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Pierre Simon Laplace.

. (page 84 of 114)

2 11

* (4162) The term â– - [9324, 9325], is equivalent to, and used instead of, T + r;

[9383n] ; b and h' being, as in [9383A] , the greatest and the least radii of curvature ; so that

the action of the fluid between the tube and cylinder is, as in [9315&], K â€” ^^'iz "f" z) >
or, according to the present notation [9381],

[94036 J K â€” ^H.-, or -K"â€” -,

[9403c] as in [9403,9404]. Substituting the value of ^ [9401], it becomes as in [9405]. We

may remark that the canal spoken of in [9402, &c.] is similar to that which is explained by
fig. 112, page 695, to which we may refer for illustration.

X. Suppl. 1. Â«Â§.7.] ELEVATION OF A FLUID BETWEEN TWO TUBES.

737

If we put q' for the elevation of the fluid in the interior branch of the canal,
above the level of the fluid in the vessel, and add gq' to the preceding action,
the sum will be in equilibrium with the action K oi the fluid in the vessel upon
the canal ; therefore we shall have *

which gives

JH" sinJ'
J'l^l'

[Elevation q' of a fluid between
two cylindrical surfaces, whose

]

We have found, in [9379], that the elevation of a fluid above its level, in a
tube whose radius is l' â€” /, is equal to this value of q' ; therefore the fluid
ascends in the capillary space to the same height as in a tube whose radius is
equal to the width of that space.

If the surface of the fluid be convex, the preceding expression of q' [9409]
will denote the depression of the fluid below the level, and the fluid will then sink
in the capillary space, in the same manner as in a tube whose radius is equal to
the width of that space.-f

[9406]
[9407]

[9408]

[9409]

Elevation
of a fluid
between
two cylin-
drical sur-
faces.

[9410]

[9411]

* (4163) The calculation in [9408] is made in the same manner as for a tube in
[9355a, &.C.], and requires no particular explanation. From this we easily deduce the value [9408a]
of q' [9409].

t (4164) This will be evident by making the calculation in like manner as in [9372 â€” 9409J,
the effect being merely to change the signs of q',6', as in [9352d, he], which does not alter
the value of q' [9409]. Therefore the elevation q' of the fluid when the surface is concave,
or its depression when the surface is convex, is given very nearly by the formula [9409],
being, as in [9410], the same as the elevation or depression in a tube ivhose radius is equal to
the width of the capillary space M'N', fig. 119, page 732. In all the calculations of <Â§> 6, 7
[9380 â€” 9412], it is supposed that the tube and cylinder are of the same substance, or that
the angle 6' [9346] is the same for both of them ; so that the fluid is either elevated by the
combined action of the tube and cylinder, or is depressed by the combined action of both of
them. In this case, if the distance M'JS' or EG be small in comparison with a, the section
of the surface M'ON' may be supposed to coincide very nearly with the circle of curvature
at the point O, whose radius is b, as in the similar case [9336^] relative to the surface in
a capillary tube, as is evident from the equation [9401]. For if we put for brevity
V â€” Z=2X, and substitute b = 26 [9383/?] in [9401], we shall get, by a slight reduction,
6.sin.^'=X; which gives the same expression of X as would be found if we were to
suppose M'ON' to be a circular arc, described with the radius C'0 = b, making the
angle OC'JS'= OC'M'=6', and its sine EF=FG = -k.

Instead of supposing the angle 6' to be the same at the points JS' and M', we shall now

VOL. IV. 185

[9410a]
[94106]

[9410c]

[9410rf]
[9410e]

[9410/]
[9410g-]

[9il0h]
[94l0t"]

738 THEORY OF CAPILLARY ATTRACTION. [Mec. C61.

If we suppose the radii of the tube and cylinder to he infinite, ice shall obtain

the result of the capillary action upon a fluid between two vertical and parallel

[9412] planes situated very near to each other; therefore the preceding theorem holds

good in this case. We shall, however, investigate this theorem, in the following

article, by a particular analysis.

