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Pierre Simon Laplace.

Mécanique céleste (Volume 4)

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[9964] a = 5^^.sinJ,.cos.w , ,

[9964'] which gives, by supposing 6^ and w^ constant,

[9965] da = ds^^ . sin.^^ . cos.w ;

[9965'] the preceding double integral, being multiplied by this expression of da, and
then integrated relatively to s^^ , will therefore become

/./•/•I I ■, . n , /\i /\i fAttraclion of the parallelopiped*!

[9966] flfdS,, . d^, . d^, . sin J, . cos. CJ, As,. "i'CSJ 4- T(sJ . I BFE upon the plane QFC in the

*■ •• *'*'•' " ' ' ' I ( II \ 11/ t \ 11/ ) Lvertical direction FE. J

[9964a]
[99646]

[9964c]
[9964rf]

[9964c]
[9964/]

[9964g-]

II X^t^.:tJA

* (42S9) Having found, in [9962], the expression of the attraction of the solid plane
J5F^ upon the vertical line CD of an infinite length, we shall now suppose that, on the
continuation of the line FC, fig. 148, we have Cc = da, and that
the infinite line cd is drawn parallel to CD. Then it is evident
that the attraction of the solid plane BFE upon the plane surface
DCcd, will be found by multiplying the function [99606] by da;
and if we integrate this, relative to a, from a = to a = a, we
shall evidently get for the whole action of the solid BFE upon
the plane QFCD, the following expression;

fffda.dd^.dvs^.cos.zj,.{s,^.1r(s,^)-j-r{s^^)l ;

s,, being the distance from the point C to the point S', where the line CS' first meets the
nearest surface of the solid plane [9956] ; and since the differentiation or integration relative
to a is considered as independent of the quantities ^^, « , so that they are considered as
constant [9964'], we must find ds^^ upon the same principle. Now this is done by supposing
the line cs' to be drawn parallel to CS', to meet the nearest surface of the solid plane in «';
since, by [9950, 9949'], 6,, zr^, will be the same for the lines CS',cs'; and if we put
CS'=s^^, we shall have cs'=is^^-{-ds^^. Moreover it is evident, as C F, FS, SS\ are the
rectangular coordinates of the point S', corresponding to x, z, y [9938', &£c.], the value FC=a
will be found by writing s^^ for s^ in the expression of x [9950^, so that by this means it
becomes as in [9964]. Its differential, supposing a, 5^, to be the only variable quantities, is
<ia = <Z«,,.sin.d,.cos.-«, [9965]; hence [99646] becomes as In [9966].

X. Suppl. 2.] ACTION OF A PARALLELOPIPED ON A PLANE. 845

The integral must here be taken from s^, = 0^^ to 5^^ = co. Now in this case * [9967]
fo\ds^r^(sJ==^s,,,r(s^^)+f-ds,^.T(s,;)==f-ds,.r(sJ, [9968]

because, 5^^ being infinite, s^^.rfsj is nothing [9240A;, &c.] ; moreover we have,

as in [9253a'], 2f,\ds^,.Y(sJ==?; hence [^68']

It

therefore the preceding triple integral [9966], will become

H /v* » 7 • o ^Attraction of the parallelopiped-i

— ./fd^ . ad . Sm.^ .cos. «,. BFE upon the plane QFC in [9970]

?r *'*' III L.the vertical direction FIE.. J

The integral relative to w^ must be taken from w^ = to w = to a right
angle ; the integral relative to ^^ must be taken from 5^ = to 5, = to two
right angles, which gives f

