594 REPORT — 1893. 



moving up to O'. The abscissa, which ought to be multiplied by the 



new ratio, is drawu from a point a;„ (fig. 13 c) with regard to its sign 



in such a way that they appear as the difierence of the abscissae a-,,— .rj; 



through all the x-i we draw verticals and join constantly those verticals 



by parallels to the corresponding t of fig. 13 b. Starting from any point, 



say from to 1, draw a parallel to ^on from 1 to 2 a parallel to t^^, 



and so on ; then these prolonged sides of the polygon cut on the vertical 



AP 

 x„ the required products {x„—Xi) -=—'. Because by the similarity of figures 



we have for APj and a;, 



AP 



1, «K-«i, (a", — a;i) .„-' in fig. 13 c, 



and 

 Hence 



1, H,, AP, in fig. 13ffl. 



Xn — ^ _ (a;„-a'i) APi-f-H) 

 H, AP, 



But as this ratio is also true for any other value of AP^, then the 



AP- 



required products (a;„ — Xi) X — r^ are intercepted on the vertical x„ at a 



distance x„ from the fixed point, in the same order in which they were 

 arranged on the line P in fig. 13 a. Hence we may get on this line the 

 sum of any number of following products : — 



« APj 



k » 



Let us imagine the a,';, that is, the relative position of the AP, as con- 

 stant, whilst the a;„ as varying, then nothing whatever will change in tho 

 polygon of fig. 13 c, only the vertical «„ will take another position. If 

 to express this generality we write x instead of a;,,, we can say that any 

 two rays of a ray pencil representing the funicular polygon (fig. 13 c) cut 

 on all verticals at a distance x the sums 



of the APj lying between these rays. 



In this result nothing is changed when the sides of the polygon in 

 fig. 13 c are drawn parallel to the dotted sides of figs. 13 a, 13 b, for the 

 above partial sums are independent of the values of the first ratio. In 

 general, all that was formerly said about tigs. 13 a and 13 b relate also 

 to fig. 13 c, all corresponding verticals being congruent. This relation 

 gives us a means of drawing two polygons through given points. For 

 instance, let 01, the first side of the polygon, pass through a point x„yg, 

 and the last 67 through the point a;„?/o ; then we commence by drawing a 

 parallel line to ^ou 3,nd finish the polygon as above described. If, finally, 

 the side 67 does not pass through the definite point a;„y„, there is nothing 

 to be done except to shift the row of points obtained in the vertical a;„ in 

 such a way that the point next in order to 6 coincides with the given 

 point x„yo. This has happened on the right side of the vertical a*,,, and 

 now the side of the polygon 01 passes through the point x„y„, and the 



