TRANSACTIONS OF THE SECTIONS. 



25 



distance between the 1st and 2nd points, &c. ; and a term with S pre- 

 fixed denotes the sum of all the terms of the same type, terms being of 

 the same type when deducible from each other by substituting one 

 figure for another, or by any number of successive substitutions. 



2 2 22 _*_* _2 2 2 2 



2 (w-4)S 12.23.34.41 -(w-4)S12.34 + 4 S 12 .23.31 .45 



\ \ 



2_ 2 4 



+ 2 S 12.23.45 



2 _ 2 2 2 



2S 12.23.34.45 = 0. 



The subscribed diagram shows how the lines are connected in each type. 



If for each line we substitute its differential of the mth order, the 

 resulting equation is true : and generally, if we substitute for each 

 line the line increased by the sum of its differentials of every order up 

 to the mth, and separate all the terms involving products of the same 

 given dimensions for- every order of differentials, their sum equals zero. 

 Representing a function of the distances and their differentials symme- 

 trical for thelst and 2nd points, and also for the rest, thus F (12 34...^^) 

 all these equations can be put under the form Zd"^ 12.F(12 34...?2) =0. 



These equations will be sufficient when the law^ of motion of the sy- 

 stem is assigned by equations involving only the distances and their dif- 

 ferentials; but when it involves the absolute velocities of the points along 

 their own paths, we must find general equations between the distances, 

 the relative velocities of the points to or from each other represented 

 by the first differentials of the distances, and their absolute velocities 

 along their own paths. For this purpose divide 2 n points into two 

 equal groups, denoting two successive simultaneous positions of the n 

 points. Draw right lines from each point to every other in its own 

 group, and to one in the other. An equation rnay be found between 

 these n- lines thus drawn between the 2 n points. When the n lines 

 connecting the two groups are indefinitely diminished, their limiting 

 values denote the absolute velocities of the points ; and the limiting 

 differences of the distances of the same two bodies, in the two groups, 

 denote their relative velocities, or the first differentials of the distances. 

 Equations involving higher orders of differentials may be deduced from 

 this as from the preceding. 



