20 
Mr. Ivory on the Method of computing 
nually exterminating dy , and integrating partially with regard 
to f, we shall obtain 
1 • P* • v' • dpt . dm' ____ S?2_ ( r -~py 1 
{ r*-zr e . y +P 2 } Ltl ~ • l rt'fi-i 
i— i 
ry 
d r(°> 
(r-p )*"" 1 d -' — ! 
i — 3 * i— i • i— 3 ' rV 
d 2 r' 
(°) 
0 — p) 
z — i 
f z — 5 
. - ■ dy — &C. j , 
z-i . z — 3 . z— 5 J 
/t -, ’ f / 
This fluent, which, it is to be observed, increases as y increases, 
is to be taken between the limits y = — i and y = i : at the 
first limit y = — i, every term of the fluent is evanescent 
when r—p — o: at the second limit, y = l and f — r — p, 
every term is likewise evanescent except the first, which is 
(r-f - 1 r(°> 
2tT X , X r— 
(r—f) 
when r — p = o : therefore 
•(r— P y— 1 
r(°) 
27 r . — , for all values of r, and even 
iT 
P* • V . dp.' . dm 2 nr (o) • 
z — i 
| r z —zr ( . y-f f 2 j l±I 
observing that we must make y = l in the function r^°\ Now 
the suppositions y = — l and y = i, correspond to pc — — pc 
and f = pc: and therefore if we put u to denote the same 
function of pc that v does of pc'; that is, if v represent what xf 
becomes when pc' = pc; then it will follow, from the nature of 
the transformed value of i/, that u = when y— l, because 
all the other terms are equal to nothing for this value of y : 
therefore finally 
fj 
( r — p) ? 1 . p* . t/ . dpt . dm 1 2 it 
|r 2 — zrp . y+ P 2 | 
*'+» 
z— i 
. u. 
We shall now pass on to the general case when i/ is a 
