408 MISCELLANEOUS PROPOSITIONS. 



and suppose /i 1} /i g , and /* a to be infinitesimal constants, 

 which are determined so that </> (x) and ty (x) may have con- 

 trary signs when x=.x lt when x = x a , and when x = x 3 : this 

 can obviously be done. For instance, the sign of ^ must be 



contrary to the sign of - - , v '' - r . Then < 



(x 1 x t ) (x l x a ) 



differs only infinitesimally from </> (x) ; but when <f> (x) has its 

 extreme values <f> (x} + -^ (x} is numerically less than <j> (x) : 

 and so <j> (x) + ijr (x) deviates less from zero than < (x} does. 

 Moreover the coefficient of a? in <f> (x} + ty (x) is unity ; so 

 that <f) (x) + ty (x) is an expression of the proper form. It 

 follows therefore $hat <j> (x) cannot be such as the problem 

 requires. 



The preceding argument will perhaps be more readily 

 understood when presented in a geometrical form. The curve 

 y = <f>(x}+^ (x) is indefinitely close to the curve y = $(x}\ 

 but where the latter curve deviates most from the axis 

 of x the former curve is nearer to the axis of x: and thus 

 the former curve deviates less from the axis of x than the 

 latter curve. 



In the same way we may treat the case in which < (x) 

 has only 2 extreme values numerically equal and numerically 

 greater than any other value ; or the case in which the 

 numerically greatest value of <f> (a) is unique. 



The considerations which we have thus employed when 

 n = 3 are applicable whatever may be the value of n. 



Hence, as we have said, to solve the problem the coeffi- 

 cients in <f> (x) must be determined so that <f> (x) may have 

 ?i + l extreme values all numerically equal. 



377. Let k denote the extreme numerical value of $ (#) ; 

 then we have shewn that the equation 



must have n + 1 values which also satisfy the equation 



(x 8 - A 2 ) f (x)=*0 ....... ... ..... . ..... (2). 



