THE SOLUTION OF EQUATIONS 107 



The root is between 1-27 and 1-28 and is nearer to 1-28. 

 To get a nearer approximation take the root as (1-28 k). 



Then A = sin x - 0-7 5x = - 0-0020 when x = 1-28 

 fi\ 



= cos x - 0-75 = - 0-4633 when x = 1-28 



z = sin x = 0-9580 when x = 1-28 

 dx* 



, dA 1 , d?A 

 But ^_*)-A-*2-+ i * aB | 



- 0-0020 + 0-4633A; - 0-4790& 2 = 

 k 2 - 0-967224 k + 0-0041754 = 

 k - 0-483612 = 0-479276 



k = 0-004336 



The root is 1-28 - 0-001336 = 1-275664. 

 Working to five significant figures the root is 1-2757. 



65. Maclaurin's Theorem. If we start with Taylor's Expansion 

 and put x = 0. 



dA z d 2 A 



d*A 

 and 



where A, -r > ^r-r . have their respective values when x = 0, 

 dx dx 2 



and these will be constant coefficients, since they are independent 

 of x and y. The expansion may therefore be expressed as 



/dA\ 



f(x) = A X= . Q + X ( -r- } 



where A =/(#). 



An alternative proof can be obtained in the following manner : 

 Let y = f(x) = A + A^x + A%K 2 + Ayi? + . . . A^x n + . . . 



where A , A lf A 2 , etc., are constant coefficients. 

 Then when x = 0, 



-j = A! + 2A.5# + SA^ 2 + . . . nA^x"- 1 4- ... 



0M? 



and when x = 0, A, = (-^ ) 



\dx/ x , 



^ = I 2 A, + 3 x 2A 3 ar + . . . n(n - 



- ' 



