78 
Mr. Ivory on the Attractions of an 
'du ! . Avt / 5 /» d/j! . du 
jr— =yj 7 
let 
1 —e 
k' z —k % . 1 -/ 
je+(i_e). . |/+(i-/).^' a j 
x /l ; and 
between the limits x = o and x—i: 
F =/— - 
v'c X + X x AT a ) . ( I — (— W) 
then observing that the preceding integrals increase as much 
from [j/— — 1 to <x' = o, as they do from [x' = o to ix' = 1 ; 
and likewise that the limits of u are from u = o to u = 27 t ; 
we shall get 
fi 2, p pdv! . dz; 2-7 t . k 1 p 
2 d S \/ g J 
It remains to find the value of 
tJX 1 o S- s.d^'.dv'.Q (2 \ 
Taking the value of Q (z) in terms of y (No. 3), we have 
= r*. (f 7 2 — ■§) : let r, denote the co-ordinates of 
the attracted point ; then a = r . [x ; b — r.*S 1 — [x 1 . cos. «r; 
c — r.s/ 1— [x* . sin. m ; therefore 
r .y=za .\k -\-b .*/ 1 — jx' z . cos. ■&' -j- c . \/ 1 — fx' 2 . sin. 
consequently 
r» . Q (,) = a- ( i vT- i) + b- . { f ( i - ^ ) cos. V - ' } 
+ t*. {f(i~V) sin.V— 
f \/ 1 — - (x /z . cos. 2/ 3^r . fx r */ 1 — ^x /z . sin. m' 
-f- 3 be . (1 — [x' 1 ) . cos. w J sin. m r : 
but log. s may be reduced into a series of this form, viz. 
-f A (l) . cos. 4- A (2) . cos. &c. 
and we may neglect all such parts of as multiplied by 
this series would produce only quantities containing sines and 
