1903.] LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. 93 
Only linear transformations which make the coefficient of 
the second term of the complete equation or of the equation 
reversed zero or small are used, as other transformations 
become too complicated to make the method practicable. 
IV.—TRANSCENDENTAL TRINOMIAL EQUATIONS. 
Let an equation of the form y+aj(y)+b=0, where }(y) is 
a transcendental function, be called a transcendental trinom- 
ial equation. Such equations are readily solved by the 
method, provided the resulting series is rapidly convergent, 
but in the absence of a transformation which insures rapid 
convergence the method has little practical value. 
Suppose the equation 2y+logy — 1000=0 to be given. 
Applying the method to 2y+zlogy — 1000=0, if the 
Napierian logarithm of y is taken, y,=5000, = = — 4.30625, 
2 ia —/0 _ 4.000215, and y=4995.69; if the common logarithm 
dz, 
ies —1.84948,1 © — + 9.00018, and 
of y is taken, Yo= 5000, | t da,? 
y= 4998. 15. 
V.—EXPANSIONS. 
If y=z+ve¢(y), where v and z are independent variables, 
Lagrange’s series expands any function of y in powers of v. 
These expansions may be obtained by writing y=z+vzr¢(y) 
and expanding [ (y), which now becomes a function of 2, by 
Maclaurin’s series and making x unity in the result. 
The method will be illustrated by obtaining two expan- 
sions which occur in theoretical astronomy. From the 
equation H=M-+e sin EH, where EF is the eccentric anomaly, 
M the mean anomaly and e the eccentricity of the orbit, it is 
necessary to find H and (1 — ecos EL)? 
To find HE, write H=M-+ex sin E, whereby EL becomes an 
implicit function of x. Differentiating twice ir succession 
with respect to x, 
dE 
j d# 
dp  ¢Sin H+ ew cos H , 
CEH dH he dk 
dg ee # ag t ex cos H daz &% sin (4 = 
