32 9 J VARIOUS EXERCISES. 357 



327. Another proof of article 303. 



As an illustration of the formulae just given we may verify a theorem of 

 303 showing that when we know the instantaneous screw corresponding to 

 a given impulsive screw, then a ray along which the centre of gravity must 

 lie is determined. 



For subtracting the second equation from the first and repeating the 

 process with each of the other pairs, we have 



- 2 ) + 2^ ( 3 + 4 ) - 2y (a-, + (i ), 

 e (773 + *7 4 ) = 26 (ots - 04) + 2x ( 5 + a ) - 2^ (! + 02), 

 e Ofc + %) = 2c (a e - ) + 2y (j + a 2 ) - 2# ( 3 + a 4 ). 



Eliminating e we have two linear equations in x^y^z^ thus proving the 

 theorem. 



If we multiply these equations by a. l + a^, a 3 + a 4 , ar, + 6 respectively 

 and add, we obtain 



e cos (OLVJ) = 2p a , 

 thus giving a value for e. 



328. A more general Theorem. 



If an instantaneous screw be given while nothing further is known as to 

 the rigid body except that the impulsive screw is parallel to a given plane 

 A, then the locus of the centre of gravity is a determinate plane. 



Let A,, //,, v be the direction cosines of a normal to A, then 



At (^ 3 + ifc) + v (%, + i/) = 0, 



whence by substitution from the equations of the last Article we have a 

 linear equation for X Q , y , z . 



329. Two Three-Systems. 



We give here another demonstration of the important theorem of .318, 

 which states that when two arbitrary three-systems U and V are given, it is 

 in general possible to design and place a rigid body in one way but only in 

 one way, such that an impulsive wrench delivered on any screw 77 of V shall 

 make the body commence to move by twisting about some screw a of U. 



Let the three principal screws of the system U have pitches a, b, c and 

 take on the same three axes screws with the pitches a, b, c respectively. 

 These six screws lying in pairs with equal and opposite pitches form the 

 canonical co-reciprocals to be used. 