8. We shall suppose, in fig. 116, page 713, that AOB is the section of the
surface of the fluid included between the two planes, by a vertical plane drawn

[9410A] take 6' for its value at the point N', and ^^ for the corresponding angle at the point M';

putting also EF=u', FG = u^, m'-}-w^=2X, and considering M'ON' as a circular arc

whose radius is h, as in [941 Og-]. Then we shall have, in the same hypothesis relative to

the nature of the fluids, as in [9410e],
19410m] u'=h.s\n.&', M^=6.sin.5^, m' + m^=2X;

so that, if 2X, 5', &^, are given, we may determine the values of h, u', u^. If we suppose
[9410n] ^'=^^=90"^, the equations [941 Om] will give u'=.u^=b=\, which correspond to the

actual form of the figure 1 1 9, where the extreme points M', N', appear to be upon the
[9410o] horizontal line M'C N', passing through the centre C of the circular arc. We may,

finally, remark that the equations [9410m] are similar to those which are given by M. Poisson,

in page 114 of his Nouvelle Theorie, S^c.
rQiiOiT These calculations are limited to the case where the fluid is acted upon by both

surfaces in the same manner; but it may happen, when the tube and cylinder are of
[9410^] different substances, that the fluid may be elevated near one of the surfaces, and depressed

near the other, so that there may be a point of inflexion of the surface, as is observed between

two parallel planes in [10158, &c.]. In the case of two parallel planes, the surface
[9410*] may be concave in one part NQO, fig. 112, page 695, and convex in the other part OM',

the point of inflexion being O, which, we shall soon see, is on a level with the fluid in the

vessel AVB\ the point O falling in H', the fluid being elevated above this level in the
[9410* ] part ON, and depressed below the level in the part OM'. For the capillary action, at

any point O of the concave surface NOO, will be represented, as in [9294, 9294'J, by

[9410u] K â€” ^H.i- + - ) ; y and h being the greatest and the least radii of curvature corresponding

to that point; and we shall evidently have, as in [9383o], &'=od ; so that this force becomes

H ff

K â€” â€” â€¢, instead of K â€” 7, which is used in [9355a] ; and if we make the calculation as
[9410i>] , ^ *

in [9355a â€” c], we must change h into 26, to conform to this alteration in the expression of

the capillary action. By this means the expression of q [9354], or 5^' in the present

HI H

[9410tÂ«] notation, becomes Â§-'=â€”. â€” ; and by putting, as in [9323o], -â€”:=a^, we shall get

[9410z] j'==â€” â– . This value of q' expresses the elevation QO^ of the fluid in the concave part
iVO of the capillary surface, above the level A VB of the fluid in the vessel; it becomes

X. Suppl. 1. ^8.] ELEVATION OF A FLUID BETWEEN TWO PLANES.

739

perpendicular to these two planes ; if we put OM==z, NM=y, z will be
a function of y only. Moreover, h and h' being the greatest and the least
radii of curvature of the surface of the fluid at the lowest point O [9316],
h will he infinite^ and h' the radius of curvature of the curve AOB, at the
point O. Hence we shall have, in the equations of partial differentials *
[9312, 9313],

dz\ â€ž _ /ddz\ 1

P = (l) = o.

q =

dy.

r = 0, s = 0,

t =

dy-

[9413']

Elevation
of a fluid
between
two paral-
lel planes.

[9414]

= 0; [9415]

negative in the convex part OM' [9411, &c.] ; and we may in both cases use the same
expression of q'=â€” [9410a;] ; considering the radius of curvature b to be positive in the

[94%]

[9410z]

concave part of the surface NO, and negative in the convex part OM'. If we suppose the
radius of curvature b, corresponding to the point O, to become infinite, we shall' have from
[9410x'] q'=0, or HO = 0; so that the point O will then fall to the level AVB of [9411a]
the fluid in the vessel; and O will be a point of inflexion, because it is the part which [94116]
separates the positive from the negative values of the radii of curvature; or, in other words, it
is the part which separates the concave part NO from the convex part OM' \ this agrees [9411c]
with what we have stated above in [9410s, &c.].

