ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 693 



point 2, whose distance from P, also measured in the direction of x, is 

 Hg. The distance of the extremity of Hi from the upper extremity of 

 AP, equals Hj^oi > therefore from the lower extremity it equals 



Hi^oi-AP,. 



But as this equals H,fi2 it follows that the ratio of the ordinates and 

 abscissa in the second case is 



t -t -^^1 



^12 — ''01 -^^ • 



Through the lower extremity of AP2 draw a third line through the point 

 2, and produce it to the point 3, whose distance from the line P equals 

 H3. If we proceed exactly as before we get the ratio of the ordinates in 

 the third case 



, _, AP2_. AP, AP„ 



*23— n-2 TT 'o\— TT- ~w- • 



±12 ill -'^2 



Proceeding in this way we get the polygon of the values of t, in 

 which the ratio of the ordinates and abscissa are in any given case 



_ 'yAP,. 



1 J^i 



We can easily see from this that <oi plays the part of an integration 

 constant with regard to 



^AP, 



-^ H ' 

 1 -^-^1 



If through any point in fig. 13 b we draw parallels to the side of the 

 polygon 1, 2, 3 . . . of fig. 13 a, then these cut the ratios 



AP 



on the vertical whose distance from the chosen point equals 1 , and 

 these ratios appear on this vertical, which we now may call the line of t, 

 added up in the same order in which the AP have been arranged. If we 

 alter the ratio of the ordinates and abscissa of the first side 01, e.g., we 

 make it ^0/ instead of /qi (as shown by the dotted lines in fig. 13 a), then 

 the whole of the figure changes, but the points on the line P remain 

 unaltered owing to their construction, and all other points lying 

 outside P move on vertical lines. Accordingly both figures have the 

 straight line P and the parallel pencil of the verticals in common. But 

 it follows from geometry of position, or even indirectly from the con- 

 struction, that in such systems all corresponding rows of points lying in 

 the same verticals are congruent ; for the rays intersecting each other, 

 say in 4, cut off on all verticals the same distances of the rays as those 

 which intersect each other in 4', because 4 and 4' ai'e equally distant 

 from P ; and as this is true for any other two rays the above proposition 

 follows from it. Besides it must be so, for these distances can be expressed 



P 



by the ratio ^, and also by the position of the verticals, both of which 



Hi 



are independent of ^oi- 



All that has been said about fig. 13 a relates also to fig. 13 b, the point 

 1893. Q Q 



