CHAP. VI.] REALITY OF THE ROOTS. 295 



d l y i,d i+l y ^d i+ *u 



w+v+v &+*%&-* 



d i+l v 

 for if we give to a positive value which makes the fluxion -^~i 



CL\j 



vanish, the other two terms -~ and -^~ receive values of opposite 



sign. With respect to the negative values of 6 it is&quot;evident, from 

 the nature of the function /(#), that no negative value substituted 

 for 6 can reduce to nothing, either that function, or any of the 

 others which are derived from it by differentiation: for the sub 

 stitution of any negative quantity gives the same sign to all the 

 terms. Hence we are assured that the equation y = has all its 

 roots real and positive. 



309. It follows from this that the equation / (0) = or y = 

 also has all its roots real ; which is a known consequence from the 

 principles of algebra. Let us examine now what are the suc 



cessive values which the term 6 ~hl or receives when we give 



to 6 values which continually increase from = to = GO . If a 



7J 



value of 6 makes y nothing, the quantity 6 becomes nothing 



7 



also ; it becomes infinite when 6 makes y nothing. Now it 

 follows from the theory of equations that in the case in question, 

 every root of y = lies between two consecutive roots of y = 0, 

 and reciprocally. Hence denoting by # t and 3 two consecu 

 tive roots of the equation y = 0, and by # 2 that root of the 

 equation y = which lies between l and 3 , every value of 6 in 

 cluded between l and 2 gives to y a sign different from that 

 which the function y would receive if 6 had a value included be 



tween 2 and 3 . Thus the quantity 6 is nothing when 0=0^ it 



y 



is infinite when = 2 , and nothing when 3 . The quantity 



y 



must therefore necessarily take all possible values, from to in 

 finity, in the interval from to Z , and must also take all possible 

 values of the opposite sign, from infinity to zero, in the interval 



from 2 to # 3 . Hence the equation A = necessarily has one 



i/ 



