1 8 THE CYLINDROID. 



The rotations are through angles equal respectively 



to 



/ cos / + 0' cos m + i// cos n 

 and 



Q' sin / + 0' sin m + ;// sin n. 



The translations are through distances equal to 



p a (&' cos / + 0' cos m + T//COS n) 

 and pft (0' sin / + $' sin ?;/ + $' sin 72). 



These four quantities vanish if 



& X V 



sin (m - n) sin (/z - /) sin (/ - m) 9 



and hence the fundamental property of the cylindroid 

 lias been proved. 



The cylindroid affords the means of compounding two 

 twists (or two wrenches) by a rule as simple as that 

 which the parallelogram of force provides for the com- 

 position of two intersecting forces. Draw the cylindroid 

 which contains the two screws ; select the screw on the 

 cylindroid which makes angles with the given screws 

 whose sines are in the inverse ratio of the amplitudes of 

 the twists (or the intensities of the wrenches) ; a twist (or 

 wrench) about the screw so determined is the required 

 resultant. The amplitude of the resultant twist (or the 

 intensity of the resultant wrench) is proportional to the 

 diagonal of a parallelogram of which the two sides are 

 parallel to the given screws, and of lengths proportional 

 to the given amplitudes (or intensities). 



18. Particular Cases. If/ a = pp the cylindroid re- 

 duces to a plane, and the pitches of all the screws are 

 equal. If the pitches be all zero, then the general pro- 

 perty of the cylindroid reduces to the well known con- 

 struction for the resultant of two intersecting forces, or 

 of rotations about two intersecting axes. If the pitches 



