348] THE THEORY OF SCREW-CHAINS. 381 



with similar values for % % , 2z, 2/ then the above equations give 

 2a? = a 2x + b % + c %z, 

 % = a 2a? + 6 % + c/ ^z, 

 2s = a&quot;2a? + &&quot;2y + c&quot;2*. 



If the system of forces in the first plane equilibrate, the following con 

 ditions must be satisfied : 



2tf = 0, % = 0, ^z = 0, 

 and from the equations just written, these involve 



2a&amp;gt; =0, % =0, 2* =0, 

 whence the corresponding system in the other plane must also equilibrate. 



To determine the correspondence it will be necessary to know only 

 the three forces in the second plane which correspond to three given forces 

 in the first plane. We shall then have the nine equations which will be 

 sufficient to determine the nine quantities a, b, c, &c. 



It appears, from the form of the equations, that the ratio of the intensity 

 of a force to the intensity of the corresponding force is independent of those 

 intensities, i.e. it depends solely upon the situation of the lines in which the 

 forces act. 



Take any four straight lines in one system, and let four forces, 

 X 1} X 2 , X 3 , X^, on these four straight lines equilibrate. It is then well 

 known that each of these forces must be proportional to certain functions 

 of the positions of these straight lines. We express these functions by 

 A ly A. 2 , A 3&amp;gt; A. The four corresponding forces will be X^, X 2 t X 3 , X 4 , 

 and as they must equilibrate, they must also be in the ratio of certain 

 functions AJ, AJ, A 3 , A of the positions. 



We thus have the equations 



Xi _ X 2 _X 3 _ -A-4 



A! A A 3 AI 



Xi _ X 2 _ X 3 _ X 



A 1 A 2 A 3 A 



We can select the ratio of X l to X^ arbitrarily : for example, let this ratio 

 be /A; then 



whence the ratio of X 3 to X. 2 is known. Similarly the ratio of the other 

 intensities X 3 : X 3 , and X 4 : X t is known. And generally the ratio of every 

 pair of corresponding forces will be determined. 



