Talks for the Solution of an Equation. 215 



'-^^r*- ****> 



writing 2h for h, this gives 



u 

 and putting 3/t for h 



~ = y. 



From these three equations u can be found in terms of a, /3, y, and 

 its value can be put into the convenient shape 



u - a -J(/3-a) + T V(y-/3) ..................... (18) 



From the sequence law it follows that 



Consequently, if values of K (a;) for seven equidistant values of x be 

 taken, the quantities a, /?, and 7 can be derived, and (18) ought to give 

 a value of u equal to that of K^z) for the middle value of x. This 

 test has been freely applied throughout Table II with very satisfactory 

 results. 



18. If the values of f(x Q + 4:h) and/(a; - 4/t) be taken into account, a 

 still more stringent test is afforded. As before, let these quantities be 

 denoted by u 4 and w_ 4 . Let z be used to denote 



jT7Y\/ vii (- r o) and let ^^ = 8. 

 Then 



z = y, 

 M + 16v + 256w + 4096s -= 8- 



whence it is not difficult to show that 



(19) 



This can be used independently, or it can be made to yield a correc- 

 tion to (18). In the latter case the quantity 



has to be subtracted from the value of u given by (18). 



