‘MATHEMATICAL ‘NOTES. 1293 
‘representing two parallel (or, it may be, coincident) ‘straight ‘lines ; 
the former reduces it to 
By? +'2 De= F, 
-and by ‘taking the origin on the curve, ‘still further ‘to 
y? — Lz, 
representing the parabola. 
3. On a method of Approximating to the Square Root of a Number : 
The following singular proposition is given by Murphy in his 
Theory of Equations, Art. 77, and is very characteristic of a mathe- 
matician, perhaps, the most original of modern timés. The demon- 
stration that follows is his own, somewhat simplified. Let NV be the 
number, and let ,/N be between'z:and xn +1. Put N— n2 =a, 
(m + 1)? — x”, or, (2x +1)= 86. Take any proper fraction 
u c : 
= , and let a series of fractions be successively formed by the law 
oO 
Ug +4 = Ae + Un, Ve +1 = be + Uz, 
then “= eonverges to the decimal part-of ./ NW. 
& 
Uae) Wy + Uz 
and isa proper fraction since a <b, 
Upil mere + Ux i Pee 
ata 
Es 
jes eee 
a 
Let thea y = Limit SE ee eco 
Ve 41° 
aty 
then ultimately y = 
ultimately y PEG 
or, y? + (aud) yiiesa 
and y? + Qny + n? = N. 
whence y = —2 + /N, 
Bince the positive sign must be’taken. 
Hence, Limit <= = SNe Nh, 
x 
lo ae 
or — converges to the decimal part of ,/.N. 
Vz 
Murphy gives as an example ,/10. Assume the traction a then 
@ = 1,6 = 7, and the successive convergents are 
1 7 25 179 1282 
6’ 437 154’ 1103’ 7900’ “""""" 
