dp^—g?.dr. [8650]
The pressure p is proportional to the product of the density p of the
particle [8649"], by its heat z [8649'"] ; therefore we shall have lS653c},
p = K^.z ; [8652]
K being a constant coefficient. Hence we obtain,* ^*
*(3987) Dividing [8650] by [8652], we get [8653]; or ^ = — JT.^; whose [8653a]
Z JJ
integral is / — = constant — KAo^.p ; and as this integral begins when p = [p) [8662a],
we get, constant =ir.log.(p); consequently /^=ir.log. — , as in [8654], We [86536]
VOL. IV. 142
[8649'"]
[8651]
666 . ON THE MEASURE OF HEIGHTS [Mec. Gel.
[8653]
dp gdr
7" "" "~ ^ '
which gives by integration, and using (p) [8649],
y^ = jr.iog.(-^.
[8654]
[8655] The relation between g and (g), gives very nearly,*
[8656] ^ = (^)._|_=(^).(^l_Er);
[8657] therefore, by putting i^ =r.(\ ^j, we shall have,
[B658] /4^ = (^)/^^ = r.Iog.^) [8654].
To integrate these functions, it is necessary to find z in terms of r' ; but
as the integral extends only through a small interval, in comparison with the
whole height of the atmosphere, it is evident that every function which
represents the temperatures of the upper and the lower stations, and makes
the temperature decrease nearly in an arithmetical progression, from the one
to the other, is admissible ; and we may select that form which most
simplifies the calculations. Therefore we shall suppose,
[8660] z = t/o2 — j/, [uxpreBBion of the heat z]
[8661] q = the temperature of the air at the lower station ;
i = an indeterminate constant quantity, which is to be taken so that the
[8^61'] expression of z [8660] may represent the temperature of the air at the
upper station. We shall then have, as in [8662e],
[8653c 1 â„¢^y remark that the equation [8652] is like that in [8400a], changing C into K, and h
into z. Moreover, the symbols used in this chapter are similar to those in
r8653<n [8288-8289', &;c.], the radius r [8138] being changed into a-\-r [8648']; hit the
differential of the radius dr is unaltered^ as in [8650, &,c.].
[8655a]
[86556]
[8655c]
* (3988) Ghanging, in [8292], r into a + r, to conform to the present notation,
[8653d], we get the first expression in [8656]. The second form is deduced from the
first, by observing that „ "^^ ( ^H ) *v^ ' — ~ ' "^arly. Substituting this in the
first member of [8658], it becomes (^)./ ■" •( 1 )> which is easily reduced to the
form in the second member of [8658], by observing that the differential of r' [8657], is
rfr' = (Zr.(l— -^"Y , _,.,^ .
X. iv. § 14.] < BY A BAROMETER. 667
r
hence we find, as in [8662g],*
form by substituting z instead of its value [8660]. Multiplying the numerator and
denominator of this last expression by q-\-z^ and substituting g-^ — z^ = zV [8660], we
get, by successive operations, the final expression in [8662e], being the same as in [8662] ;
[8662]
r'=(l±i) ^ loe ^-Pl. [8663]
2 (g) ' ^* jp '
we shall use, in this equation, the tabular logarithms instead of hyperbolic
logarithms, which only affects the constant quantity K. We shall put I for [8664]
the temperature corresponding to that of melting ice ; and shall suppose,! l-
q = l^t; z = l+t'; [8665]
* (3989) Substituting the value of z [8660] in the first member of [8662c], and then
integrating, we obtain the third expression in [8662c], as is easily proved by differentration.
