positions of the several screws on 

 the cylindroid may be concisely defined by the intersections of the pairs of 

 planes, 



y = x tan 0, 



z = in sin 26. 



In these equations, 9 varies in correspondence with the several screws, while 

 in is a parameter expressing the size of the cylindroid. In fact, the whole 

 surface, except parts of the nodal line, is contained between two parallel 

 planes, the distance between which is 2m. 



The pitch of the screw corresponding to 9 is expressed by 



P=pn + cos 20, 

 where p is a constant. 



* See papers in Proceedings of the lioyal Irish Academy, Ser. n. Vol. iv. p. 29 (1883), and the 

 Cunningham Memoirs of the lioyal Irish Academy, No. 4 (1886). 



t I may refer to a paper by Professor Mannheim, in the Comptes rendits for 2nd February, 

 1885, entitled &quot;Representation plane relative aux deplacements d une figure de forme invariable 

 assujettie a quatre conditions.&quot; Professor Mannheim here shows how the above plane repre 

 sentation might also have been deduced from the instructive geometrical theory which he had 

 brought before the Academy of Sciences on several occasions. 



