12 THE CYLINDROID. 



of jy, and a line perpendicular to x and y for z. If we re- 

 solve the wrench on ]3 into forces JT, Y, Z, parallel to 

 the axes, and couples of moments Z, M,N, in planes per- 

 pendicular to the axes we shall have 



X = /3" cos O ; Y= /3" sin ; Z = o 

 L=" cos O - 3"<*sin O ; J/= 3"j sin O + 3"<J cos ; 



We thus reduce the given wrench to four wrenches, 

 viz., two forces [and two couples, and we reduce the 

 given twist to two twists, viz., one rotation and one 

 translation. By the principle ofio the work done by 

 the given twist against the given wrench must equal the 

 sum of the eight quantities of work done by each of the 

 two component twists against each of the four compo- 

 nent wrenches. Six of these quantities are evanescent. 

 In fact a rotation through the angle a' around the axis 

 of x can only do work against Z, the amount being 



The translation p a a' parallel to the axis of x can 

 only do work against X, the amount being 



a'/3"/ a cos O. 

 Thus, the total quantity of work done is 



"' ft" \ [fa + fp] cosO-dsinO}. 

 The expression 



[f a + fp) cos O - d sin O, 

 is of great importance in the present theory.* It is 



* The theory of screws has many points of connexion with recent geome- 

 trical speculations on the linear complex, by the late Dr. Pliicker and Dr. Felix 

 Klein. Thus the latter has shown, (Mathematische Annalen, Band II., p. 368), 



