88 LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. [Apml3, 
equations. The convergency test shows that the series found 
by introducing the variable factor in the second term is con- 
vergent. Now the mathematician is satisfied when the con- 
vergency of the infinite series he uses is established, but the 
computer desires that the infinite series he is obliged to use 
shall converge rapidly. By transforming the equation 
x? — 2x — 2=0 into another lacking the first power, which is 
accomplished by placing += ya the equation 
54y3 — 18y —23=0 is found. The series found by applying 
the method to 54y?— 18%y— 23=0 converges much more 
rapidly than the series obtained from the original equation. 
Differentiating 54y? — 18%y —23=0 four times in succession 
and making x zero, 
Yo 7524, — 8762 + .8762V —3 
@Yo 4477, -- 0738 = .0738// 3 
dx 
i oe 
21 da? %, 4 
1 BY = — ar ae 
5 dag) = 0019, + .0010 = .0010/ —3 
1 dy, = as 
cam ae = .0004, — .0002 = .0002/ —3. 
The three values of y are .889 and —.4492 + .3012/ —3, 
the corresponding values of x are 1.768 and 
— .8847 + .5898Y —1. If the computations are made by 
logarithms they are not very lengthy. 
The equation y!— 11727 y+40385=0 occurs in a paper by 
Mr. G. H. Darwin ‘‘On the Precession of a Viscous Sphe- 
roid,” published in the Philosophical Transactions of the Royal 
Society, Part II, 1879, page 508. The convergency test shows 
that the factor x must be introduced in the last and in the 
first terms. The equation therefore has two real positive 
and two imaginary roots. Applying the method to 
