DYNAMICS OF A RIGID BODY. 135 



and these three screws will be co-reciprocal, and also 

 conjugate screws of inertia. They will, therefore ( 57), 

 form what we have termed the principal screws of 

 inertia. When the screw complex reduces to a pencil 

 of screws of zero pitch passing through a point, then the 

 principal screws of inertia reduce to the well-known 

 principal axes. 



121. Lemma. If from a screw complex of the n ih 

 order we select n screws A ly . . . , A n , which are conju- 

 gate screws of inertia ( 57), and if Si be any screw 

 which is reciprocal to A 2 , ..., A ny then an impulsive 

 wrench on Si will cause the body, when only free to twist 

 about the screws of the complex, to commence to twist 

 about A i. Let RI be the screw which, if the body were 

 perfectly free, would be the impulsive screw correspond- 

 ing to A i as the instantaneous screw. jRi must be reci- 

 procal to A 2 , . . . , A n ( 54). Take also 6 - n screws of 

 the reciprocal system B l9 . . . , B 6 _ . Then the 8 - n 

 screws R i9 S lf ly . . . , 2? 6 _ w must be reciprocal to the 

 n - i screws A z , . . . A n , and therefore the 8 - n screws 

 must belong to a screw complex of the (7 - n) th order. 

 Hence an impulsive wrench upon the screw ^i can be 

 resolved into components on R l9 Bi 9 . . . B* _ . Of 

 these all but the first are neutralised by the reactions of 

 the constraints, and by hypothesis the effect of an im- 

 pulsive wrench upon R is to make the body commence 

 to twist about AI, and therefore an impulsive wrench 

 on Si would make the body twist about A\. 



122. Relation between the Impulsive Screw and the In- 

 stantaneous Screw. A quiescent rigid body which pos- 

 sesses freedom of the third order is acted upon by an 

 impulsive wrench about a given screw r\. It is required 

 to determine the instantaneous screw 0, about which the 

 body will commence to twist. 



The screws which belong to the complex, and are at 



