70 
Mr. Ivory on the Attractions of an 
B (0) = • -d\A . dm' = ~ .J'J'dvf . dm -j- x . a 3 -J'J'y* • 
d[d . dm ' . ^ . a 3 -f- « . 27r . a 3 . U (0 ^ Therefore the 
value of V for a point without the surface of the spheroid, will 
be found by this series, viz. 
V= x | U (o) + ± . U (,) + . u' 2) + &c. } .(s) 
If the attracted point is within the surface, we must operate 
upon the series investigated in No. 8 , of which the general 
term is, 
r* /y »Q ^•dyf.dvt' # 
i—z R 1 ” 2 ’ 
and if we substitute a . (l -f- x .y') for R, and reject the term 
which is evanescent as before, and likewise all the terms which 
are above the first order with regard to * ; it will become 
simply, 
2 7T . xC? 
U (i) : 
with regard to the particular term JT log. R . d\t! . dm ' . Q^ 2 ', 
we have only to substitute for log. R, its value log. a -f- a, . y ; 
and it will become 
x . r* ■ <j £f 'y* . d\f . dm ' . = 2?r . eta * 
also the term iff R ' . d\d . dm 1 . Q (o) will become by substi- 
tution, 
V ’dii • dm 1 . -j- xa x y' . d p/ . dm ' . = 27ra*-{- 2 tt . aa\ : 
these things being observed, the value of V relative to a point 
within the spheroid, will be expressed by this series, viz. 
