DYNAMICS OF A RIGID BODY. 



163 



152. Definition of the Sexiant. When six screws, 

 A ly &c., At, are reciprocal to a single screw T, a certain 

 relation must subsist between the six screws. This 

 relation may be expressed by equating the determinant 

 of 41 to zero. The determinant (called the sextant 

 may be otherwise expressed as follows : 



The equations of the screw A k are 



a* /3* 7* 



We shall presently show that we are justified in 

 assuming for T the equations 



= ~ = (pitch = p). 

 o p 7 



The condition that A k and T be reciprocal is 



(p 4- p*) (aa* + /3/3* + 77*) 4 Xk(yfik - fiyk) 4 yk(ayk - yak) 

 + 2*(|3a* - a/3*) = O. 



Writing the six equations of this type, found by 

 Iving k the values i to 6, and eliminating the six 

 quantities 



pa, p/3, p7, a, /3, 7, 



we obtain the result : 



t 73^3 - 



+ 75^5 ~ 



+ 

 ftp4 + 



ftps + 

 ftpe + 



- 73*3, 73P3 + 

 ~ 74^4, 74P4 + 



~ 7s^5> 75P5 + 



,, , 72 

 , ft, 73 

 4> ft, 7* 



s, ft, 7s 

 e, ft, 7 6 



By transformation to any parallel axes the value of 

 this determinant is unaltered. The evanescence of the de- 

 terminant is therefore a necessary condition whenever the 

 six screws are reciprocal to a single screw. Hence we 



M2 