* (4290) From [9960'] we have 7^' <^«//. *(«,,) = c"—r(*^,), whose differential is
ds^^.ir(Sff) = — d.r(s^^). Multiplying this by s^, and integrating, we get

as is, easily proved by differentiation. This agrees with [9968], the limits of the integrals
being from s^^s=0 to s^,= oo [9967]; and at both these limits we have s^,.r{s^)=iO
[9240A:, &c.] ; hence /o%«?«/,.*(5j==/o'"<?s,,.r(sJ [9968]. Substituting this value of
f^dsj^.r(s^^), in the first member of [9969], we get

but by [9253a'], we have 2/o"5^^(Zs^^.*(5j=- , as in [9968']; hence the preceding [9967(fl
integral [9967c] becomes as in [9969], and by substituting it in [9966], we get [9970],

f (4291) We have fd6^.sm.6^=l — cos.iJ,, which vanishes when ^^ = 0; and at the

second limit, where fl^=180''=«- [9971], it becomes f^d6^.sm.6^==:i2; hence the first

member of [9972] becomes /cZ«^.2cos.2^^=/«Zzrf,.(l + cos.2s<,) = w,4- Jsin.2itf^. This [99726]

vanishes when ot^ = 0; and at the second limit, where w^ = ^t, it becomes equal to ^t,

as in the second member of [9972]. Substituting this in [9970], it becomes |fl, as in [9972c]

[9973]. In this calculation the attracting body and the attracting plane are considered as

being of the uniform density unity ; and as the symbol H is sometimes used instead of H' [9972rf]

[10032, &c.], we shall, to avoid confusion, suppose that, when the fluid is homogeneous, we

shall have

hH=:q, or H=2q; ' [9972e]

and in this case, we get

The vertical action of the homogeneous rectangular solid BFE upon the rectangular plane QFCD=q. [9972/*]

VOL. IV. 212

[99676]

^46 THEORY OF CAPILLARY ATTRACTION. [Mec. C61.

[9972] f^ft^^, ' d^, • sin.4, . cos.^x^, = ^* ;

therefore we shall have
[9973] ^jff=the whole vertical attraction of a solid parallelopiped upon a plane surface.

This attraction is what we have before called p, or p', if the plane is of the
^ ^ same nature as the [homogeneous] fluid ; therefore we shall have *

[9973a]

[9973/]

* (4292) The expression ^H=q [997 2e,/], represents the action of the rectangular solid
BFE, fig. 148, page 844, upon the rectangular plane QFCD, extended infinitely in the
directions FC, FQ; and if we suppose its thickness, in the direction perpendicular to the

r9973 1 P'^"^ Q,Fd^) to be dc, the whole action of the solid upon the plane will be ^H.dc; and its
integral ^H. c represents the whole action of the solid upon the plane whose thickness is c,

r997S//l *^'" action being in the vertical direction FE [9940'J. This is the quantity which is called
Q' or p'.c [9915, 9929]; hence we have p'.c = JjH'.c. Dividing bye, we get p'=|fl,
as in [9974].

Pggy„ , Wc havc supposcd, in the formula [9973], that the dimensions of the attracting solid, in the
directions FB, FE, fig. 148, page 844, are infinite ; and in like manner that the dimensions
of the attracted plane, in the directions FQ^, FC, are infinite ; but it is evident that we may
decrease these quantities very much, without affecting that formula, and that we can suppose

[9973/"] tiie limits, in the directions FB, FC, to be equal to the insensible quantity X [9173n],
which expresses the distance to which the corpuscular action extends ; so that, in the formula
[9973], we may suppose BF=FC = \; FC being on the continuation of BF, and the
angle BFE= QFC=:90''-. What we have said relative to the limits of the attracting solid,
in the direction FB, holds good relative to the direction FE, and also in the direction
perpendicular to the plane of the figure, either above or below it ; so that we may consider

[997%] i\^Q attracting solid as being composed of two cubes, whose sides are represented by X,
supposing the base to be included in the angle BFE; one of the cubes being above the plane
of the figure, the other below it. We shall hereafter have occasion to ascertain the action of
an attracting solid in the form of a wedge upon a plane, or triangle, &c, ; we shall, therefore,
in this note, investigate several formulas of this kind.

