M. Poinsot on the Percussion of Bodies. 355 



the diameter 28 conjugate to 28'*; at the other extremity of 2S 

 the body strikes with the same force, but in an opposite direction. 



The magnitude of this percussion is measured by that of the 

 couple divided by the diameter 28 which the plane of the couple 

 determines in the central ellipse. 



32. A close relation, too, may be established between this 

 theorem and that in art. 20, so as to include both in one enun- 

 ciation. For without distinguishing the two kinds of impulsion 

 which the body may have received, and regarding merely the 

 motion it possesses, it is evident that in both cases we have to 

 consider a body which actually turns around a spontaneous axis 

 OS situated in one of its principal planes. Now in art. 20 we 

 found the centre of maximum percussion D to be situated in the 

 direction GO of the diameter 28 conjugate to the direction of 

 the spontaneous axis, and its distance \ from the point to be 



X=± \/A2+AH; 

 or, since AH = 8^ (see art. 3), 



\= ± Vlf^^, 



where the letter A represents the distance GO. 



But, ^ being the inclination of the diameter 8 to the sponta- 

 neous axis OS or to its parallel, the conjugate diameter 8', 8 sin ^ 

 will be the arm of inertia of the body around 8', and consequently 



'/8-sin2</) + A2sin2</, 



the arm of inertia around OS ; hence representing this line by 

 K, as usual, 



\ sin </)= sin <^ \/A2 + 82=K. 



Hence we may say, in general, if a body is actually animated 

 by a motion of rotation around a spontaneous axis situated any- 

 where in one of its three principal planes, the two points of this 

 plane at which the body strikes with the greatest possible force 



* The couple L tends to make the body rotate around the axis of y 



with an angular velocity equal to — oj, and the couple M, alone, would pro- 



M 

 duce a rotation around the axis of x whose velocity would be — 5. By the 



composition of rotations, therefore, the axis 28', around which the body 

 will actually rotate, makes an angle with the axis of x whose tangent is 



L M «' L 



m^ ' ma^ " 1* ' M ' ^^^ diameter 28, parallel to the intersection 



of the jjlane of the couple, is inclined to the axis of x at an angle whose 



M «2 



tangent is -jj ; hence, the product of both tangents being — -m, 28 and 28' 

 are conjugate diameters (sec art. 3). 



