161] THE GEOMETRY OF THE CYLINDROID. 155 



In Fig. 36 will be found a drawing of this curve. The following are the 

 values of the constants adopted : 



a = 25; /3 = 26; h = lS; wi = 28 9; 

 with which the equations become 



= % tan (0-25), 

 y = 20 - 66 sin 20. 



The curve was plotted down on &quot; papier millimetrique,&quot; and has been copied 

 in reduced size in the figure. The constants were selected after several 

 trials, in order to give a curve that should be at once characteristic, and 

 of manageable dimensions. 



The distribution of pitch upon the screws of the cylindroid is of 

 fundamental importance in the theory, so that we must express the pitches 

 appropriate to the several points on the cubic. 



Let p denote the pitch ; then, from the known property of the cylindroid, 



p=p + mcos 26, 



where p Q is a constant. Transforming this result into the co-ordinates of 

 the point on the cubic, we have 



(x 2 if cos 2 ft) cos 2a + 2xy cos 8 sin 2a 



/\1 /i /vvi v_ * 



P ~ P ^ + 2/ 2 cos 2 /3 



161. Chord joining Two Screws of Equal Pitch. 



As the pitches of the two screws, defined by + and 6, are equal, the 

 chord in question is found by drawing the line through the points x , y and 

 x&quot;, y&quot;, respectively, where 



x = y cos ft tan (9 a), 

 y = h sec ft m cosec ft sin 20, 

 x&quot; = y&quot; cos ft tan ( a), 

 y&quot; = h sec ft + m cosec ft sin 20. 



After a few reductions, the required equation is found to be 

 xm (cos 20 + cos 2a) + y (h sin ft + m cos ft sin 2a) 



- h- tan ft + m 2 cot ft sin 2 20 - 0. 

 If this chord passes through the origin, then 



- h* tan- ft + ??i 2 sin 2 20 = ; 

 or, h tan m sin 20 = 0. 



But this is obviously necessary ; for from the geometry of the cylindroid it is 

 plain that must then fulfil the required condition. 