Instead of supposing the line FC to be on the continuation of the line BF, as in the
preceding calculation, fig. 148, page 844, we shall now suppose these

[9973fc] lines to coincide, as in fig. 149 ; then, taking in the plane of the figure the
horizontal line BF^:^^, we shall consider the vertical lines EFQ, GB H,
as being drawn in that plane perpendicular to the line BF, and continued
infinitely, on both sides of it. We shall also suppose that the attracting
solid is bounded by planes drawn through BF, BG, FE, perpendicular to
the plane of the figure, and continued infinitely above and below it; the

[9973*] attracted plane being the rectangular parallelogram HBFQ^, situated in
the plane of the figure, and having the infinite sides BH, FQ. We shall
suppose, as in tlie preceding calculation [9941(^J, that the rectangular

[9973» ]

X. Suppl. 2.] ACTION OF A PARALLELOPIPED ON A PLANE. 847

f^i T^ J-f r Vertical action of a parallelopipedT \nc\fA-\

r 5 ' ' Lupon a homogeneous plane surface. J (_yy/4J

coordinates of any point S' of the attracting solid, are CZz=z, SZ=x, SS' = y:
CP = z', P S' = s', the origin of the coordinates x,y,z, being the point C, corresponding to
FC=a. The calculation of the action of the solid upon the plane differs but very little from
that in [9941—9973]. In the first place, the vertical action of a particle, dx.dy.dz,
situated at the point S\ upon a particle at the point P, is represented in like manner as in

[9941^], by dx.dy.dzS—^^.c^{s)y whose triple integral gives the whole action of the solid [9973m]

upon the point P, as in [9941]. In the next place, integrating, relative to z', it becomes, as [9973n]
in [9949],

fffdx.dy.dz.n{s)', . [ggrsni

\$1 being the value of «, when the point P falls in C, making CS'==s/, and putting, in like [9973o]
manner as in [9946'], n{s^) = c^—fQ'ds^.cp(s^). Then supposing, as in [9949'], that w^ [9973p]
represents the angle which the line s^ forms with the horizontal plane drawn through the
origin of the coordinates C, and d^ the angle which the projection of s^ upon the horizontal
plane, makes with the axis of y, as in [9950], we shall have the same values of x, y, as in
[9950', 9951]. Substituting them and the expression of dx.dy.dz [9951'], in [9973n'], we
get, as in [9952], for the action of the solid upon the fluid situated upon the line CP,

fffsfds^ . dd^ . dvs, . cos.zs^. n(s,). [99739]

Assuming the expression of the function 'sr(s^) = c' — y^*5^ds^.n(sj, as in [9953], and then
integrating [9973g'], relative to s^, we get, as in [9955],

/o */'^. • " W = — »r *(«/) +fo'^, • ^(«/) ; [9973r ]

the constant quantity in its second member being neglected, because «^.y(5J vanishes at
the commencement of the integral where «^ = 0. Substituting the value of

f^AM';) = c"-r(s,) [9960'], [9973.]

and putting s^, for the value of s^ at the second limit of the integral, corresponding to pohits
of the solid which are situated on the vertical side of it whose section is EF, we get

J^"s;'ds,.Tl{s;)=^s^,.^{sJ — T{sJ+c". [997dt]

Hence the expression [9973^] becomes

ffdd^.d^,.cosM,.\-s^^.^(s,)-T(s^,)-{-c"\. [9973U]

The part of this expression depending on c" may be cleared from the signs of integration,
because J^d^^^-r, /<Zcy^.cos,«,=sm.«^, and f^'d-si^.cos.zs^=l; hence the expression [9973t>]
[9973u] becomes

c"*— j7aa,.rftrf,.cos.«,.|5,,.*(«,,)-|-r(0|. [9973H

Multiplying this by da, and integrating from a = 0, at the point P, to a=X, at the point B, [9973r]
we get the action of that part of the solid corresponding to values of s^^, which terminate at
the surface EF; and as the values of 5^^, terminating at the surface GB, must produce a
similar expression, their sum will be doubled, and by this means the whole action becomes '• ^^

848

THEORY OF CAPILLARY ATTRACTION.

[Mec. Cel.

[9974'] as we have found, in [9936], by the comparison of the results of

t9973z] ^\'^Jda — 2fffda.d6,.d^^.co3M,.{s^,.^{sJ + r(s,)l.

