Ball — Contributions to the Tlieonj of Screws. 665 



could not be fulfilled. We therefore see that the co-ordinates just 

 written can only denote those of a screw of infinite pitch parallel to a. 

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



x clR X dR 



ai + - . -— , . . . as + — . -7- 



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

 are thus conducted by a more direct process to the results previously 

 obtained {Screivs, p. 100). 



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. Eeferred to any co-ordinates, we denote this 

 function by F" expressed in terms of Aj . . . Ag. If we express the same 

 function by reference to six coreciprocal axes with co-ordinates ai . . . a^, 

 we have the result 



Piai^ + . . . iho-i = y- 



Forming now the first emanant, we have 



2piaifdi + . . . + 2pea6(3e = /^i ^tT ' ' ' "^ '^^ ^ ' 



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

 intensities into the vertical coefficient of the two screws ; hence the 

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

 fererdiations Ave make the intensities equal to unity, we thus have the 

 following expression for the virtual coefficient between two screws A 

 and /A referred to any screws of reference whatever : — 



dV dV 



d\i dXa 



Suppose, for instance, that A is'reciprocal to the first screw of refe- 

 rence, we have 



dXi 

 This can be verified in a somewhat instructive manner. We have 



