596 REPORT— 1893. 



The addition of these columns gives at once 



^r ^^P* 



The author goes on to remark that it has been often said, and 

 specially emphasised, that the products ought to be arranged with regarct 

 to their signs, and proceeds to point oat how to deal with the construc- 

 tion for every possible condition of signs, concluding by a remark that, 

 exactly as fig. 13 c was obtained by means of fig. 13 h, a new polygon can 

 be constructed from fig. 13 c, giving products of the form 



or even of the form 



^M„ — % Z^ — Zi X^-Xi 



AP 



and so on. 



The method of using two diagrams, one of which is derived from the 

 other, forms the basis of all constructions in connection with problems in 

 mechanical science. 



There are, however, methods in which they are not directly used. 

 Thus to find the value of the expression of the form of 



2Aia;"-'=AX + Aa;"-i+ . . . +A„_,a;-1-A„ 



the construction given by Lill for the solution of a numerical equation 

 may be used (Cremona, chap, vi.), or that given by Egger, which is 

 modified by Culmann, who uses the sine instead of the tangent, and 

 is as follows. To find the value of 



where pi=a given length, 



a=a positive ratio ^ of two segments m and n. 

 The foregoing equation may be written 



2/=| n (ni2^i+Pi-l)"i-l +i'i-2 «i-2+ • • • +i'2J"2 + 2'l j«i+i'o. 



and the quantities in brackets may be replaced by 2/._i2/(-2 ; 

 so that 



yi-\=^hPi-\-Pi-\ 



yk-i = "kPk+lh~\ 



yo=y- 



The problem consists in finding the different values of ij. 

 Take two axes 0„ 0„, fig. 14, and draw from the original lines at 

 angles 6^, (^;_j, O^.o, such that 



sin di=:ai 

 sin 0j_,=a,_, 