* (4165) As z does not contain x [9413'], the values of p, r, s [9312, 9313], must
vanish, and those of q, t [9312, 9313], become q = (f). ^=(^2)* Substituting [9415o]

these and 4 = 0, also Â§==Â«â€¢ [9328], in [9318], we get [9416]. Multiplying this by

-_-, ^

dz, and taking its integral, we obtain [9417]. At the point O, where z = 0, we also have

â€” = 0, because the tangent at O is parallel to the horizon; substituting these in [9417],

we get â€” 1 = constant; and by substituting this last expression in [9417], we obtain

1

1/

â– OU'

r-1'

[94156]

[9415c;

which, by changing the signs of the terms, and making a slight reduction, becomes as in [9419].

We may here remark, that the author supposes, in this article, that the angle 6' is the same

for both planes, so that the fluid is either elevated near both planes or depressed near both. [9415i]

The case where the fluid is elevated near one of the planes, and depressed near the other, is

treatedof in [10158â€” 10257].

The author simplifies the equation [9416], by taking, as in [9323m], the origin of ar at
the level surface of the fluid in the vase, instead of at the point O, fig. 116, page 713 ; or.

in other words, by changing z into Â« â€” ^/j ^or ^7 ^^^^ means the differential equation
[9416] changes into the following, which is the same as [10164] ;

[9415e]

740 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

[9415'] hence the equation [9318] becomes

ddz

Multiplying this by o?, and putting aa2=l, as in [9323/?], it becomes

â€ž ddz

[9415Â£-] . ^^^ = 2;r,

being the same as is used by M. Poisson, in page 174 of his Nouvelle Theories ^c.
[9415A] Multiplying [9415^] by â€” dz, integrating, and adding the constant quantity b, we get

[9415t]

^ ^ df

2

b â€” z2.

Putting h for the elevation of the point O above the level of the fluid in the vase, we shall

have, at this point, whei
this in [9415i], we get

[9415&1 dz

have, at this point, where 7' = 0, o?=h â€” W [9415i], or h = 0L^-\-h^\ substituting

Â«2 4- A2 22

[94i5n \/'r7^^~

Deducing the value of dy from this equation, we obtain

[9415m] ^i/ = /(22_;,2).(fc2_|_2a2_22)'

which is of the same nature as the equation [10172], and is easily integrated by means of
esiiptical functions. It is rather remarkable that these functions are not mentioned by La Place
throughout his whole work, which can be accounted for upon no other principle than his
dislike to Le Gendre, the great promoter and improver of this calculus ; since there are many
parts of the Mecanique Celeste where it would have been very advantageous to have used
this method of integration. The process of reduction to elliptical integrals, in the present
case, is easily obtained from what Le Gendre has published on the subject, and we shall here
give the calculation, with all the necessary details.

We shall assume for z^ an expression of the form [94l5p], which, by dividing the
[9415o] numerator and denominator by cos.^9, or multiplying by 1 -f- tang.^ 9, gives [9415^];
whence we easily deduce the value of tang.^cp [9415r].

- (/i2-|-2a2)./i2

[9415n]

[9415n']

/i2-|-2a2.cos.2(p

( A2 J^ 2a2) . ^2 . ( 1 _j_ tang.2^)

[94]5/,l i

[9415ol

^ (;i2-]-2Â«9) + /i2.tang2(p

VQAyK ^ ' o (/i2 + 2a2).{z9 â€” ;i2)

[9415r ] tang.2 m = ^â€”^ '-^ ' .

