309.] BODY WITH FIXED AXIS. 



mined separately; the other components of the reactions can 

 be found from the first, second, fourth, and fifth equations. 



The impulsive stress to which the axis is subjected by the 

 impulse, or the so-called percussion of the axis, instead of being 

 represented by two impulses A, B as above, can also be regarded 

 as composed of an impulse whose components are 



-A X -B X = X, -A y -B y = Y- , 



and an impulsive couple whose vector has the components 



o. 



The last component being zero, the resulting couple lies in a 

 plane passing through the axis of 2. 



If there were any number of impulses acting on the body 

 simultaneously, the effect on the axis could be determined in 

 the same way, except that the quantities X, Y, Z, y^Zz^Y t 

 z^Xx^Z, must be replaced by the corresponding sums. 



309. It follows from the preceding article that the conditions 

 under -which a single impulse acting on a rigid body with a 

 fixed axis will produce no stress on the axis are 



Z=o, -z l Mx+E=o, D=o. (15) 



If these conditions are fulfilled, the resulting motion will be the 

 same even when the axis is free. 



The first and third equations show that the impulse must be 

 perpendicular to the plane passing through axis and centroid. 

 The meaning of the fourth and fifth conditions becomes appar- 

 ent if the .rj/-plane be taken so as to pass through the point of 

 application P of the impulse. The new origin O' is the foot of 

 the perpendicular let fall from P on the fixed axis. To trans- 

 form the conditions (15) to the new system it is only necessary 

 to substitute z+z l for z\ the first three conditions are not 

 affected, and the last two become 



z-^Mx + 3 mzx + z^mx = o, *%myz + z^my = o, 



