2 
2 ny“ 
4 n 2 y 2 4 n 2 y 2 \ . . . v 
' -J7 — J’ or ' initially =g—( i _2« — 2« a -2M 1 _4M 1 - 4 « l ) - 
i — 2n — i2nn 
mp-s x 
v. In the 
next place, calling this fluxion H(K— L — M — N — P — Q) we obtain, for the fourth, H (K — L— M — N 
HN (-&-42-1 
dn 
- P -Q>^ 
4HK 
j dy 
\ ydr 
_ 3 dg | 
zdr / 
-HL (; 
—HP ( 2d y . 
7d2 
H n ( 2 ' 
y 5 d« 
\ ydr 
zdr j 
Q l yd 
r zdr 
n yy \ \ 
... r , v 1 
’ 2 
2 ny 
2 ny \ 
Z 4 / / + ps 1 
1 « 
z 3 
* ) 
\ydr zdr 
4 W 
— HN — \-^L - 
ps \ z 3 
x 2 I y 7 - 2 ny 3 in % y 
m z p s s 3 l z s z* 
6ny 
z 
*a ,4 
7 
ps \ z 
— HP — 1 - 2 . — -± 2 ?- — 
ydr 
Any \ 
Z 
zdr j 
-Q) 
ydr 
± + 
zdr i 
nyy 
mps \zz 
j_HM — (-1 ^21 
I ps\ Z Z 3 
zz 
4 
6ny 
z 
ps 
4 »y 
-HQ — 11 ) — 
ps \ z z 3 z I 
2 jpy‘ t 
4 n z y 3 
zs 
4>i 2 y 3 i ny 3 2n 2 y 4 
Ti TT” Ts 
2 » 3 J 5 
4 » 3 y 4 
ny 3 
2 n 2 y 4 
z 7 
+ z 7 
z 7 
2 n 2 y 3 
4 n 2 y 2 
4 «V\ 
+ 
z 3 
z 7 
z 3 J 
1 2 n 3 y* 
I 2n 3 y 4 
i 2 n 3 y 4 
4 
— i — 
z 8 
Z 6 
+ 6 
z 6 
2 n 3 y 
3v5 2n 3 y 3 
4 n 3 j 4 
4 « 5 J 
z 
H 4 
zi 
2 y 
V X 
Wp 3 s 1 
nn 3 y* 20 n 2 y z 
\ J / _y^ 2 »y a _ zn 2 y 3 
j mp z s \ z 3 z 3 z s 
zny‘ 
zny 
zny 
Wy 7 _ 4^y 3 , 
Z 6 X 4 Z 6 + 
i6n 3 y 3 ^ i6n 3 y 3 
1 2« a yi 2 
- 4 
l6n 3 y 3 ( i6n 3 y 3 
r "4 
}• 
It 
z* z° Z‘ w a" 2” z° Z“ 
will be unnecessary to continue the whole series any further; but it will be satisfactory to obtain 
that part of the sixth term, which is independent of v ; and for this purpose we must take the 
fluxion of the first part with respect to y and z, and then with respect to v ; and that of the second 
twice with respect to v only; and it will be sufficient in this case to employ the initial values of — , 
dr 
dz , dn , . , — v ( i +4«) — v . 1 4 - 2« , ... , , 
— > and which are , — — , and ■ — s ; and calling i4-4« — k, the part required will be 
dr dr ps ps mps r n 
( — ik +5 +6&n — ion +8 kn 7 — 14« 2 +8 kn 7 - — ion 2 + i2kn 7 — 36n 2 4 i2#n 2 — 2 8n 2 — }kn +5 n 
\ m z p*s* 
+ 8kri 1 —ion 2 + 10 kn 3 — i4n 3 4 iokn 3 — ion 3 4 i6An 3 — 3 6n 3 + 1 6kn 3 — 28n 3 — ^kn 4 yn 4 8 kn 7 - — 1411 2 
+ iokn 3 — i8n 3 4 iokn 3 — I4n 3 -J- 16 kn 3 — 44 « 3 4 i6kn 3 — 36n 3 ) — [ — £ + 3 44^1 — 6n 4 6kn z 
—ion 2 4 6 kn 7, — 6 n 2 + 8&n 2 — 28n 2 + 8 kri 1 — 20 n 2 ] 
I -J-2H 
mps 
— s +2 
mp 3 s z 
1+2 n 
mps 
mp 3 s 3 
(2 — 2 n — 2 n 
2 n +8n 2 4 8n 2 — 4M 2 4 i2n 3 4 12 n 3 4 I2n 3 4 i2n 3 — 20 n a 4 i6n 3 4 i6n 3 — i2n 2 4 i6n 3 4 i6n 3 ) — 
(3 — 6n — 56H 2 4 i28n 3 4 4i6n 4 ) rnr [2 — 6n — 2on 2 4ii2n 3 ] 
( m 2 /> 4 s 4 a t t ~r / mp 3 s z 
— s| — ^ — ,(2—6 n — 2on 2 4ii2n 3 )4 
j mp 3 s 3 
142 n 
mps 
— s) 42 
1 4 2 n 
mps 
1 /i 4 2n s \ 1 
6, The whole equation becomes therefore ultimately ps — vr 4 j ~ 2m p' s ~ j n 2 4 - 
( 
— in — linn 
6 mp z s 1 
1 — i6n 2 — 24n 3 i-in—imn 2 — 6n—20n z +ii2n 3 
i^m z p 3 s 3 24 mp x s 24 mp*s l 
2— 6 ra — 2on 2 4U2n 3 l fi 4 2B \ , f 3 4 zn 
-'I +2 l 
yzomp 3 s 
mps 
mps 
v i U., ! ( p— 6 n-s 6 n 2 4 i 28 n 3 44 i 6 n 4 
j \ L 720 m z p*s* 
2 — 6n — 2on 2 4 3 12 n 3 
720 mp 3 s 3 
72 om 2 /) 4 s 4 
4 . . . J r 6 4 . . . We also 
obtain, for finding, on this hypothesis, the height x, corresponding to the pressure y and the density z, the 
expression mx — m — 1 — — 4 hi \ » y ^ e3n S — “7 ZZZ 5 anc ^ ^ — 1 + 4 n2 * ^ ut 
r z q 2 zz — </y(i — z) z+n — nzz 
the utility of the Taylorian theorem, thus applied, in obtaining a series, is not confined to Professor 
Leslie’s hypothesis: it is equally well adapted to that of Laplace, or to any other admissible supposition 
respecting the distribution of temperatures : and we may therefore employ it in examining the comparative 
accuracy of the results of these different hypotheses. 
