OPERATIVE DIVISION 575 



and the expressions on the right denote the straight lines in 

 operative notation. If m and n are not equal, the lines meet 

 at some point. Let xi be the abscissa of that point. Then 

 {figure 1) 



[£#0 — mxo + mojxi = [£# — nx -f no]xi, 



and 4^=0+ \x . 



|_ n — mj 



Next, through the same points (£# , x ) and (^x 0> x ) draw 

 tangents to f and £ and let these tangents meet at the point 

 of which the abscissa is x\. Then (figure 2) 



*-. = [0 + Ml 



#0- 



Now if (£# — %xo)/( n — m) an d (£#0 — £#o)/(|:'*o — f^o) have 

 the same sign, and also n — m > Z'x Q — ^Xo, then x\ 

 lies between x and a^j. Hence (changing their order) if 

 %'x — %'x is always > m — n between x and X, then the 

 iterants x x , x 2 , x 3f . . . will always be less than the abscissae 

 of the intersections of the corresponding tangents. But very- 

 near to X the common-points of the tangents are very nearly 

 the same as the common-point of £ and £ themselves ; so that 

 if this condition holds the iterants must always be < X if 

 x Q < X, or be > X if x > X, but in both cases will infallibly 

 approach X when £jc — gx becomes very small — as it must do 

 if £ and f meet at all. The only exception to this is when the 

 curves cross each other at a right angle ; but in this case 

 vicarious curves Z and E can be found which have the same 

 common-point but which cross each other at another angle. 

 If £ and I approach each other at a very small angle, as when 

 they merely osculate, the iteration may be very slow. 



We may denote the iterand by /, and its tangent at the 

 common-point by \'X ; so that 



/=0 + (£-£)/(»-w); l'X=i +{?X-?X)/{n-m). 



We may call l'X the root-tangent, and m and n the iteration- 

 tangents. 



Observe that the latter are arbitrary numbers which we 

 may select as we please so as to make the iteration valid or 

 rapid. They need not necessarily be constants since the 

 arbitrary curves &o//x and ^x l/x need not necessarily be 



