68 
MATHEMATICS: J. H. McDONALD Proc. N. A. S. 
of the roots of f n = 0. Assuming n + 1 > 0 as before, so that the roots 
of g* = 0 are all real and negative, let z h z 2} Zz be the k th roots of g? + i g? 
and gp + 1 = 0, then Zi>z 2 >z z and the functions g all change in the 
same sense from negative to positive or the reverse when z passes through 
the corresponding root. If z' is the k th root of (g? + x )' = 0, from the 
above equation g* + 1 (z f ) = g" and it can be seen that (g" + i(22))'>0 
or < 0, according as g? + i changes from negative to positive or the 
reverse when z passes through z%. This allows a series of approximations 
to a root of f„ = 0, say the first one, as follows: Let z 2 be the root of 
g 2 = 0; form the quantities z 3 = z 2 - g ±^L, z 4 = z z - Cli^, etc. These 
gz(z 2 ) g 4 7 fe) 
z's converge to the first root of f n = 0. In the case of the first root the 
approximations are rational functions of n. 
