96 DYNAMICS OF A RIGID BODY. 



tion of screws whose co-ordinates satisfy one homoge- 

 neous equation of the second degree, and which may be 

 defined to be a screw complex of the fifth order and second 

 degree. We shall develop a few propositions on this 

 subject, which will be useful in what is to follow ; but 

 the general discussion of this species of complex, though 

 apparently of great interest, lies beyond the scope of the 

 present volume. 



91. Polar Screws. Let us denote a screw complex 

 of the fifth order and second degree by the equation 

 Ue = o, where U Q is an homogeneous function of the 

 second degree in the six quantities ft, . . . , ft. 



Let rj and denote any two screws. If then (to 

 adopt the fertile principle used by Dr. Salmon*) we sub- 

 stitute in U = o for ft, ft, &c., the expressions /^ + m%i, 

 ,<i-> &c., we obtain the equation : 



lm UT& + m z U$ = o 

 where U^ denotes the expression : 

 -dU 



Solving the quadratic equation for / : m y we obtain 

 two values of this ratio, and hence ( 89) we see that 

 two screws belonging to the screw complex UQ = o can 

 be found on any cylindroid (rj, ). 



If the relation between rj and be such, that 



the two roots of the equation will be equal in magni- 

 tude, and opposite in sign, and hence we deduce the fol- 

 lowing theorem : 



* Conic Sections, 3rd Edition, p. 134. 



