386 Mr. Airy on Laplace’s Investigation of the 
Differentiating with respect to r, 
dV U™  2U" | 3U% 
7 pare ul = + a + &e. 
Now if we neglected quantities of the order a, or neglected 
the difference between the sphere whose radius is a, and the 
spheroid (the radius of the spheroid at this point being a (1 + ay)), 
Ara’ 
or 
we should have V= Hence in the preceding expression 
for VY, U® = as in@ a small quantity ef the order a, which we shall 
call U': and U®, U®, &c., are small quantities of the order a. 
Substituting a (1 + ay) for r in the expressions above, and neglect- 
ing quantities of the order a’, we have 
1 ora uo UM U® 
ws = Ogaay) Scaatact Mage Tne: 
dV Ara? Ue 20% 302 
—b7—= SU Tay) eet = + &e. 
Substituting these values in the equation, 
View 25 eal 
we have at length 
+ &e. 
U'o 3U» 5 U® 
EN ae + a ic ate 
Hence it follows that y is of the form Y" + Y + V+ &e., 
¥ being subject to the equation 
_d _ a¥o ee ee 
o=s-{i-#- St + a tt TFT Ve: 
taking it for granted then, that y can be expanded only into 
one such series, and supposing y so expanded, we shall have 
