MISCELLANEOUS PROPOSITIONS. 407 



hence A t , A 6 , A s , ... vanish, 



A 2 A 2 



^ = 5^ = -3T5' 



A - A - ^'^ 



7 ~7 5 ~ 3.5.7* 



Put ^for sin" 1 a;; thus we deduce 



See Quarterly Journal of Mathematics, Vol. 6, page 23. 



375. Let <j)(x) denote 



where n is a positive integer. It is required to determine 

 the coefficients p t , p^, ...p n so that the numerically greatest 

 value of </> (x) between the given limits h and h for x shall 

 be as small as possible. 



If we give a geometrical form to the problem, we may say 

 that the curve y = (f>(x) between the limits h and h is to 

 deviate as little as possible from the axis of x. 



The maxima and minima values of <> (x) will be deter- 

 mined by the equation </>' (x) = 0, which is of the (n 1)* de- 

 gree ; and therefore there cannot be more than n 1 of such 

 values. These values, together with the values of $ (#) when 

 x = h, and when x = h, will be called extreme values. 



376. Now we admit as sufficiently obvious that there 

 must be some definite values of the coefficients in <f> (x) which 

 solve the problem ; and we shall first shew that there must 

 be n + 1 extreme values all numerically equal. 



Suppose, for instance, that n = 3 ; then there must be 4 

 extreme values all numerically equal. 



For if possible suppose that there are only 3 extreme values 

 of < (x) all numerically equal ; namely, corresponding to the 

 values x v # 2 , and x a of x. Let ty (x) denote the expression 



