5 8o SCIENCE PROGRESS 



be easily verified from figure 7 — all the curves being continuous 

 and the roots real roots. 



(a) A partial root can refer only to one actual root, if to 

 any. For if it lies between two actual roots, it must be 

 separated from one of them by another partial root, either 

 of the same partial curve or of the other one. 



(b) If a partial curve possesses only one positive root not 

 zero, then if this partial root lies between two actual roots 

 it must refer to one of them. 



(c) If £(0) and f(o) have different signs, the odd positive 

 roots of both counted upwards will refer only to odd positive 

 actual roots ; and the even positive roots of both, only to even 

 positive actual roots. But if they have the same sign this will 

 hold true only of the function which is numerically the greater 

 when x = o, while the converse will hold true regarding the 

 other partial curve, zero roots being neglected. 



(d) The question whether a partial root is prospective or 

 retrospective, that is, less or greater than the actual root to 

 which it refers, can be determined by the sign of fx at the 

 partial root, since fx changes sign at each actual root. 



(2) With these and similar theorems, partial roots derived 

 even from a single well-chosen setting will often give easy 

 analyses of equations {Note IV) ; but our present subject 

 requires rather the use of partial roots of different settings. 

 I have space only for power-series equations. Consider 



fx^p x n -\-p 1 x n - 1 + . . . +p n - l x+p n = o. 



This may be set in many ways by putting any two of the 

 terms on one side of the equation, and the other terms on the 

 other side. Such a pair of terms may be called a critical pair ; 

 the term with the higher power of x may be called the superior 

 term ; and the real root or roots (other than zero) of the pair 

 may be called critical points. Thus if p r x n ~ r + p q x n ~ q be a 

 critical pair, the corresponding critical point or points will be 

 ( — pq/p r ) 1{q ~ r) ', an d this cannot possess more than one real 

 value of one sign. Then we have the following propositions. 



(e) Every positive critical point which lies between the least 

 and the greatest positive actual roots must refer to one of the 

 positive actual roots next to it. From (a) and (b) of the 

 previous subsection. 



(/) A critical point is prospective if at that point fx has the 



