| J( THE CYCLIDE. 



aenta such a cyclide. The outer sheet with its circles of contact and re-entering 

 conical points, and the inner spindle with its conical points meeting those of 

 tlu- outer sheet, have a certain resemblance to the outer and inner sheets 



f FrvsMfl's Wave-Surface ; and, if we bear in mind that the wave-surface 

 has four such singular points while the cyclide has only two, we may find 

 Figure III. useful in forming an idea of the singular points of the wave-surface. 



If we give to r all values from -f to o> , the cyclide assumes the 

 forms (3), (2), (1), (-1), (-2), (-3) in succession, and every point of space 

 w traversed four times by the surface. For when r is infinite, any given 

 point R is within the spindle or inner sheet of (3). As r diminishes, the 

 spindle contracts, and when r = c it vanishes; so that for a certain value, r,, 

 greater than c, the surface of the spindle passes through the point R. 



At this instant the outer sheet of the cyclide is still beyond R, but as 

 r diminishes, the surface contracts, and finally vanishes when r = b, before 

 which it must have passed through a value ?,, for which the surface passes 

 through R. At this instant the surface may have the form either of the outer 

 sheet of (3), or of the ring cyclide of one sheet (2), or of the negative lobe 



"I" (I)- 



The positive lobe of (1) begins to appear when r becomes less than b, and 



increases as ; diminishes, till when r= b it becomes a ring, and when r= c 

 it becomes the outer sheet of the cyclide ( 3). 



This surface, therefore, for some value, r t , of r, passes through the point R. 

 This value r, is necessarily less than ?v 



When r = c the interior sheet of ( 3) is developed, and increases Ln- 



Infinitely as r diminishes, so that for some value, r v of r, which is less than 

 < the point R is on the surface of this interior sheet. We thus see that the 

 cyclide may be said to have four sheets, though not more than two can be 

 real at once. These four sheets touch at three conical points. 



The first sheet, corresponding to r,, is the interior lobe of the cyclide (3), 

 ;md always touches the second sheet at a conical point on the positive branch 

 of the hyperbola. 



The second sheet, corresponding to r t , has three different forms, being either 

 the outer sheet of (3), the ring cyclide of one sheet (2), or the negative lobe 

 of (l). When the first sheet exists, it meets it at a conical point on the 

 |Kitive hyperbola, and when the third sheet exists, it meets it at a conical 

 point on the ellipse. 





