[deville] 



ABACUS OF THE POLE STAR 



5 



To shorten writing, let /"„, ç5„, ^■,j, be written instead of f„ («'„), 

 <f>n (^n), '/'n (^n), and let us consider the particular case when equations 

 (1), (2), (3), assume the form : 



^ A + y ^1 + ^'i = o 



(4) a; ^2 + y ^2 + ^-2 = o 



•Î-' fs + y <l'3 -\- fa = o 



Each of these equations defining a system of straight lines, their re- 

 sultant after the elimination of x and y : 



(5) 



4\ 



— o 



is represented by three systems of straight lines. Thus an abacus con- 

 sisting of straight lines only can be constructed whenever the equation 

 to be represented can be put in the form of equation (5). 



By the application of the principle of duality, this figure can be 

 transformed into a correlated one such that to straight lines shall cor- 

 respond points. Each of the equations (4) which, in the first figure» 



Fig. 2. 



defines a system of straight lines tangent to their envelope, defines, in 

 the second figure, points distributed upon a curve, their bearer, as in 

 Fig. 2. Equation (5) which in the first figure means that three straight 

 lines are copunctal, means in the correlated figure that three points are 

 coHiraight. Instead of following as in Fig. 1 the lines {a^ and 

 (o-j) to their intersection and finding the line of the system (0-3) 

 which passes through this point, the mode of employment of the new 

 kind of abacus (Fig. 2) consists in joining by a straight line the points 

 («jj and (a-,) and i-eading the graduation at the intersection of the 

 bearer of (a^. The abacus has gained in simplicity, consisting only 

 of three lines, and the interpolation by estimation instead of being simul- 

 taneous between three pairs of lines is now made three times in succes- 

 sion between two divisions of a graduation, a process susceptible of con- 

 siderable precision. 



