86 LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. [April 3, 
Differentiating (2) twice in succession 
dy 
(3) 4y° OU __ ay? + Toy — Gay SH 4 + 75x tr = 
Ae dy 9,, UY dy dy\? __ 
(4) dy Gat ey (4) —12y "+ 1504 — br (S*) 
a Be te ey 
6ay — da? = + 5a qa 
Making x zero in (2), (3) and (4) 
iy | 10 S40 ey 100 0 
20 — — 1195, — 2625, +. 1875 — .075 /—1, + .1875 + .075/—1 
0 
oy °—__ 0029, — .0029, — .0000015 + .00391/ —i,— .0000015 — .0039/ —1, 
Ls 
Substituting these four sets of values in Maclaurin’s series 
and placing « —1, the roots of equation (1) are found to be 
Y, = + 9.886, y, =— 10.261, ys = + .1875 + 9.927 Y—1, 
Y, = + .1875 — 9.927 V—1, 
all correct to the last decimal. 
This method will be applied to the solution (II) of tri- 
nomial algebraic equations, (III) of general algebraic equa- 
tions, (IV) of trinomial transcendental equations, and finally 
(V) the method will be applied to obtain expansions com- 
monly obtained by Lagrange’s series. 
II.—TrRinoMIAL ALGEBRAIC EQUATIONS. 
The general trinomial equation of degree n has the form 
(1) y®—nay>*—b=0. 
Introducing a factor x in the second term of (1) 
(2) y®—naay?*—b=0. 
Applying the method and denoting the n“ root of b by w 
2 
(3) y=o + ot* a + ot (1 2 4-0) 
3 
+ ol (1— 3k +n) (1 —8k + 2n) aT 
4 wise (1 —4k-+ n) (1 — 4h + 2n) (—ae-+ Bm) sey 
