ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 595 



point 01 in the vertical ai,j ; the side 12 passes through the intersection 

 of 01 with x^, and the point similarly denoted in Xn, and so on. This 

 construction can be executed still more easily by means of the first 

 vertical Xg ; for this line does not even change its position, and all sides 

 of the polygon are intersected by it ; therefore the last side of the polygon 

 ^7 is given by the point x^i/g, and by the point 67 in Xg, viz., the intersec- 

 tion of the first polygon with x^ ; the second is given by the intersection 

 of this side with the intersection of the coiTesponding side of the former, 

 and so on. Owing to the want of space this construction is indi- 

 cated only from the side 34. 



Expressions are next found for the ordinates i/i. The differences of 

 two following coordinates are denoted by A^^ j+i and Ay, ;^.j, and possess 

 two indices, for it is impossible to denote successive coordinates by 

 continuous indices. At the same time the difference should be positive 

 when the ordinate of the second index is the larger one. 



We have therefore 



and hence 

 or 



1 1 ^i 



We can get another expression for ?/„ if we take notice that the first 

 chosen side of the polygon with the ordinate a^„ equals yo + *n^oi> ^od ^^^ 

 then the segment intercepted by the sides of the funicular polygon 



= ^{x,-x^-^. 



Hence we have also 



" AP 



1 -tli 



and therefore 



1 1 -"^i 1 -tij 



By putting down the second sum of the first expression we can easily 

 see that the second expression is a partial integral of the first, viz., if we 

 put the real differences instead of Aic we get 



A.,,^=(.,_.0^ 



^^ V^P^-/^ _a, >) /AP, AP., AP3\ 



A "^'AP, . ./AP,,AP2, ,^P«-A 



Q Q 2 



