ON QUADRAIUKES AND INTEKPOLATION. 351 



The general problem of maxima and minima is, in interpolation as in 

 ordinary analysis, to determine u and x so as to make -^ = 0. It is a 



particular case of the more general problem in which y- = a, but it is 



practically much simpli6ed by the consideration that Am is generally very 



small when — vanishes, so that the approximate position of the maximum 



or minimum is visible at sight ; but there is no such help in the general 

 case. 



The process for determining a maximum or minimum is to expand 

 (1 + £^y Uq as a rational integral function of x, and also such that the 

 ftinctions of A appearing in it shall be capable of interpretation ; then to 

 differentiate with regard to x, and equate the result to zero. The appro- 

 priate root of the resulting equation thus gives the value of x, and that 

 of ttj. is then found by interpolation. As already stated, there is practi- 

 cally an approximate value, obtained at sight, to start the more exact 

 approximation. The most obvious course is to take the ordinary bi- 

 nomial expansion, namely, 



« = .,0 + f ^ ^'0 + ^ A^«o + '^^ - ^^^ + ^' A3.0 + &c. 



whence 



flu r. . , 2a; — 1 . 2 , 3x^ — 6x + 2,. 



_ = = Azio + — 2 — ^ ^'o + Q ^^0 + &c. 



and this equation has to be solved with respect to x, preferably by succes- 

 sive approximation, after which u is determined. But any other expan- 

 sion, such as that by symmetrical differences, in which the expansion 

 variable is A : y (1 + A) may also be taken. When the solution is 

 obtained otherwise than by successive approximation, as, for instance, by 

 solving as a quadratic, care must be taken to select the proper root. 



Values corresponding to inflexional points In a curve, are of course 



determined by operating in like manner upon ■ — ~. 



dx^ 

 In the use of equidistant ordinates, no diflBculty can arise from % and 

 X reaching a maximum together. But when the ordinates are not equi- 

 distant, this point requires attention. It presents no other difficulty than 

 is met with in the ordinary theoiy of implicit maxima. 



The case of -^ = a, only differs from that of ^ — 0, as far as work 



ax clx 



is concerned, by its being less easy to see what the first approximation is 



to be. Graphical processes, however, or trial and error, soon remove any 



difficulty. ^ 



It must be remembered that the determination of a tangent is of a 

 higher order of precision than the determination of the point of contact. 

 It follows that the determination of the argument corresponding to the 

 maximum or minimum value of a tabulated function Is less precise than 

 that of the corresponding value of the function, and also less precise 

 than Its determination generally. 