Substituting in the first term f^da = \ and in the second da = ds^^.sm.6^.cos.-a^ [9965],
it becomes

[9974a] 2cVX — 2/7/rf^^.<?w^.sinJ,.cos.2a^.(^*^,.|5^,.>F(*J + ^(5J^

We may here observe, that, while the angles 6^, w , remain unaltered, we can suppose the
origin C to move from the point F, where a = 0, and «,,=0, to the point B, or to the

[99746] second limit of s^^, where a=X, and s,^ is equal to, or exceeds X. Now as the corpuscular
action vanishes at the distances which are equal to X, or exceed X, we may extend the second
limit of s^^ to s^^=z=oo ; and by substituting H=2q [9972e] in [9969], we get

hence the expression [9974a] becomes
[99746"] 2cVX— l?.j7a5,.e?«^.sm.d^.cos.2«^;

and, by using the integral [9972], we finally get, for the whole action of the solid upon the
plane, the expression

t9974<;] 2c"«X — 2g' = the action of the solid EFBG upon the plane QFBH.

Now by [9973s], we have c''==f^ds,.^{s)==f^dz,^{z) =^ [9253a'] j whence we get
^''ir=K', substituting this in [9974c], we obtain . "

[9974rf] "KK — 2g=tlie action of the parallelopiped EFBG upon the plaine QFBH, in a vertical direction parallel to FE or BG.

In the preceding calculations [9937 — 9973], the author supposes the upper line FC of

the plane Q^FCD, fig. 148, page 844, to be horizontal, and the surface EF of the attracting
[9974c] body BFE to be situated in a vertical plane perpendicular to the line FC. We shall now

suppose that the attracting body is in the form of a wedge AFBj fig. 150, limited by
[9974/*] two infinite plane surfaces drawn through the lines

FA, FB, perpendicular to the plane of the figure, and

continued infinitely above and below it ; the attracted

plane being the surface Q^FCD, which is limited
[9974^'] ^y ^^^ vertical line jFQ, and the line FC on the

continuation of BF. We shall also suppose that the

vertical lines QF, DC, are continued upwards

towards E and Z; that the lines FC, CGF'H, are

horizontal, and that the line AF, being continued,

meets CH in G. We shall take C, for the origin of

the rectangular coordinates; the vertical line CZ, for
[9974g'] the axis of z ; the horizontal line CH, for the axis of

x; and for the axis of y, the horizontal line drawn through C, perpendicular to the plane of

X.Suppl.2.]

ACTION OF A WEDGE ON A PLANE.

849

the two methods. We see evidently by both, not only the identity of [9974']

the figure, and in an upward direction ; so that if an attracting Tpa.TUc[e be at S', we shall have
for its rectangular coordinates the vertical line CZ=z, and the horizontal lines Z S=x,
SS'=y. We shall also put CP = z', P being the place of an attracted point as in
[9941e]; the angle FCH=90^'—w, the angle FCZ=w, the angle EFA=ix,
CF' = a, and PS'-=s. Then the vertical zcuon of the attracting particle at iS' upon the
attracted particle at P, is the same as in [9941] ; the value of s^ is as in [9943] ; and the
integration of this attraction relative to z', being taken as in [9944-— 9947], produces the
expression [9947] ; so that the whole of this attraction finally becomes, as in [9949], equal
to fffdx.dy.dz.Jl{sy, s^ being, as in [9948], the value of CS\ corresponding to the
origin C. Substituting in this the values of x, y [9950', 9951], it becomes, as in [9952],

f/A^ds, . dd, . dzr^ . cos.zi^ . n(5^) ,

OT^, 6^, being defined as in [9949', 9950] ; considering C as the origin of the line s,, and the
horizontal plane to which 6^, zs^, are referred, as that drawn through CH, perpendicular to the
plane of the figure. Integrating this relative to s^, as in [9953 — 9962], we finally get, as in
[9962], by putting s^^ equal to that part of the line CS'=:s^, which is contained between
the point C and the surface FA,

ffd6, . dw, .cos.^^ .{s„. ^-(5^ + r{s,,) I .

