314 Mr. J. Sylvester : Meditation on the 



maximum value of ( rr — 1 ) for values of a? intermediate between 



a and b shall be the least possible. Some little way, but only a 

 little way, into the solution of this problem we can look in ad- 

 vance. In the first place, if we seek for the maximum values of 



( jj — 1 \ , we obtain the rational equation 



derivatives of the odd order (subsequent to the first) are concerned, from 

 the consideration that the limits of error in excess and in defect will be 

 actually attained for values of x lying within the prescribed limits ; but 

 these errors, c* and rji (when f=»7, which is true by hypothesis), are 

 never equal, the former (the extreme error in defect) being always the 

 greater of the two ; but if any such derivative were the best of its kind, 

 the absolute values of the extreme errors of excess and defect ought to be 

 equal to each other. But more generally, if possible, let the ith deriva- 

 tive to L(x) (where L(x) represents the radical linear approximant fx+b 



to */a + bx+cx\ say Q(x)), viz. V ' } J- — , f- Q(x), be sup- 



7 (Lx+Qx) 1 — (LxQx) % 



• . , . 2(L# — Qx) 1 

 posed the best of its kind : then the relative error is * — . '- ,-, 



v (La?+Qa?) , -(La?-Qa?) 



and the maximum value of this must be equal (to the sign prhs) to the 

 value which it has when we give to x either of its extreme connecting values* 



Now obviously the above is a maximum only when - _q is a minimum, 



and therefore when ^- is a maximum ; but by hypothesis, the value of x, 



La? 



say m, which makes this a maximum, gives to ^ — 1 the same value with 



the opposite sign to that which it would have in writing for x either of its 

 limiting values, say k or k'. 



Thus we have two equations for determining -=-7;, t^-{, viz. 



q,k y(wi) 



Lm_._ 1 Lk 

 Qro ~QP 



and 



(&'M£-')'-<~H(^0'-(S-')T 



Thus, suppose i = 2, we should obtain from the second equation 

 — = — ^rp which is inconsistent with the first ; so if z=3, we should 



obtain ( — — ) =( — - ) , and therefore, on account of the first equation, 

 VQm/ \QkJ 



7^7^=1 j and so in like manner for any value of i, we should derive one 

 Q(ro) * 



Lwi . ... 



or more numerical values for q—, which is absurd, since this quantity is a 



function of k, k' t the two connecting values of x. 



