THE CYLINDROID. 15 



shall first examine a special case, and from it we shall 

 deduce the general solution. 



Take, as axes of x and y y two screws a, )3, intersect- 

 ing at right angles, whose pitches are / a and pp. Let a 

 body receive twists about these screws of amplitudes 

 0' cos / and 0' sin /. The translations parallel to the axes 

 of x and y will then be p a W cos / and p$' sin /. The re- 

 sultant of the two translations may be resolved into two 

 components, of which Q' (p a cos 2 / + pp sin 2 /) is parallel to 

 the direction of that axis, a rotation about which is equi- 

 valent to the two given rotations, while 0' sin / cos l(p a -pp) 

 is perpendicular to the same line. The latter component 

 has the effect of transferring the resultant axis of the 

 rotations to a distance sin / cos / (p a - pj^ y the axis 

 moving parallel to itself in a plane perpendicular to that 

 which contains a and /3. The two original twists about 

 a and /3 are therefore compounded into a single twist of 

 amplitude Q' about a screw 9 whose pitch is 



p a cos V + p ft sin V. 



The position of the screw 9 is defined by the eq 



/ z - \fm -ft) sm * cos L 



Eliminating / we have the equation 



z (x* +y) - (p a -pp] xy = o. 



The conoidal cubic surface represented by this equa- 

 tion has been called the cylindroid.* 



* This surface has been described by Pliicker (Neue Geometric des Raumes, 

 p. 97) ; he arrives at it as follows : Let Q = o, and Q' = o represent two linear 

 complexes of the first degree, then all the complexes formed by giving /i dif- 

 ferent values in the expression Q + /zQ' = O form a system of which the axes lie 

 on the surface z (x 9 + y*) - (& - k*} xy = o. The parameter of any complex of 



