332 THE THEORY OF SCREWS. [307- 



Let the centre of gravity be constructed as in the last section ; then an 

 impulsive force through the centre of gravity will produce the velocity of 

 translation on 8j. Let us denote by % the screw of zero pitch on which this 

 force lies. 



We thus have % as the impulsive screw corresponding to the instan 

 taneous screw S^ while i/r is the impulsive screw corresponding to the 

 instantaneous screw S . 



Draw&quot; now the cylindroids (%, i/r) and (8 1} S ). The first of these is 

 the locus of the impulsive screws corresponding to the instantaneous 

 screws on the second. As already explained, we can completely correlate 

 the screws on two such cylindroids. We can, therefore, construct the 

 impulsive screw on (^, ^) which corresponds to any instantaneous screw 

 on (S 1; S ). It is, however, obvious, from the construction, that the original 

 screw B lies on the cylindroid (S 1} S ). Hence we obtain the impulsive screw 

 which corresponds to B as the instantaneous screw, and the problem has 

 been solved. 



308. Another method. 



We might have proceeded otherwise as follows : From the three given 

 pairs of impulsive screws and instantaneous screws rja, %J3, 7 we can find 

 other pairs in various ways. For example, draw the cylindroids (a, #) 

 and (, ); then select, by principles already explained, a screw B on the 

 first cylindroid, and its correspondent 6 on the second. In like manner, 

 from the cylindroids (a, 7) and (17, ), we can obtain another pair (&amp;lt;/&amp;gt;, e). We 

 have thus five pairs of correspondents, ??, /3, 7, 08, &amp;lt;f&amp;gt;e. Each of these 

 will give a diameter of the momental ellipsoid, and the radius of gyration 

 about that diameter. Thus we know the centre of the momental ellipsoid 

 and five points on its surface. The ellipsoid can be drawn accordingly. 

 Its three principal axes give the principal screws of inertia. All other 

 pairs of correspondents can then be determined by a construction given 

 later on (311). 



309. Unconstrained motion in system of second order. 



Suppose that a cylindroid be drawn through any two (not lying along 

 the same principal axis) of the six principal screws of inertia of a free rigid 

 body. If the body while at rest be struck by an impulsive wrench about 

 any one of the screws of the cylindroid it will commence to move by 

 twisting about a screw which also lies on the cylindroid. For the given 

 impulsive wrench can be replaced by two component wrenches on any 

 two screws of the cylindroid. We shall, accordingly, take the component 

 wrenches of the given impulse on the two principal screws of inertia which, 

 by hypothesis, are contained on the cylindroid. Each of those components 