As the integral commences when r = 0, it is evident that the constant quantity must be [8662a]
2a
-r ; hence we obtain the first expression in [8662c?], which is easily reduced to its second
[86626]
^^IZip- — 7•vr-^r' [8662c]
2 2.(9-2)
= 7 •{?-/92_ir'} = —7—^ ^ [8662rf]
2.{q^—z^) 2ty 2/
= ., _,, = rj—r-, = — r- • ' [86626 ]
1.(3+2) t.(54.z) g+2 â– â– J
Substituting this last value in [8658], we get,
rgdr 2.{g)y {p)
yT=l+r = ^-^^S.-. [8662/]
Dividing the two last expressions in [8662/] by the coefficient of /, we obtain r' [8663]. [8662g]
It is proper to state that the hypothesis for the temperature z, assumed by La Place in
[8660], is substantially the same as the ancient hypothesis of De Luc, modified by
La Grange ; as has been shown by Plana, in vol. 27, page 194, of the work mentioned [8662^]
in [8540A].
t (3990) The heat is supposed to be expressed in degrees of the centigrade
thermometer, supposing Z to be its measure when the temperature is equal to that of l8665a]
melting ice. Then the temperature at the lower station being t [8668], the corresponding
heat will be expressed by q==l-\-t [8661, 8665] ; and at the upper station, where the
temperature is i' [8668], the heat becomes z=Z-f"^'> as in [8649'", 8665]. Substituting
the values of q^z [8665], in [8663], we get [8666]. ] s''''*"'?)f:t:-
[8668]
568 ON THE MEASURE OF HEIGHTS [Mec. Cel.
then we shall have,
[8666] / _ . ; 1 _!. ^ ' ^ ^ . log. ^i-^ .
The comparison of a great number of measures of the heights of mountains,
by the barometer, with their trigonometrical measures, has been made by
Ramond ; who found, in the parallel of 50°, that the coefficient,
[8667] _ =18336"^*^^^
(g)
To determine the coefficient /, we shall remark that t and t' denote the
degrees of the centigrade mercurial thermometer, counted from zero. If we
consider an invariable mass of air at the temperature zero, each degree of
increment in its temperature increases equally its elastic force or pressure,
the increment of pressure, corresponding to a degree of the thermometer, is
very nearly 0,00375 [8488] ; so that if we put (p) for the pressure, or the
[8670] elastic force of a mass of air at the temperature zero, we may suppose that
[8670'] for each degree of the thermometer the pressure increases by (j?).0,00375 ;
but this pressure is, by what precedes, equal to* Kp.(l-\-t); hence we
[8671] have (p) = KpL The increment of one degree in the temperature gives
an increment in the pressure equal to Kp, or to KpL—; or lastly to
[8672] (p), — ; putting this quantity equal to ('p).0,00375 [8670], we obtain
[8673] I ;=; = 266,6.. . Therefore we shall have, upon the parallel of 50°,
[8674] / = 1 8336•»«"■''^ fl + ^^±^ . 0,00375\ log. -^-^ ;
the pressures (p) and p are determined by the heights of the barometer ;
[8669]
[8671o]
* (3991) Substituting in [8652] the value of q [8665], we get, at the lower station,
{p)—Kp.{l-\-t); and when <=0, (p) becomes (p) [8649,8670]; hence (yj=Kpl,
[86716] as in [8671]. JMoreover when i = 1, the preceding value of (jp) [8671fl] becomes
r^)
(p) — Kpl-\-Kp = (v)-{-Kp;theTe(oie(p) — (p)=Kp = ~- [8671al, represents the
[8671c] ^
increment of (p), arising from a variation of 1°, in the centigrade thermometer; being
the same as in [8672]. Putting this quantity equal to fp) .0,00375 [8670'J, and then
[8671rf] dividing by (jp), we get I [8673]. Substituting this value of /, and that of
— = 18336°'«tres [8667], in [8666], it becomes as in [8674].
X. iv. Ǥ> 14.] BY A BAROMETER. 569
but we must reduce the mercury in the barometer to the same temperature. It
has been found, by an accurate experiment, that mercury increases in bulk rggyci
TTTa''' part,* for each degree of the centigrade thermometer ; therefore, in
the station corresponding to the lower temperature, we must increase the
observed height of the barometer, by as many times its -J4Y2*'' part, as
there are degrees of difference in the temperatures of the mercury in the
barometer at the two stations. Moreover, the temperature of the mercury [8676]
in the barometer is not always exactly the same as that of the surrounding
air ; therefore we must use a thermometer attached to the barometer.
