466 



THE THEORY OF SCREWS. 



[420- 



From the first formulae the equation for p is, as before, 417 



(11) -p (12) (13) (14) 



(21) (22) -p (23) (24) 



(31) (32) (33) -p (34) 



(41) (42) (43) (44) -p 



From the second, the equation for p must be 



=0. 



= 0; 



but we may interchange rows and columns in a determinant so that the last 

 may be written, 



= 0; 



whence we see that the equation for p must be unaltered, if for p we sub 

 stitute - . It must therefore be a reciprocal equation of the type 



p 4 + 4Ap* + GBp 2 + 4&amp;lt;Ap +1=0, 

 and the roots are of the form 



, 1 ]_. 

 and as 



this transformation fulfils the fundamental condition ( 417). 



421. Quadrics unaltered by the Orthogonal Transformation. 



The special facilities of the orthogonal transformation in the present 

 subject arise from the circumstance that it is the nature of this transforma 

 tion to leave unaltered a certain family of quadrics. This is as we have 



