a Problem in physical Astronomy. 54$ 
tution for these quantities has been made in the same equation 
in which it occurs, and consequently that they are no longer to 
be considered as part of that equation. This I have found to 
be better than cancelling, as it answers the same end without 
obliteration. 
We may now proceed to the summation of the series before- 
mentioned, in which the utility of what has been premised will 
quickly appear. 
3. As it does not seem necessary to set down the operations 
of computing the sums of all the series which arose in the pre- 
ceding paper, I will make choice of the summation of those 
which occur in Art. 11, they being the most difficult, as the 
properest examples to illustrate this method. 
It is well known that the expression ■- * . is =2 y y~~ 5 
4 ii£2 4. 4 j- z -3-s-yyf 1 2-3.5-7.9jy 
" ' A ~ . c . o 2.4.6.8.IO 9 * 
2.4 
2.4.6 
2.4.6. 8 
v,— 5 
from which equation we have ~ zyy JL 2 y y~ 5 y y~ 3 — 
^ Vc- Now the fluents of 
the terms on the 1 
first side are ^ 
f —W* —yy) 3x/(l—yy 
zy 4 
1 
4 y 
1 3 h.L. y 
4 ‘*4- v' ( 1 —yy) 
+ T^T + 
L T 
2 yy 
4 J 
+AH.L.- 
1 4 1 
+ V(i—yy ) 9 
on the second side, the fluents are -^21 4. LL :? y l r 3 - 5 -7-9/ 
4.6.2 1 4.6. 8.4 * 4.6.8.10.6’ 
&c. And, to find whether these two expressions are = each 
other, or have a constant difference, we may compute their 
numerical values, y being put = any small simple fraction, 
such as t L 5) or y^ 1 — , either of which values of y is a very 
convenient one for the purpose. But an easier method to dis- 
