375] THE THEORY OF PERMANENT SCREWS. 411 



This is obviously true unless it were possible for the determinant 



dd? 



d*T 



to become zero. Remembering that T is a homogeneous function of the 

 quantities l) ... n in the second degree, the evanescence of the determinant 

 just written would indicate that T admitted of expression by means of n 1 

 square terms, such as 



This vanishes if 



= 0, &c.; 



each of these is a linear equation in lt ... n , and consequently a real system 

 of values for 6^ ... n must satisfy these equations, and render T zero. It 

 would thus appear that a real motion of the mass-chain would have to be 

 compatible with a state of zero kinetic energy. This is, of course, im 

 possible ; it therefore follows that the determinant must not vanish, and 

 consequently we have the following theorem : 



If the screw-chains of reference be co-reciprocal, then the necessary and 

 the sufficient conditions for 6 to be a permanent screw are that its co-ordinates 

 l , 2 , ... n shall satisfy the equations 



dT =o- ^=o 



dOl d0 n 



There are n of these equations, but they are not independent. The cmanant 

 identity shows that if n 1 of them be satisfied, the co-ordinates so found 

 must, in general, satisfy the last equation also. 



375. Conditions of a permanent Screw-chain. 



As the quantities #/, . . . tl are small, we may generally expand T in 

 powers, as follows : 



T=T + 1 T 1 +...0 n T n 



The equation 

 therefore becomes 



dT 



