18 THE THEORY OF SCREWS. [10- 



coefficient of the two screws a and ft, and may be denoted by the 

 symbol 



11. Symmetry of the Virtual Coefficient. 



An obvious property of the virtual coefficient is of great importance. If 

 the two screws a and ft be interchanged, the virtual coefficient remains 

 unaltered. The identity of the laws of composition of twists and wrenches 

 can be deduced from this circumstance*, and also the Theory of Reciprocal 

 Screws which will be developed in Chap. III. 



12. Composition of Twists and Wrenches. 



Suppose three twists about three screws a, ft, 7, possess the property 

 that the body after the last twist has the same position which it had before 

 the first : then the amplitudes of the twists, as well as the geometrical rela 

 tions 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 : 



Since the three twists neutralize it follows that the total energy ex 

 pended in making those twists against a wrench, on any screw 77, must be 

 zero, whence 



a trar, + ft ^ftr, + J^yr, = 0. 



This equation is one of an indefinite number (of which six can be shown 

 to be independent) obtained by choosing different screws for 77. From 

 each group of three equations the amplitudes can be eliminated, and four of 

 the equations thus obtained will involve all the purely geometrical conditions 

 as to direction, situation, and pitch, which must be fulfilled by the screws 

 when three twists can neutralize each other. 



But now suppose that three wrenches equilibrate on the three screws 

 a, ft, 7. Then the total energy expended in a twist about any screw 77 against 

 the three wrenches must be zero, whence 



&amp;lt;*&quot;^a,, + ft ^fr + j ^yr, = 0. 



An indefinite number of similar equations, one in fact for every screw 77, must 

 be also satisfied. 



By comparing this system of equations with that previously obtained, it 

 is obvious that the geometrical conditions imposed on the screws a, ft, 7, in 



* This pregnant remark, or what is equivalent thereto, is due to Klein (Math. Ann., Vol. iv. 

 p. 413 (1871)). 



