THE CYLINDROID. 13 



called the virtual coefficient of the two screws a and /3, 

 and may be denoted by the symbol 



14. Symmetry of the Virtual Coefficient. One property 

 of the virtual coefficient is of the utmost importance. 

 If the two screws a and j3 be interchanged, the virtual 

 coefficient remains unaltered. The identity of the laws 

 of composition of twists and wrenches can be deduced 

 from this property,* and also the Theory of Reciprocal 

 Screws. 



15. Composition of Twists and Wrenches. Suppose 

 three twists about three screws a, ]3, 7, possess the 

 property that the body after the last twist has the same 

 position which it had before the first : then the ampli- 

 tudes of the twists, as well as the geometrical relations 

 of the screws, must satisfy certain conditions. The 

 particular nature of these conditions does not concern us 

 at present, although it will be fully developed hereafter. 



We may at all events conceive the following method 

 of ascertaining these conditions : 



It follows from i o, that the sum of the works done 

 in the twists about a, /3, 7, against a wrench, on any 

 screw i), must be zero, whence 



a' w an 4 /3V0 n + 7'^,, = O. 



This equation is a type of an indefinite number (of 

 which six are independent) which may be obtained by 



that if/ tt and ^3 be each the "hauptparameter" of a linear complex, and if 

 (P a + Pp) cosO d sin O o, 



where d and O relate to the principal axes of the complexes, that then the two 

 complexes possess a special relation and are said to be in "involution." 



* This remark, or what is equivalent thereto, is due to Dr. Felix Klein 

 (Math. Ann., vol. iv., p. 413). 