By continuing the same process of calculation as in [9963, he.], it becomes necessary to find
the value of a in terms of s^^, 6^, zi^. Now in the rectangular triangle CF'F, we have, by
using the notation [9974^], Fl«"=C-F'.tang.F'CF=a.cot.t^; and in the rectangular
triangle FF'G we have F'G = FF'.tSing.F'FG==FF'.tang.a.=a.cot.w.tang.^. We
shall now suppose that the extreme point of the line s^^, wliere it meets the surface of the wedge
whose section is FA, is projected perpendicularly upon the line FA in the point A, and upon
the line CH'm the point H; then we shall have, by using the notation [9914m', m", 9949', 9950],

CH= Si, . sinJ, . cos.-s^ , A H= s^i . sin.-ss, , HG = AH. tang.a = s,, . sin.«^ . tang.a ;

and since a or CF' is represented by CF'=:CH — HG-}-F"G, we shall have

0=8^,. sin.^^ . cos.isj^ — s^i . sin.w^ . tang.a -j- a . coUw . tang.a.

Transposing the last term of this expression, and then dividing by the coefficient of a, we get
the value of a [991 4t], using for brevity the expression of m [9974s],

sin.^^ . cos.trf^ — sin.cr^ . tang.a

OT:

a=.m.s

1 — cot.io . tang.a

[9974A]

[9974t ]
[9974;k]

[9974/ ]

[9974m]
[9974m']
[9974m"]

[9974n]

[9974o]
[9974p]

[9974?]
[9974r]

[9974«]
[9974f ]

The differential of this expression of a, considering a, s^,, as the only variable quantities,

is da = m.ds^,, which is to be used instead of [9965], when multiplying [9974nl, to obtain [9974u]

the expression of the attraction similar to that in [9966], which we shall represent by JZ"; so

that we shall have

Z=fffds,,.dd,.d^,.m.cos.r,,.\s,.*{Sii) + r(Si)]. ' [9974.]

VOL. IV. 213

850 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

[9975] the forces p' and ^H, upon which the capillary phenomena [of a

[9974tf>] The integration of this expression relative to s^^ is made as in [9966 — 9969] ; and by
substituting the result of this integration [9969] in [9974t'], we get

[9974r] Z= — 'ffd^i .d-si^.m. cos.ra^ .

Resubstituting the value of m [9974s], it becomes '^

i9974«l Z= — -; ■• rrd&,.d^.jsmJ,.cos.^zs, — sin.'ztf,.cos.'zs,.tanff.a?.

Substituting cos.^ot^=^-j-^<^os.2zj,, sin.'S^.cos.'5J^=|sin.2t!*/, then integrating relative to
■cj^ from its least value -a^ssx-a^ to its greatest value ■5*^=7^3; we get

r9975al Z= g ^ +M . sin.^, . (-f ^vs, -f- i sm.^zi^) + i tang.a .fd&^ . cos.2«3 ) ^

2 ■' ^.(l—cot.ti*. tang.a) ' ^ -\-fdd^.smJ^ . ( — ^zs^ — l^sin.Sxrfj) — :|- tang.a ./^^^ . cos.2w2 5

The value of -m^ evidently corresponds to the angular edge of the wedge, or the line fFc ,
[99756] drawn through jP perpendicular to the plane of the figure; and if we draw the line//'
parallel and equal to FF', we shall have the angle /C/'='cf2, the angle Cf'F'=d^i

[9974z]

[9975c]

ff = FF'=:a.cot.w [99740], C/'=-7-^^-— = -^ ,

ff sin /I

tanff.«, = r^, r=: a . QX3li.w X — — = sm.5. . cot.t(> :

whence we get, by using [30"], Int., "
[9975rf] " «2 = arc . (tang. = sin.d^ . coX.w) ,

r«««,- n • Ow Stang.-sJa 2sin.^,.cot.«?
[9975c] ^ sm.2trfa= ^ / = ^ ^ -^-,

[9975/]

[9975g]

r» J^ ■■ — st: 1— sin.2^,.COt.2tO

C0S.2^2 = V/l-«^"-^2t.2 = i.,;„2,...nt.Q,. '

l-j-sin.2d^.cot2u>

a

The greatest value of ■z^^, represented by -2^(3, is evidently found by putting —=00, or — =0,

which gives m = [9974^]; and by substituting the value of m [9974*], we obtain

[9975g^] Oassin.^^.cos.tzfg — sin-i^Tg . tang.a.

