MATHEMATICS: H. B. FINE 547 
the equations f) = 0 have one and hit one solution in S, and an approximate 
value of this solution, as close as may he desired, will he obtained by de- 
terminations first of . . . , and then of xi^^\ • • • ? 
Xn^^\ for j = 0, 1, 2 , . . successively, by the equations 
/•, (xS^ x^\ . . ^) + 2) 14) = {i=\,2,,,., n) 
tri (b) 
(where = the solution being lim^ = ^ {x^^\ x[^', . . . , x\i^). 
Moreover the equations (h) will yield a similar sequence of approximations 
to this solution if instead of (xf X2\ • . . xl^^) any other point in S be 
chosen as the point of departure. 
For the case of a single equation f{x) =0 the condition (a) reduces to 
\nxo)\<-, (aO 
V 
where X (>0) is the lower bound in the interval S of the values of \f' (x)\ . 
For let 
Then, if both (4^^ x^^\ . . xf) and {x['"''\ 4'■+'^ . . . , 4''+'0 are 
in S, 
/u'+i) _ /(i) 1 'V— ^ i^l-'^-J- i'V — (i=l 2 n) (\) 
where each Xk^^ lies between Xk^ and xi^'^^K 
Hence if the numbers be so taken as to satisfy the equations 
(»'=1.2,. . .,«), (2) 
we shall have 
■'4 
Solving the equations (2), we find 
^P" = iy; ^ 0)^ {i=\,2,. . .,n), (3) 
^ = -2=^ , {k = \,2,. . .,n), (4) 
