278 THE THEORY OF SCREWS. [265, 



If a? be a variable parameter, then the co-ordinates 



x dR a; dR 



Oti + -. ~ 7 i #6 &quot;T ~i ~9 



4p, da, 4p 6 da 6 



must denote a screw of variable pitch x on the same screw as or. We are 

 thus conducted to a more general form of the results previously obtained 

 ( 47). 



These expressions may be written 



! + COS flj , or a + COS O.J, . . . 



^ 2j? 2 



where a lf a 2 , ... are the angles which a makes with the screws of reference. 



266. A general Expression for the Virtual Coefficient. 



We may also consider that function of the co-ordinates of a Dyname 

 which, being always proportional to the pitch, becomes exactly equal to the 

 pitch when the intensity is equal to unity. More generally, we may define 

 the function to be equal to the pitch multiplied into the square of the 

 intensity, and it is easy to assign a physical meaning to this function. It 

 is half the work done in a twist against a, wrench, on the same screw, where 

 the amplitude of the twist is equal to the intensity of the wrench. Referred 

 to any co-ordinates, we denote this function by V expressed in terms of 

 Xj,... X 6 . If we express the same function by reference to six co-reciprocal 

 axes with co-ordinates cfi, ..., we have the result 



p 1 a. 1 s + ...&amp;gt; 6 a 2 = V. 

 Forming now the first emanant, we have 



2pi i A + . . . + 2p, 6 /3 6 = ^ -^- . . . + ^ 6 ; 



but the expression on the left-hand side denotes the product of the two 

 intensities into double the virtual coefficient of the two screws; hence 

 the right-hand member must denote the same. If, therefore, after the 

 differentiations we make the intensities equal to unity, we have for the 

 virtual coefficient between two screws X and yu, referred to any screws of 

 reference whatever one-half the expression 



dV dV 



Suppose, for instance, that X is reciprocal to the first screw of reference, 

 then 