Besides this correction for the temperature, there is also another which is
required, in order to reduce the observed heights of the barometer to the
gravity (g), corresponding to the lower station. The gravity at the
of the mercury, become (/t) and ^ ; hence we have,t
[8677]
[8678]
[8679]
[8680]
superior station is (g), . [8656] ; therefore, by putting (h) and h
for the observed heights of the barometer at the two stations, reduced to the
same temperature, we find that these heights, reduced to the same gravity
.og.(j)=log4)+2.Iog.(, + 0.
— being a very small fraction, the hyperbolic logarithm of l-\ — is very
nearly equal to — ; consequently its tabular logarithm is —.0,4342945;
therefore we shall have,
log. ^ = log. ^^ + - .0,868589.
The coefficient 1 8336'°^*'^' is exact only upon the 'parallel of 50° ; it varies [868I']
iviththe latitude f and is inversely as the gravity (g) [8667]. If we put [J] [8682]
* (3992) The late experiments of Dulong and Petit [8490a], make this increment j-g^ -
Wbtyj instead of -^â– ^.
f (3993) The pressures (p)f p, are as the corrected heights of the barometer (A)
and A. ( 1 -j — ) ; hence we have — = -7-.n-|- — j ; whose logarithm is as in [8679o]
[8679]. This is easily reduced to the form [8681].
VOL* IV. 143
[8681]
670 ON THE MEASURE OF HEIGHTS [Mec. Cel.
for the gravity at the equator, and ^ for the latitude corresponding to (g),
we shall have, as in [2054, 1770a],*
From this we readily see that the coefficient 1 8336"*®*''®% corresponding to
50° of latitude, is for any other latitude y, equal to,t
[8684'] 18336'"^*'^\{ l-}-0,002845.cos.2^5. ^
This being supposed we shall have, to determine the heights by a barometer,
the following formula ; J
* (3994) In [2054] we have, for the length of a pendulum vibrating in one second,
[8683a] 0-'-,739502. ^ I + ^^ . sin.^* ^ ,
[86836] in the latitude y. At the equator, vfheie ^ = 0, it becomes O'"«*'%739502. These
r8683cl lengths are proportional to the gravities (g) and [J] [8682], respectively ; as we have
seen in [1769"] ; hence we get the expression of (g) [8683]. For greater correctness, we
[8683i] may change the coefficient of sin.**- into ' =0,005333 [2056q,j?], corresponding
to the best observations of the pendulum.
/
t (3995) Putting for brevity 2& = '^,,^^ = 0,005690, we find that the factor of
[8684a] ' ^ ^ & J' 0,739502 ' '
[8684&]
[^]' ^" ^^^ second member of [8683], becomes 1-f 26.sin.^^=l-[-& — 6.cos.2^ [1] Int.;
hence the expression of (g) [8683], is (g") = [2i]*{l+^ — 6.cos.2*j. Substituting
Kl
this in the factor — [8666], we get,
raraA ^ Kl Kl ( , 6 ^ ) "^ Kl C , , 6 ^ >
^"^'^ (F)=(Tw[i]-r-i+^""-''^l -iH^m-v^^^-"^-^^r^''-
Kl
When ^ = 50°, it becomes , ^ -, ; and if we put this factor equal to 18336"*®*'"",
(1+6).[1.] ^ ^
as in [8667,8681'], we get, for the general expression of this coefficient in any latitude,
the same value as in [8684']. For greater correctness, we may use,
[8684rf] 5 = i X 0,005333 = 0,002666 [868Sd],
for the coefficient of cos.2^.
t (3996) Substituting in [8666] the value of p^ [8667,8684']; also for ^-^ its
[8685a] ^ (^) ^
value deduced from [8673], namely ^-?±1^ 0,00375; and for log.— its value [8681],
2 p
it becomes,
X.iv.<§.14.] BY A BAROMETER. 671
Formula
to find the
Dividing the first member of this expression by 1 , it becomes equal to r [8657J.
a
In like manner we must divide the second member of [86856] by the same divisor
1 — — , which is very nearly the same as to multiply it by \-\ ; then connecting this
factor with the last factor of [86856], we get, by neglecting terms of the order r^, the
expression [8685].