Dividing by cos.'sJg . tang.a, we get tang.'zrf3==sin.5^.cot.a, or

[9975A] *3 = arc. (tang. = sin.^^.cot.a}.

r9975^'l ^o'^paJ'ing this with [9975t?], we find that -m^ can be derived from zi^, h'^ merely changing
w into a; and the same substitutions can be made in [9975e,/]. From the expression
of ?*a [9975(ZJ, we easily perceive, that, at the upper extremity / of the line cFf where

[9975t] Ff=i 00, ^,==0, we have zi^^s=0', and at the point jP, where ^/^ J*, we have -ui^^=9Qi^- — w.
At the lower extremity c of the line cFfy where Fc= — go, and d^*, we have
1^2=0. Hence we see that, after substituting the values [9975d!, e,/] in [9975a, line 2],

[9975A;] and taking the integral relative to &, from ^^ = to ^ = 'n', we shall have, at both these
limits, Wj=sO. Now substituting the values [9975dlj e], in the first member of [9975/], we

X. Suppl.2.] ACTION OF A WEDGE ON A PLANE. 851

homogeneous fluid] depend, but also their derivation from the attractive [9975']

get its second member; and its integral is expressed by the function [9975m], as we shall
soon show ;

fdd, . sm.&, . (— htz^—ym.'H'ui,) ^fdd^ . sin.^, . \$ — ^arc . (tang.=^sin.d^ . cot.w)^ ^^^^^^^ I [9975/ ,]

a=^cos.^/ . arc . (tang. =sin.^^ . cot.t^) -f- 2<jos.i(; . arc . (tang. =:cot.^^ . sin.t^^) — - . cos.tv. [9975wi]

That the expression [9975»i] is the integral of [9975?], is easily perceived by observing,

that, if we consider cos.^^ as the only variable quantity in the first term of [9975m], its

differential will be the same as the first term of the second member of [9975/], so that

it only remains to show that the differentials of the two arcs in [9975m], namely,

arc. (tang. = sin. ^^.cot.ttf), and arc. (tang. = cot.5^. sin. w), considering 6, as the variable [9975n]

quantity, produce the second term of the second member of [9975/]. Now the differentials

of these two arcs produce in the differential of [9975m] the following terms, using [54], Int.,

and the similar expression d . coU&^ =s — dd .{l-\-cot.^d^),

- , 5 cos.^S,-cot.tv (1 -{- cot.^^J . BJn.to . cos.u> } rQQ7tAl

2^"/'^l+sin.2^^.cot.2u> l + cot.2^,.sin.2u> 5* •• ^

Multiplying the numerator and denominator of the second of these two terms between the
braces by ^.^gj , or sin.^d^.(l-j-cot.^i^'), it becomes

[9975p]

(sin.2^^-|-co8.2^^).cot.tg {sin,2^^-f-cos.a^^).cot.tc .

8in.2^^.(l-|-cot.2tf))-}-cos.2^^^^ l-{-sin.2d^.cot.2u> '

substituting this in [9975o], and then reducing, it beconries — ^dd^. ^^.' ' — -rg-j

being the same as the second term of J9975/]. The integral [9975m] vanishes at the first
limit ^^=0, where arc. (tang. = sin.^^. cot.t^) =0; and arc.(tang.=3C0t.^/.sin.w)=^'jf. [9975</]
Then, while the arc ^^ increases from ^^ = to 6,=^'^, the arc. (tang. E=cot.fl^. sin. w)
decreases from A* to 0, and then becomes negative, so that, at the second limit of ^^ = *, it
becomes equal to — i-jr; moreover as w^ = arc. (tang. = sin.4^.cot.i/;)=0, at this second [9975r]
limit of 6^, the first term of [9975m] will vanish, and the complete integral [9975m],
depending on the two remaining terms, will become - - , .