X. Suppl. 1. <^8.J ELEVATION OF A FLUID BETWEEN TWO PLANES.

741

ddz
If

(

^'-v

[9416]

The limits of the value of the first member of [9415?] being a^ and 0, it follows from the
second member of the same equation, that the limits of z^ must be h^ and o?-\-h^) so that
the limits of z^ â€” h^ must be and a^ ; therefore the value of tang.2(p [94l5r] must
be always positive, and <p a real angle. Now substituting the value of z"^ [9415p], in the
first member of [9415m], we get, by a slight reduction, its second member; and by using
this in [9415v], it is reduced to the form [94151^)]. The square root of the product of the two
expressions [9415m, z^] gives [9415a:], using for brevity the symbol w = \/h^-\-2a^ . cos.2(p.

2a2A2 _ 2a2A2 . cos?q) 2tt^h^ . sin 2 (p

h^ =

A2 4- 2ofi â€” z^=: 2a2 â€” (z2 â€” h^) = 2a2

Â«;2

2a2;i2 . ain.2 (p 2a2 . (^2 _}_ 2a2 . cos.2 (p) â€” 2a^h^ . sin.a cp

v^

2a2A2 . (1 - sin.2 (p) -|- jgi . co3.2 cp 2a2.(fe2-|-2a2).cos.8(p ^

m;2 Â«,2 '

, 2a2A.(A2_|_2a2)i.sin.(p.cos.(p

^(z2_/,2).(/i24.2a2_z2)= J -^â€¢

Substituting w [9415f] in [94l5p], and taking the square root, we get -,

whose differential is

and as the differential of

U)2 = ^2 _j_ 2a2 . cos.^ (p

[9415^] is

2wdw = â€” 4a2.d(p.sio.<p.cos.(p,

we shall have

dz = (A^ + 2a2)* . w~^ . 2a2A . dcp . sin.(p . cos.(p.

Dividing this by [9415a?], we obtain

dz Â«^ df

y/(2a_;i2) . (/i2-j_2a2â€” 22) w v/^2_[_2a2. cos.2 (p Â»

multiplying this by a2 + A2 _ ^2 ^ â€ž2 ^ ^2 _ LJL_:i^ [941 Sp, ^], we get the value of

dy [9415?n], under the following form ;

{a^-\-lfi).d(p

^y=]

(fc2_|_2a2).fe2.rf,p

â– (/l2-[-2a2.C09.2(p)i (A2_|_2a2.COS.2g.)2 *

To reduce the radical v/A^4-2a2. cos.2 cp to the usual form of the elliptical functions

2a2

[9415Â«]

[9415f ]
[9415u]

[9415u]

[9415to]

[9415a:]

[94%]

[94152]

[9416a]

V/lâ€” c2.sin.2(p' [891 Oi, fee], we may put cos.2(p = l â€” sin.^cp, and (^==j^T^sl then [94166]
from this value of c^, we easily deduce

1 â€” c^ =

s

h"^

/i2 + 2a2

VOL. IV.

2â€” C2:

186

2.(a2+ft2)
^2_^2a2 '

A2=

2a2.(lâ€” c2)

[9416c]

742 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

[9416'] Multipljing this by dzy and integrating, we shall get

[9416.] â€ž,4.i. = fJ?=3; AÂ» + 2aÂ» = ?|\

Now substituting cos.2(p = l â€” sin.^cp in [9416a], and using the values [9il6b, c,d], we

erQt

'â–  ^ ^ C.v/2 Vlâ€” c2.sin.2(p c.\/2 *(1 â€” c2.8in.29)i"

Bin-(p . co9.(p

[9416/'] If we put for a moment, for brevity, A=^i_c2.sin.2(p, and X= , we shall

have, by taking the differential of X, and dividing by dcpy then putting cos.2(p=:lâ€” sin.2(p,
/dX\ cos.2<p â€” sin.2(p c'^.sin.^tp.cos.^cp 1 â€” 2a'm.^q) c2.8in.2<p â€” c^.einAq)

mm (f)='

A3 A ' A3

[9416fc]