[8686]
[8687]
r===18336^J14^,002845.cos.2T|.$l+^^lo,00375^.$^l+^Ylog.^~^ [86851
^ •' V. \ / -' height by
It is sufficiently exact to substitute, in the second member of this equation, terf'""*^"
the value of r, computed upon the supposition that r = 0, in the second
member. We may also suppose, without any sensible error, that
a = 63661 98â„¢"''"' [8278]. The corrections depending upon the latitude and
upon the variation of gravity, are very small ; but as they really exist, it is
best to notice them, so as to leave in the calculation no other imperfections
than those which arise from the inevitable errors of observation ; or from
the effect of the unknown attractions of the mountains ; or from the
hygrometrical state of the air, which ought to be noticed ; or finally from
the error arising from the use of the hypothesis [8660], relative to the law
of the diminution of the heat. We may satisfy in part for the state of the
hygrometer, by increasing a little the coefficient 0,00375 in the term
J.(^ + i'). 0,00375, in the formula [8685]. For the aqueous vapor is lighter
than the air [8526], and the increase of temperature increases the quantity
of vapor, all other things being equal ; so that we can very nearly satisfy the
observations which have been made by the barometer, by changing
^^±^.0,00375 into ^^^^\ in the formula [8685] ; * and by this means [8689]
it becomes,
/= 18336"».{l+0,002845.cos.2*|. J 1+ ^i±^^ 0,00375 \ . \ log. ^^ + ^ .0,868589 | . [86856]
[8688]
[8685c ]
[8685rf]
[8690o]
* (3997) The quantity of vapor, in the column of air of the height r, increases with
the mean temperature of the air ^.(^-j-^) [8508] ; producing a corresponding increment
in the pressure, and in the term depending on \'{t-\-t') in the formula [8685]; so that
the numerical coefficient ^X 0,00375 [8685], is found by observation to become nearly [86906]
equal to 0,002, as in [8690].
The formula [8690] may be reduced to English fathoms and to Fahrenheit's scale in the
following manner. If t, t', be expressed in degrees of Fahrenheit's scale, the
[8690c]
572 ON THE MEASURE OF HEIGHTS BY A BAROMETER. [Mec. Gel.
The same
[8690] ^^ i8336-«'-.| l+0,002845.cos.2^}. ^ 1+^3^^ 1 • [ (^^)-l^S- ^ "^ 7-0,868589 1.
corrected
for the
humidity.
corresponding degrees of the centigrade thermometer will be {t — 32°). |^, (<'— 32°). jfg,
respectively. These are to be used instead of t, t', in [8690] ; so that t-\-t' will
[8690rf] ^,^.^,.
become (<+<'— 64°). |§^ ; and the factor 1-f- -\^ will change into,
[8690e] 1 + ^-^^^^^^ X 4t^ = ^^.f836°+^ + ^'} ;
moreover, as a metre is equal to 0f^^°'",54681.. [2017p], the factor,
[8690e'] 18336'»«»'«^ ^ 1 + ?^^^^^ \ ,
changes into,
[8690c"J 18336"' X 0,54681 ^ |836°+ ^+<'| = ll*tS1404.|836°+< + <'|,
and the expression [8690] becomes, by using the corrected factor relative to S'F [8690e],
[8690/] r= 1 lf»t'>-,1404. \ 1-}-0,002845.cos.2y \ . {836°+ t-\-l!\.\ (l+-^).log. j + f • 0,868589! .
Finally, the correction ajx^j which is used in [8675], must be reduced to
[8690g] = wh^i to correspond to Fahrenheit's scale; if we use -^^ [8675a],
[8690A] instead of -^-^^^ , the correction becomes -^-^ X j|^ = ^^Vtt 5 instead of ^^^ .
X. V. «^ 15.] DESCENT OF BODIES FROM A GREAT HEIGHT. 673
Deviation
til tlie east
of the
vertical.
[8691]
equator
there is no
deviation.
CHAPTER V.
ON THE DESCENT OP BODIES FALLING FROM A GREAT HEIGHT.
