484 THE THEORY OF SCREWS. 



But since the seven screws are independent both C7 a ^ and trr^ must be, in 

 general, different from zero, whence by the former equation we have 



a&quot; ft&quot; 



sin (ftp) sin (pa) 



Thus we obtain the following theorem ( 28). 



If seven wrenches on seven given screws equilibrate and if the intensity a&quot; of 

 one of the seven wrenches be given then the intensity of the wrench on any one 

 ft of the remaining six screws can be determined as follows. 



Find the screw ty reciprocal to the five screws remaining when a and (3 are 

 excluded from the seven. 



On the cylindroid (aft) find the screw p which is reciprocal to i/^. 



Resolve the given wrench a&quot; on a into component wrenches on ft and on p. 



Then the intensity of the component wrench thus found on ft is the required 

 intensity ft&quot; with its sign changed. 



NOTE II. 



Case of equal roots in the Equation determining Principal Screws of 



Inertia, 86. 



We have already made use of the important theorem that if U and V are both 

 homogeneous quadratic functions of n variables, then the discriminant of U + A V 

 when equated to zero must have n real roots for X provided that either U or V 

 admits of being expressed as the sum of n squares ( 85). 



The further important discovery has been made that whenever this deter- 

 minantal equation has a repeated root, then every minor of the determinant 

 vanishes (Routh, Rigid Dynamics, Part II. p. 51, 1892). 



This theorem is of much interest in connection with the Principal Screws of 

 Inertia. The result given at the end of 86 is a particular case. It may be 

 further presented as follows. 



Taking the case of an n system each root of A. will give n equations 



l dT ! dT 



Of these n - 1 are in general independent and these suffice to indicate the values 

 of *,...*,,. 



But m the case of a root once repeated the theorem above stated shows that we 

 have not more than n - 2 independent equations in the series. The principal 

 Screw of Inertia corresponding to this root is therefore indeterminate. 



But it has a locus found f rom the consideration that besides these u - 2 linear 



