1903.] LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. 87 
To determine when series (3) is convergent, group the 
terms numbered 1, n+1, 2n+1, 3n+1,.... , then those 
numbered 2, n+2, 2n+2, 3n+2,.... , finally those num- 
ered, 27; Bn, 41,6... wn. Each of these n partial series 
is found by Cauchy’s ratio test to be convergent when a” is 
numerically less than k*(n—k)**b*. When this condition 
of convergency is satisfied series (3), by substituting for w 
in succession each of the n values of the n” root of 6b, deter- 
mines the n roots of equation (1). 
By introducing the factor x in the third term of equation 
(1) and applying the method a series is obtained which deter- 
mines k roots of equation (1), and by introducing the factor 
x in the first term of equation (1) a series is obtained which 
determines n —k roots of equation (1). The two series thus 
obtained are convergent when a" is numerically greater than 
k*(n—k)**b*. When a*=k*(n—k)**b* equation (1) has 
equal roots. There is therefore developed a complete theory 
of trinomial equations. 
The general fifth degree equation can, py Tschirnhausen 
transformations requiring the solution of equations of the 
second and third degrees only, be transformed into the tri- 
nomial equation y’+ay+b=0. If a@ is numerically less 
625 
256 
ing the method to y?+ary+b=0. If a’ is numerically greater 
625 
than 356 
to y+ay+br=0 and zy’+ayt+b=0. If a numerically equals 
6255.4 
206 ” 
moval of the equal roots makes the solution of the fifth 
degree equation depend on the solution of an equation of a 
degree not higher than the third. A third degree equation 
becomes trinomial by removing the second term, which is 
accomplished by a linear transformation. The method of 
this paper therefore effects "the complete solution of the 
general fifth degree equation in infinite series. 
In Weber’s Algebra, volume I, pages 396-399, the real and 
imaginary roots of the equation 2? —2x —2=0 are computed 
by a method invented by Gauss for the solution of trinomial 
than 
b*, the five roots of this equation are found by apply- 
b‘, the five roots are found by applying the method 
the fifth degree equation has equal roots, and the re- 
