PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
545 
NOTE (B). 
The following method of obtaining an expression for dt equivalent to that in 
art. 9, was communicated to me by Sir John Lubbock after the reading of my Paper, 
and appears to be well worthy of being recorded in connection with the process of 
solution I have adopted, as, on a resumption of the reasoning for the purpose of 
carrying the approximation farther, it might considerably abbreviate the analytical 
details. 
\i dt=^^-rdv, and v be taken for the independent variable, the equation 
becomes 
and if 
ar 
r=a — ae cos v + cos 
and 
Hence 
t>=cos' 
1 
L 
1 cos V 
ae ' 
^ ae 
uj Q sin vdv — sin Q cos vdv, 
Jq sin cos vdv^- 
dt 
fa ^ fa '>'dT + r sin vdvf Q sin vdv + r cos vdv f Q, cos vdv 
V ft ^ gjjj ^y’Q cos 
Neglecting powers of Q above the first, 
dt 
fa J rdr 
^ l(«V-(a- 
-.4 
rdv (sin vj' Q, sin vdv + cosvJ'( 3 i cos vdv) 
-VH 
-VH: 
(a — r) {a^e^— {a— rYY 
r[a — r)dr[co^ vJ'Ql sin sin vJ'Ql cos Wv)'| 
{aV-(a-r)2)§ / 
Q sin yc?y4“Cos v i Q cos vdv) 
f rdr 1 
1 rdv 
[a — rY)^ 
' ae sin v 
r cos idv , — sin z;fQ cos vdv) 
ae sin^^t; ^ J J 
rdr 
2 o 2 . 
a^e 
■ (a— r)^)- 
ae sin 
^^iQsinfG^ylj 
iin'^wj ^ j 
which equation is true to all powers of the eccentricities and inclinations, v being the 
eccentric anomaly.” 