1 c2

adding to both sides of this equation, and making successive reductions, we get

'rfX\ 1â€” c2 1â€” 2sin.2g) (1 â€” c4.sin.4<p)â€” c2(lâ€” c2.sin.2<p)

rfy/ ' c2.A3 A ' c2.A3

1â€” Ssin.gy (1 â€” c2.8in.2(p).(l-|-c2.8in2y â€” c2)
A ' c2,A3

1â€” 2sin.2(p (14-c2.sin.2<pâ€” c2) c2.(lâ€” 2sin2(p)-f-(l-|-c2.sin.29â€” c^)
+

A ' c2.A c2.A

1â€” c2.sin.2<p A2 A

[9416Â»] c2.A "~c2.A 7^'

Multiplying this by â€” â€” -jrâ€” , and transposing the first term, we obtain

roiiMi 2(1â€” c2).a.rfqp 2a , . , /^ j^

[9416*1 - CV2-A3 =-'^^'d<P'^ + '^cV2.dX;

substituting this in [9416fi], it becomes, by resubstituting the values of A, X [9416/*],

roiiiftn J (2 â€” c^)-a dcp 2a , . , /- -, f sin.cp . cos.<p \

^ C.v/2 V^lâ€” c2.Bin.(p c-v/a ^ ^ ^ ' *^ Vv/lâ€” c^-sm-W

Integrating this expression, and using the elliptical symbols [8910A:, ?], we get

[9416m]

^Â°'^"-'-(->^)-?^-^(-^)+";f-tr;^

no constant quantity is added, because we have y = 0, at the point O, where 2r = A
[9415fc], which corresponds to <p = in [9415r] ; multiplying the value of y by iâ€” ,
and making a slight reduction, we finally obtain
[9416., . ?^ = 2^).F(câ€ž)-?.E(..)+?^^^^.

which is the same as the equation (3) of M. Poisson in page 177 of his Nouvelle Theorie^
fyc, changing x into y, to conform to his notation.

X. Suppl. 1. Â§8.] ELEVATION OF A FLUID BETWEEN TWO PLANES. 743

I

^z^ = r,-\- constant. [^^17]

Substituting cos.2(p = l â€” sin.2(p in [9415p], then dividing the numerator and
denominator by As _j_ 2â€ž2^ ^^ get, by using c2, h^, h^ + Sa^ [94166, c],

2a2.(l_c2)

[94160 J

c2.(l â€” c2.sin.2<p)'

which is the same as the equation (4) page 177 of M. Poisson's work. Hence we see that
the values of the coordinates y, z, may be ascertained in functions of the angle 9, by means
of the formulas [9416m, 0]. From these expressions we may trace the form of the curve
surface, and the relation of the coordinates. If we suppose, as in [9346/], that zs is the [9416p]
acute angle formed by the lower part of the plane and a tangent to a point of the section
placed at the limit of the sphere of sensible activity of the first plane, we shall have, by the
usual differential formulas,

dy 1

sm.w =

[94%]

and if we suppose the value of z near the fixed plane to be z = q, we shall have, by
substitution in [9415/], a^.s\nM = a^ -\- h^ â€” q^; hence we easily deduce the expression [9416r]
of q^ [9416Â«], and by substituting the value of h^ [9416c], we get [9416<]

q^=:h^ -\-a^. {I â€” sin.zi); [9416Â«]

92 = ^.(2 â€” c2 â€” c2. sin.-n), ^ [9416i]

The first of these values of q^ gives

^2 â€” Aa = aÂ«.(l â€” sin.-n); h^-{-2a.^ â€” q^ â€” a^.{l-\-s\n.'a). [9416f]

Now if we suppose to be the value of 9, corresponding to z = q, we shall have, by
substituting these values in [9415r], and reducing by means of â€” â€” â€” = â€ž [9416c, rf], [9416u]

