252 THE THEORY OF SCREWS. [233- 



233. Analytical Method. 



A quiescent rigid body which has freedom of the fifth order receives an 

 impulsive wrench on a screw 77 : it is required to determine the instantaneous 

 screw 6, about which the body will commence to twist. 



Let X be the screw reciprocal to the freedom, and let the co-ordinates be 

 referred to the absolute principal screws of inertia. The given wrench com 

 pounded with a certain wrench on X must constitute the wrench which, if 

 the body were free, would make it twist about 6, whence we deduce the six 

 equations (h being an unknown quantity) 



Multiplying the first of these equations by X 1} the second by X 2 , &c., adding 

 the six equations thus produced, and remembering that and X are reciprocal, 

 we deduce 



i/&quot;2%A.i + X&quot; 2V = 0. 



This equation determines X &quot; the impulsive intensity of the reaction of 

 the constraints. The co-ordinates of the required screw 6 are, therefore, 

 proportional to the six quantities 



Pi Pe 



234. Principal Screws of Inertia. 



We can now determine the co-ordinates of the five principal screws of 

 inertia ; for if be a principal screw of inertia, then in general 



. 



whence 



with similar values for ,, ... 6 . Substituting these values in the equation 



and making ^- = #, we have for sc the equation 



fv 





p l -co p 2 -x p 3 - x pt- a; p 5 - x p 6 - x 

 This equation ts of the fifth degree, corresponding to the five principal 

 screws of inertia. If x denote one of the roots of the equation, then the 

 corresponding principal screw of inertia has co-ordinates proportional to 

 Xj X-s X3 X^ _ X 5 X 6 



^~- x&quot; PI-X&quot; p 3 -x&quot; pi-&quot; Ps-x&quot; PS- 



