226 SCIENCE PROGRESS 



tion of simple ascending division for this purpose. It is suitable 

 chiefly for equations of low degree, such as those which are 

 mostly dealt with in text -books on the theory of equations. 

 But I am by no means sure that it is the best method even for 

 these ; while owing to the fact that the method generally 

 requires at least one transference of origin, it often becomes 

 laborious for equations of high degree or transcendental equa- 

 tions — which, however, can frequently be readily solved by 

 other kinds of operative division, or by the arithmetical pro- 

 cesses which they represent. Simple operative division (when 

 the first term of the divisor is o) is, however, easy to remember 

 and to apply. 



(2) For a first example we will find the lesser positive root 

 of the equation o = a! — b'y + c'y n . 



Observe that it is written with the absolute term a' positive 

 and leading. We can always reduce the coefficients of the 



first two terms to unity by putting y = j-, x and dividing by a', 



which gives for solution an equation of the form 



o = 1 — x + cx n . 



When x — o, this function = 1 ; that is the function cuts 



the axis of jy at a unit's distance above the origin ; and as the 



absolute term of the tangential, — 1 + ncx n ~ x is also unity 



when x = o, the function begins by descending towards the 



axis of x at an angle of — 45 °. But if c is large, the positive 



term cx n soon exerts its influence and causes the curve to turn 



upwards before it can reach the axis of x at all ; so that the 



two possible positive roots become unreal. Obviously, by 



Descartes' Rule the equation has no negative root when n is 



an even number ; but certainly has one, and only one, when 



n is odd. It is easy to ascertain whether the two positive 



roots are real or not. For the tangential vanishes when 



n ~ 1 / T~ 

 x= V — . If 1 — x + cx n is positive at this value of x, the 

 v nc 



roots are unreal ; if it is zero, the roots are equal ; and if it is 



negative, the roots are real. That is, the roots are unreal if 



(n — i) n - 



c> «» 



as is well known. 



