592 
MATHEMATICS: A. A. BENNETT 
By some reaction of this type, if proper protection against intramo- 
lecular rearrangement be provided, we may predict that a di-aryl sub- 
stituted hydrogen peroxide will be prepared, which in turn will dissoci- 
ate into mono-aryl oxide, ArO, the odd molecule, or free radical, of 
univalent oxygen. Such a peroxide would probably be less dissociated 
than the similarly substituted hydrazine, just as the latter is less dis- 
sociated than the corresponding ethane. 
^Gomberg, /. Amer. Chem. Soc, 38, 770 (1916). 
^Piccard, Liebig's Ann. Chem., 381, 347 (1911;. 
' Schmidlin, Ber. D. Chem. Ges., 41, 2471 (1908). 
4 Lewis, /. Amer. Chem. Soc, 38, 770 (1916). 
sWieland and Offenbecher, Ber. D. Chem. Ges., 47, 2111 (1914); Meyer and Wieland, 
Ibid., 44, 2557 (1911). 
6 Chichibabin, Ibid., 40, 367 (1907). 
^ Schlenk, Weickel and Herzenstein, Liebig's Ann. Chem., 372, 1 (1910). 
8 Wieland, Ibid., 381, 200 (1911). 
NEWTON'S METHOD IN GENERAL ANALYSIS 
By Albert A. Bennett 
DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY 
Received by the Academy, August 8, 1916 
The present paper is essentially an extension of the methods and re- 
sults given by Dean H. B. Fine, On Newton's Method of Approxima- 
tion,^ and the results there obtained need not be explained here. 
An illustration of some general notions. — As might be expected, 
Newton's Method is of very wide application and may be used, for ex- 
ample in the following three cases to find a real 'root' of the real func- 
tionelle where F has the property that in a certain domain, and 
for every x{s) and x{s) + h{s) in this domain, there exist functionelles 
Fi and F2 for which 
F[x{s)+h(s)]^F[x(s)] + fo'FMs),r]h(r)dr 
+ fo' i l^(s), fi, f J h (n) h (r^) dr,dr„ (A) 
(1) where ^(s) is such that max^ | ^(s)—x{s) | ^ max^ \ K^) \ and max^ 
x(s) -\-h (s) — ^(s) I ^ max^ | h(s) \, if by 'max/ is meant the maxi- 
mum as varies, and by a 'root' of the above equation is meant a func- 
tion x(s), such that max^ | F[x(s)] \ =0; or (2) where ^(s) is such that 
VJl'[^ (r) -X {r)]'dr ^ V fo'h' {r) dr and V fo\x {r) + h {r) -^{r)]Hr ^ 
'x/ Jo^ (r) dr, and by a 'root' is meant a function x(s), such that 
Vfo^F[x(r)Ydr = 0; or (3), where ^(s) is such that Jl^ \^{r)-x(r) \dr^ 
