350 KEPOKT— 1880. 



Section 6. — Interpolation of direction : maxima and minima. 



It frequently happens that it is required to find, by means of given 

 ordinates, whether differenced or not, the value of some particular 

 difi"erential co-ejB&cient, or else the value of the variable corresponding 

 to some given value of the differential coefficient. These ultimately 

 depend upon the symbolic equation 



suitably transformed and duly interpreted, or solved. In certain cases, 

 the desired result may be obtained by mere algebraical transformation, 

 by indeterminate coefficients or otherwise. 



As an example, let it be required to obtain a formula for the tangent 

 at the head of the middle ordinate of the set 



Mq Ui «2 "^^s ^4 



the common interval being Ji, The analytical problem is to determine 



= A(1 +A)2 1og. (1 + A)«o 



du2 



dx 

 in such a form as to stop the difference series at A^Wq. The work is 



i*' = (l+;A)'(A-iA'+|A.-ii.)«. 



2 1 



The process is perfectly general, and needs no further remark, except 

 that the work might have been made a little more symmetrical by taking 

 the middle ordinates as origin. This will give considerable simplification 

 in the algebraical solution. To obtain this, 



write M = aa; + cx^ 



omitting the even powers, which evidently disappear in the result. 



Then ^ = a+ 3ca;2 (= a when a; = 0) 

 ax 



«! = — «_i =-^ («! — M-l) = ah + cA' 



v.< 



= — M_2 =-^ (u2 — «_2) = 2aA + ^cW 



and Xj and Xg are to be so determined that 



Xj (Mj — u_{) + X2 (1I2 - «-2) = «^ 

 This gives 2\i + -iXj = 1, 2Xi + 16X2 = 



2 1 



whence Xj = — , Xg = — ~2. This, allowing for the change of origin, 



is the same result as that already obtained. 