. Q^ (;i2_|.2a2).a2.{l â€” sin.-Â») â€ž ., . _

tanK.2 0= â€” â€” -^ [9416v]

1 â€” 8m.-si

_ â€ž . . [9416wl

(1â€” c2).(l+8ill.TO) "^ ^

â€¢If we suppose that y==^, corresponds to z==q, and to <p = 0, near the first plane, [94l6x]
we shall have, by substitution in [941 6n],

a .t/2 2 c2 _, , . 2 _ , . , 2c.8in.0.cos.0

If we denote by o- and -a' the distance and angle relative to the second plane,
corresponding respectively to a, -a, for the first plane; we shall get the following equation^ [9416z]
which is similar to [9416y], and may be deduced from it^ by changing a^ into a^, e into e\
and w into w'j -

744 THEORY OF CAPILLARY ATTRACTION. [Mcc Cel.

dz
[9418] At the point O, â€” = ; therefore constant = â€” 1 ; consequently

[9419] 7=^^ + ^^ = -6^*

[94176]

[9417/]

[9417t ]

[9417f ]

a'v/2 2 â€” c2 -r, / ,^ 2 _. ... 2c.sin.0'.cos.0'
'â– â– ' a c \Â»/cV'>'' ^1â€” c2.sm20'

Adding these two equations, and putting o-^-\-^^-='2l for the distance of the two planes, as
in [9443e, 10236], we get

+

c
2c.sin.0.cos.0 . 2 c. sin. 0'. cos.Â©'

\/l â€” c2.sin.20 ^ v/lâ€” c^-sin.20' *

Hence it appears that, if c, is, w', are given, we may deduce Â©, Â©', from the value of tang.^Â©
[94l6tr], and the similar value of tang.20' [9416z] ; substituting these in [94176], we

'â€¢ ^ obtain the distance of the two planes, 2Z. If this distance be given instead of c, we may
deduce, by an inverse process, the value of c from the same three equations ; but this is

[9417c] a much more laborious process than the preceding one, where c is given; and it will be
necessary to form a little table, giving the expressions of 2Z for values of c, increasing by
small differences, which is easily done by means of Le Gendre's elliptical tables ; and by
entering this table, we may find by inspection the value of c, corresponding to any proposed

[9417g-] value of 2/} a being considered as a given quantity. We may remark that the equations
[9416/, w, y, 94176], are equivalent to the equations (5), (6), (7), (8), in pages 177, 178,

r9417W Â°^ ^' Po'sso'^'s Nouvelh Theorie, Sfc. The equation [9416w] being subtracted from
[9416y], gives, by putting a^â€” y = y.

2 c . sin.0 . COS.0 2c.sin.()D.cos.(jo

y being the distance from the first plane to the point of the surface corresponding to the
[9417A:] angle 9, and being the value of 9 at the first plane. In like manner, if y' be the distance
from the second plane to the point of the surface corresponding to the angle 9, and Â©' be
the value of <p, at the second plane, we shall have, from [9417i], by changing y into /,

2 c . sin.Â©' . COS.Â©' 2 c . 8in.<3D . co3.q)
"^ v/l â€” c2sin.^ v/^â€” c2.sin.2g) *

When the second plane is at an infinite distance from the first, the equation of the surface
rQ41'~ 1 [9'^^^*] comes under the case which is treated of in [9435], and integrated by means of
logarithms in [10208]; the elliptical functions which occur in [9417iJ being, in this case,
reduced to common logarithms, as is seen in [10208].

