94 LAMBERT—MACLAURIN’S SERIES OF EQUATIONS.  [April3, 
dk, 
: @E ‘ 
Making x zero, H,=M, =e sinM, dn? = 2 cos M sin M. 
0 0 
Substituting in Maclaurin’s series and making x unity, 
2 
E=M+esnU+ > sin(2M) +.... 
To find (1 — ecosH)-, write H=M-+ezxsin EH and 
y=(1—ecos£)~. Since y is a function of x through EH, 
1 
pt ey (1 — e cos #)—* sin yy 
dx dz 
Ly RE 8S os #)—+sin? wo 
ae 6e? (1 —ecos #)—‘sin i. 
2 
— Re (1—ecos H)—*cos #( ) 
dx 
— Re (1—ecosH)—'sin # ue 
dx* 
Placine: 2—0; when w=; Le =esinM and 
CH Lone 
ap 26 sin M cos M, 
Y = (1 — e cos M)—’, 
Wo _ __ 96 (1 — ¢ cos M)— sin? Wf 
dX 
2 
ayo = Get (1 — e cos M)-—‘sin‘ I, 
dxy as 
— 6e (1 — e cos M)—‘sin? YU cos M. 
Substituting in Maclaurin’s series and making z unity, 
(1 — e cos H)— = (1 — e cos M)— — 2¢ (1 — e cos M)— sin? M 
+ 8c ( — ¢ cos M)—‘sint M 
— 3é (1 — e cos M)—sin’*M cos M+ .... 
In like manner all expansions obtained by Lagrange’s 
series may be obtained by a direct application of Maclaurin’s 
series. Of course it is evident that if e is considered a varia- 
ble the derivatives with respect to e may be formed and the 
introduction of x is unnecessary. 
