4 ROYAL SOCIETY OF CANADA 



In the Bame way, the equations : 



(2) f^ {x, y, a,) = o 



(3) Ta (x, y, «'s) = o 



define the systems of curves (a^) and (0-3). When three of these curves 

 taken respectively in each of the systems intersect in one point, the cor- 

 responding values of the variables a^, a,, a^, satisfy the equation : 



F (o-j, «2, a^) = o 



resulting from the eliminati(m of x and y between the equations (1), 

 (2) and (3). The value of any one of the variables can thus be ob- 

 tained by means of the other two. For instance, if we wish in Fig. 1 to 



Fig. 1. 



find the value of a^ corresponding to o-j = 2 and o-j = 4, we follow to 

 their intersection the curves marked " 2 " in the system (^a^) and "4" in 

 the system (^2) : the curve of the system (a^) passing through this point 

 being marked "5", this number is the required value of 0-3. 



This kind of abacus is the one most frequently met with, although 

 by no means the best. Usually one of the variables, a^, is taken as 

 X and another, a^, as y; a^ is thus represented by a series of parallels 

 to the y axis, a^ by a series of parallels to the x axis and a^ by a series 

 of curves. The use of this abacus requires simultaneous intei'polation by 

 estimation between three pairs of lines, an operation not susceptible of 

 much precision. The accuracy may to some extent be increased by 

 drawing more lines, but a limit is soon reached beyond which the num- 

 ber of lines becomes confusing. 



