92 LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. [April 3, 
y' — 107?+52y+8xr=0 and xy! — 10y?+5a4y+8=0. From 
each of the two equations two real roots, one positive and 
one negative, are found. 
(c) Tat + 202° + 382? — 162 —8 = 0. 
Here the terms to be underscored are probably the first, 
second and last, indicating the existence of two imaginary 
and two real roots, one positive and one negative. All doubt 
is removed by transforming by x=y — .7 into 
Ty + Ay? — 18 42y? — 1.404y — .5093 = 0. 
The transformation x=y —.7 is selected because it is a 
simple transformation which makes the coefficient of the 
second term very small. 
(d) x + 12a + 592° + 1502? + 2012 — 207 = 0. 
Here probably only the first and last terms are to be 
underscored, indicating the existence of four imaginary roots 
and one real positive root. Transforming by r=y — 2, 
which makes the coefficient of the second term small, 
y° + By -b Sy? =P ay? by — Bal 0. 
The roots are found by applying the method to 
y? + 2ay*t + Bry? + 4ay? + dry — 321 = 0 
(e) xt — 80a* + 19982? — 149372 + 5000 = 0. 
Here probably every term should be underscored, indicat- 
ing four positive real roots. Transforming by the substitu- 
tion z=y+20, 
y* — 402y? + 983y + 25460 — 0. 
Here the terms to be underscored are the first, second and 
last. More rapidly convergent series are found by reversing 
the last equation, 
234600 + 9830? — 4020 + 1=0, 
and making the substitution »=z—.01, whence 
2546024 — 35.42% — 416.2142? + 8.233062 + .9590716 = 0, 
When z has been computed, zx is found from 
20002 + 80 
Fe Ne ee a 