15. A BODY, beginning to fall from a state of rest at a point considerably
elevated above the surface of the earth, will deviate sensibly from the vertical
line, on account of the rotatory motion of the earth ; an accurate observation
of this deviation vi^ill therefore be useful in rendering this motion manifest ;
and although the rotation of the earth is now established, with all the
certainty which comports with the state of the physical sciences, yet a
direct proof of this phenomenon must be interesting to mathematicians
and astronomers. For the purpose of comparing the theory with such
observations, we shall here give the expression of the deviation of the body, m the
to the east of the vertical, ivhatever be the figure of the earth, or the resistance
of the air. We shall also show that the deviation is greatest at the equator,
[8736', 8760, 8769]. The following symbols will be used : g^^,^,^
X, Y, Z, are the rectangular co-ordinates of the point, from which the body rgeosi
begins to fall, at the commencement of the time t ;
X, y, z, are the rectangular co-ordinates of the body, after falling from rest,
during the time t. The fixed axis of a: or X is the same as
the axis of rotation of the earth. The fixed axis of y or Y is
in the plane of the equator, and coincides with one of the
principal axes of the earth, at the commencement of the time t ;
r = the primitive radius, drawn from the centre of the earth to the rgggoi
point where the motion commences ;
r — as = the variable radius, drawn from the centre of the earth to the r8694i
place of the body, at the end of the time t ;
6 = the angle formed by the radius r and the axis of rotation x ; [8695]
VOL. IV. 144
574 ON THE DESCENT OF BODIES [Mec. Gel.
[8696]
[8697]
d-\-o.u = the angle formed by the radius r — as and the axis of rotation
X, at the end of the time t ;
z3 = the angle formed by the two planes xr, xy^ intersecting each
other in the earth's axis of rotation x. One of these planes xr
passes through the axis of rotation x and the radius r at the
commencement of the motion ; the other plane is the fixed
plane oi xy\
[8698] nt = the rotatory motion of the earth during the time t ;
nt-}-'^ = the angle formed by the fixed plane xy, and the revolving
plane xr ; this last plane being that which passes through the
revolving radius r and the fixed axis x, at the end of the time t ;
nt-\-'Ci-\-o.v = the similar angle, formed by the plane xy, with the plane
passing through the fixed axis x, and the variable radius r — ol5,
at the end of the time t ;
V = the sum of all the particles of the earth, divided by their
distances from the falling body ;
f-^j, (-7-)j {'l~p represent the forces acting upon the body in the
directions parallel to the co-ordinates Xj y, z, respectively,
and tending to increase them [455'"].
From this notation it follows, that after the body has been falling during
the time t, the radius r changes into r — ^s ; the angle ^ changes into
d-j-aw ; and the angle ts changes into zs-\-^v. Then vre shall have,*
[8703] Z=r.cos.d;
[8703'] y= r.sin.^.cos.(7J^4-«) ;
[8703''] Z=r.sin.^.sin.(7t^ + ^). r coordinates.]
[8704] X = (r — a 5) .COS. (^ -f- a m) ;
[8704'] y = (^ — a,5).sin.(^ + 0L|^).cos.(n^-f-^-}-av);
[8704"] z = (r — a5).sin.(5-j-az<).sin.(ni4-OT4-av).
[8703al * (3998) The notation [8692—8700], is precisely lilce that in [323^—324], except in
the sign of s [8694] ; r — as being used instead of r-f-as [323''], because the radius
decreases as the body falls. This change being made in a?, y, z [324], they become as
[87036] ^^ [8'704 — 8704"]; and at the commencement of the motion, when s, m, v, vanish, these
values change into those of X, Y, Z [8703—8703"].
