14] THE CYLINDROID. 21 



Pe = Pa cos 2 / + p ft sin 2 1, z 1 = (p a pp) sin I cos I, 

 P&amp;lt;t&amp;gt; = P* C s 2 wi + Pft sin 2 m, z z = (p a pp) sin m cos m, 

 A = I m, k = z l z z , 



_ 



Pa JJB A &amp;gt; 



sin A 



Pa + pp=pe+p&amp;lt;t,-h cot A, 



+ 5, 



A 



with similar values for m and z. 2 . It is therefore obvious that the cylindroid 

 is determined, and that the solution is unique. 



It will often be convenient to denote by (6, &amp;lt;) the cylindroid drawn 

 through the two screws 6 arid &amp;lt;. 



On any cylindroid there are in general two but only two screws which 

 like a. and /3 intersect and are at the same time at right angles. These two 

 important screws are often termed the principal screws of the surface. 



14. General Property of the Cylindroid. 



If a body receive twists about three screws on a cylindroid, and if the 

 amplitude of each twist be proportional to the sine of the angle between the 

 two non-corresponding screws, then the body after the last twist will have 

 regained the same position that it held before the first. 



The proof of this theorem must, according to ( 12), involve the proof of the 

 following: If a body be acted upon by wrenches about three screws on a 

 cylindroid, and if the intensity of each wrench be proportional to the sine of 

 the angle between the two non-corresponding screws, then the three wrenches 

 equilibrate. 



The former of these properties of the cylindroid is thus proved : Take 

 any three screws 0, &amp;lt;/&amp;gt;, i/r, upon the surface which make angles I, m, n, with 

 the axis of x, and let the body receive twists about these screws of amplitudes 

 , &amp;lt;/&amp;gt; , -\Jr . Each of these twists can be decomposed into two twists about the 

 screws a and /3 which lie along the axes of x and y. The entire effect of the 

 three twists is, therefore, reduced to two rotations around the axes of x and 

 y, and two translations parallel to these axes. 



The rotations are through angles equal respectively to 



cos I 4- &amp;lt;f&amp;gt; cos m + ty cos n 



and sin I + &amp;lt;/&amp;gt; sin m + ty sin n. 



The translations are through distances equal to 



p a (0 f cos I + &amp;lt; cos ra + -&amp;gt;/r cos n) 

 and pp (0 sin I + &amp;lt;/&amp;gt; sin m + -fy sin n}. 



