40 
ARCHDliACON PRATT ON THE EFFECT OF LOCAL 
In order that the series above may be convergent, 
1 , 1 
^ ^ 2 n + 1 ^ 2 « — 2„_2 
must be less than unity 
^ 111 2/Z J 
must be less than ; tt — - 
2n-l 
2n(2n+ 1) 
7/r 
(2?2 — i)(2n — 2) 
The least value of this is when w=co , when it =1 ; 
m must not be less than h. 
This then is the value we shall give to m ; and the attraction of the whole mass on 
A parallel to x 
=^h|l— ^2 1^3—4 KsH !“(— l)”^K.2n+iH--*)j ; 
or, calling the series S, =:g>H(L — S). 
14. If BCD'E' be a tabular mass lying symmetrically on the opposite sifle of the 
axis of X, the attraction of this mass on A parallel to x will obviously be the same 
as the attraction of the mass BCDE. This, indeed, our formula shows ; for if — Y 
be put for Y it does not change the value of L. The same is not true if we change 
the sign of X ; the reason of which is, that negative powers of <r occur in the inte- 
gration, and these all become infinite as we pass across the axis of y from x—m to 
X— —X. On account of this we must calculate the attraction of the masses all lying 
on one side of y and add them together, and then separately those on the other side 
and add them together, and take the difference of the results. 
15. The tabular mass has been hitherto supposed to be always in contact with the 
axis of X, and at a small distance m from that of y, and is therefore restricted in its 
position. I will now, however, deduce a more general formula. 
Draw ah and a!h' at equal distances parallel to x on opposite sides ; and let Bff=j/. 
Then the attraction of either of the masses aC or a'C on A parallel to x, 
' 
^ X ^ 
■ 
log'. 
m 
— S 
S being independent of the coordinates X, y. ^Subtracting this from the attraction 
of EC, and adding it also, we have, 
Attractions of tabular masses E6 and E6' 
v 
. I 
1 H — ^ + ] 
V 
I , 
1 + Y2"^ ^ 
and ^11 
loge 
X2 
m 
Y2 
-2S 
log. 
