380 THE THEORY OF SCREWS. [348 



the outlines of this theory, with the view of generalizing it into one adequate 

 for our purpose. 



We can easily conceive of two systems of corresponding forces in two 

 planes. To each force in one plane will correspond one force in the other 

 plane, and vice versa. To any system of forces in one plane will correspond 

 a system of forces in the other plane. We are also to add the condition that 

 if one force x vanishes, the corresponding force x will also vanish. 



The fundamental theorem which renders this correspondence of im 

 portance is thus stated : 



If a group offerees in one of the planes would equilibrate when applied 

 to a rigid body, then the corresponding group of forces in the other plane 

 would also equilibrate when applied to a rigid body. 



Draw any triangle in each of these planes, then any force can be de 

 composed into three components on the three sides of the triangle. Let 

 x, y, z be the components of such a force in the first plane, and let x, y , z 

 be the components of the corresponding force in the second plane ; we must 

 then have equations of the form 



x = ax + by + cz, 

 y = a x + b y + c z, 

 z = a&quot;x -f b&quot;y + c&quot;z, 



where a, b, c, &c., are constants. These equations do not contain any terms 

 independent of the forces, because x, y , z must vanish when x, y, z vanish. 

 They are linear in the components of the forces, because otherwise one force 

 in one plane will not correspond uniquely with one force in the other. 



Let a? 1} y-i, z^ ac 2 , y 2 , z. 2 ; ... x n , y n , z n be the components of forces in the 

 first plane. 



Let Xi, yi, Zi] #/, y 2 , z.\ ... x n , y n , z n be the components of the corre 

 sponding forces in the second plane. Then we must have 



k = a-r k + by k + cz k , 

 yk = a xk + b y k -f c z k , 

 z k = a&quot;x k +- b&quot;y k + c&quot;z k , 

 where k has every value from 1 to n. If therefore we write 



and 



