THE CYLINDROID. 17 



sin ,4 



A + A = A + A ~ ^ cot 



-A) cot A + 



with similar values for / and 22- It is therefore obvious 

 that the cylindroid is determined, and that the solution 

 is unique. 



It will often be convenient to denote by (0, 0) the 

 cylindroid drawn through the two screws and <. 



17. General Property. The general property of 

 the cylindroid, which is of importance for our present 

 purpose, may be thus stated. 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 occupy the same position 

 w r hich it did before the first. 



The proof of this theorem must, according to 15, 

 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-corre- 

 sponding screws, then the three wrenches equilibrate. 



The former of these properties of the cylindroid is 

 thus proved : Take any three screws 9, 0, i/>, upon the 

 surface w T hich make angles /, m, n, with the axis of x, 

 and let the body receive twists about these screws of 

 amplitudes 0', $', ;//. Each of these twists can be de- 

 composed 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. 



c 