•J

COS.tf'.

2

7t

j.cos.wssz — ^le.cos^my [9975rQ

hence the expression [9975m] gives

/;(i(),.sin.d,.(— ^«, — isin.2ts,)=« — ^*.cos.ty. [9975,3

Again, if we substitute the value of cos.S^tf^ [9975/"], in the first member of [9975^], and

then integrate it, we shall get ^

y, M sin ^/1 cot ^w^

/c^, .cos.2z.,=/rf^, . \^3.„'2^;;^^t.2u; =— ^/ + asin.tg. arc . (tang. =: tang. Vcosec.to), [997ai

9» is easily proved by taking the differential of its laist member^ relative to 6^; for it becomes,

852

THEORY OF CAPILLARY ATTRACTION.

[Mec. Cel.

[9975"] forces of the particles of the bodies which produce their affinities. The *

by successive reductions, and putting sin.tf?.cosec.u?=l, cosecfiw = \-\-cot,.^w , &ic.

.d&

'/ "T 1-|- tang .2^^ . ( l-j- cot.2M>)

[9975m]

= ~da,-f.

2rf^,

= c?4,.

(1 — sin.2^^.cot2«j)

l-}-sin.2()^.cot% '• l-|-8in.2^^.cot2tc '

which is the same as the proposed differential in [9975^]. At the first limit ^^= of the integral
[9975f], it vanishes. Moreover, as 6^ increases from to <r, the arc. (tang. = tang.^^.cosec.M>)
also increases from to *; so that, at the second limit of the integral, we shall have ^^^flf,
and arc . (tang. = tang.fl^ . cosec.zij) = *. Substituting these values in [9975^], we get

[9975»1 /o''(Zd^.cos.2t32 = 3-.(— l + 2sin.«;).

Multiplying this by — l^tang.a, and adding the product to the integral [9975s], we obtain

[9975w] f^d^^.s\n.di.{ — Ins^ — Jsin.22OT2)— :|-tang.a.y^''<Z^^.cos.2OT2=^ff.(— cos.'W-f-^tang.a— tang.a.sin.iij).

If we change -a^ into zs^ in the integrals [9975s, v], the effect will be to change w into a
in the second members of these expressions, as we have seen in [9975A']. Making these
changes, and multiplying the integral derived from [9975u] by — ^tang.a, we get for
the sum
[9975x] f^d&,.svc\.&,.{—^TSs — |sin.22trf3)— itang.a./o''t?^,.cos.2^3=^'r.( — cos.a+^tang.a — tang.a.sin.a).

Subtracting [9975a?] from [99752^;], and multiplying the remainder by
we get the expression of Z^ [9975a], which becomes

H

7r.{l — cot.u> . tang.a)

[9975y]

[9975z]
[9976a]

z==

!i/f.|(cos.a — cos.w) -f- (tang.g . sin.cn. — tang.ct . sm.w) \

1 — cotto . tang.a

Multiplying the numerator and denominator of this expression by sin.t^ . cos.a, and reducing
by means of [24, 22, 1, 31, 34'J, Int., we get, by successive operations,

y j.H".sin.io.|(co8.2a-|-Bin.2a ) — (co5.to.cos.a-f~s^"-"'-^^"-'^)^ ^Jf. sin.w .\\ — cos.(tt> — g) ^

sin.to . cos.a — cos.to . sm.a

sin.(w — a)

[99766]

[9976c]

2sin.a(u;— aj-coMtttf— a) ^ s 2\ j

This expresses the upward vertical action of the wedge AFB, included by the angle w — a,
upon the plane Q-FC, corresponding to the angle w.

If we change a into a', in the expression of Z [9976a], it will become

Z^=|fl'.sin.w.tang.i(zy — a') ;

subtracting this from the expression [9976a], we get the value of Z, corresponding to the
action of a wedge included by the angle a' — a; the vertex of the wedge being the line

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