X. Suppl. 1. <Â§>8.] ELEVATION OF A FLUID BETWEEN TWO PLANES. 745

Now putting

Z= â€”r, az^, [9420]

we shall have *

, Zdz

^y = ^i^rZ2' [9421]

and this is the equation of the elastic curve, as ought to be the case, since, as in

the elastic curve, the force which depends upon the curvature is inversely [9422]

proportional to the radius of curvature.f At A^ fig. 116, page 713, the most

dz
elevated point of the curve ANO, we havet j- = tang.^', 6' being, as in [9423]

ay

* (4166) Transposing the term ouz^ [9419], and substituting [9420], we get

squaring and reducing, we get the value of dij [9421].

f (4167) Substituting [9326] in [9416], after changing m into y, we get - â€” 2(\.z==-;
and if we put for brevity ^ = ^~T/} it "^^y b^ put successively under the forms [9423a]

and, as the curvature at any point is inversely as the radius of curvature, it is evident that
the curvature at any point iV, fig. 116, page 713, will be directly as the quantity e-\-z, that
is, directly as the absciss z, augmented by the constant quantity e, being the principle upon
which the properties of the elastic curve are founded in [9422].

[9423i]

% (4168) This is similar to [9383/], changing u into y, and taking rfy positive, because
y and 2; increase together. Substituting this value of dz = dy. tBng.6' in [9421], dividing
by (?y, and reducing, we get 1 â€” Z2 = Z2.tang.2^', or Z2.(l-|-tang.2d') =1; whence ^^^^^"^
Z=cos.d', as in [9425]. Substituting this in [9420], we get [9426], and then dividing [94256]

11 2

by â€” a, we obtain z^-{-â€”,.z = - .{I â€” cos.d') =:-.sin.2^a'. From this quadratic [9425c]

equation we get z [9427], and when 6'= 00, it becomes as in [9429]. Now from [9430] [9425rf]

we have a = fr= = = :, ; hence we get i/2a=-7^ . , . [94^5e j

we nave ^ h iq iq Iq.t^ng.h^' ^ ^ ^lq.tang.hd'

Substituting this in [9429], it becomes as in [9431]; and if in this we put Z=1""S

5 = 6'"S784, d'=90^ or tang.J^'=l, we obtain ^r = /6^V84 = 2""-,6046, as [9425g-]

in [9433].

Substituting 0. = â€” [9323p] in [9429], it becomes Â« = a. v/2. sin. Jd', which is the same [9425^]

ft
VOL. IV. 187

746 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

[9346], the complement of the angle ivhich the extreme side of the curve makes
tvith the plane ; therefore we shall have, at this point,
[9425] Z=cos.^' [94256],

which gives, to determine the extreme value of z [9420, 9425],

[9426] â€”Y-, a.z^=COS.d\ or

[9429]

from each

[9434]

1 >/2sin 2 14' 1

[9427] z = -^, + ^_-^ + ^-^. [BÂ«..Â«â„¢,Â«..r.,

[9428] If the two planes are at an infinite distance from each other, b' will be infinite,
and we shall have

2 sin. A 5'

When the ^ '2" rExtreme value of z, when the"l

parallel ' ^ /^ * Ldistance of the planes is infinite. J

planes are V '*'*

at a very â€ž

groat dis- -y^g \idi\Q "5 = ^L [9328] ; moreover, in a capillary tube whose semi-diameter

H

other. 0. sin. 4'

[9430] is /, we have â– ^=â€” â€” [9379], q being the height to lohich the fluid in
the tube rises above the level [9378] ; therefore we have

rntnii -T 1 /Â«7 .â€žâ€žâ€ž 1 a' T Extreme value of i, when the T

[9431] -^ Y qi.iang.^O . LÂ«listanceoftheplanes is infinite. J

If we suppose 6' to be equal to a right angle, which appears to be the case for ivater
[9432] ^and alcohol] relative to glass, I being a millimetre, we shall have q = 6'"',784
[9364], which gives for the height to which the water is elevated, by a glass plane
dipped vertically into a vessel filled with that fluid, 2^,6046 [9425^]. This
ought to be rather less by experiment, because the point which we take for the
[9433] origin of the curve can become sensible only by being at a distance from the
sides of the tube ; it must therefore be a little below the point A, fig. 116, page
713. We may observe that, by the extreme point A, we always mean the point
nearest to the tube, but situated without its sphere of sensible activity; and as this