[8699]
[8700]
[8701]
[8702]
[8702']
X. V. § 15.] FALLING FROM A GREAT HEIGHT. 575
We shall notice the resistance of the air, supposing it to be represented hy* j^g^^g,
<p.( fl.5, ^'4-\ (^nd that the body falls from a state of rest. For the relative ^""«^^°°
, . » expressing
velocity of the body through the air, considering the air as at rest, is ti^je-^
1 • I supposing
evidently much greater in the direction of the radius r, than in the l^^^^^^y
. /. J . ff om a
direction perpendicular to r ; hence it will follow that the expression ot this *tateof
ds
relative velocity is very nearly represented by <*-• x • ^^5 ^^^ greater [8706]
simplicity, we put r = 1 , the relative velocity of the body, in the direction
du
6, will be cL. — [22096] ; and the relative velocity in the direction ro, will ^^^^
dv
Jt
dv
be equal to a. — .sin.^ [2209c] ; therefore, if we put for brevity ,t
ds\
/ ds
5f.
S = -:^ ^^ [8707]
^•dT
[8705a]
* (3999) The body falls from rest very nearly in the direction of the radius r, through
the space as, in the time t; therefore its velocity, at the end of the time tj will be
ds
nearly represented by a. — [8706J. Now the resistance must be as a function of this [87056]
velocity and of the density of the medium ; moreover the density of the medium depends
on the radius r — as, or as; therefore the resistance must be a function of as and
a.—; which is represented by 9. fas, t* - ;^), in [8705].
f (4000) The radius r — as [8694], at the end of the time t, will be varied by the
quantity — ac?s, in the element of time dt ; making the velocity, in the direction of this
[8705c]
ds
[8707a]
radius, equal to — a. — , or «-. — , towards the centre of the earth. The velocity, i
dt ' dt
m
[87076]
the direction of the meridian aw, is a. — [22096] ; and in the direction a«, perpendicular
to the meridian a. — .sin.^ [2209c]. The sum of the squares of these three partial
rectangular velocities is equal to the square of the whole velocity ; and as the parts
depending on du^, dv^f are extremely small in comparison with that depending on ds'^, r8707cl
ds
we may consider the whole velocity to be very nearly equal to the part a. — [8707a], as
in [8706]. Now if we divide the whole resistance [8705] by the whole velocity [8707rf]
676
ON THE DESCENT OF BODIES
[M6c. C^l.
Jholifin we shall have, for the resistance of the air m the directions r, 6, «,
the direc-
tiona
r, 6 J «.
[8708]
[8709]
[8710]
respectively,
5'. a.
ds
dt'
— o. a. — ,
dt '
— o. a. — . sin.^
at
[ResiBtancn in the direction of the *!
raditu r. J
[Resistance in the direction of the 1
arc of the meridian dg. J
[Rnsistance in the direction of the "I
parallel of latitude <Xv,»iaQ. J
[8711] Then we shall have, by the principle of virtual velocities,
ds
tt. — , we shall obtain tlie quantity 5 [8707] ; and multiplying it by the three partial
ffo tjhti /^i^
velocities a. — , o-. -r > a. — .sin.4 [8707a, 6], we shall evidently obtain the three
dt at at
relative resisting forces [8708,8709,8710], which tend to decrease s, &, zs; or, in other
words, the forces tend to increase the radius r, and to decrease 6, -a ; agreeably to the
signs which are used in [8708 — 8710]. For convenience of reference we have inserted
[87075"] the symbol S, in the expressions [8708 — 8710], instead of its value [8707], which is used
in the original work ; having transposed the definition of S from [8711] to [8707].
[8707e ]
[8707/]
[8712a]
[8712&]
[8712c]
[8712i]
[8712e]
[8712/]
[8712^]
* (4001) The principle of virtual velocities is expressed in the equation [37], which
may be put under the following form ;
»=^-f+^^-^f+^-
ddz
1
The first line of this expression is the same as in [8712 line 1] ; the second line produces
that in [8712 line 2], depending on the attractions of the earth ; also that in [8712 line 3],
depending on the resistance of the air. For we have seen, in [41], that the function
P.^x-\-^.^y-\-R.^z can be reduced to the form 2.S.<5s, representing the sum of the
products, formed by multiplying each force S, by the element of its direction 5 s. Now the
attraction of the earth produces the forces i-j-p (aT)' \d~) t^^^^]' '" ^^® directions
X, y, z, respectively; and by multiplying these forces by the elements Sx, Sy, Sz, then taking
the sum of the products, they produce, in [87120^, line 2], the same terms as in [87121ine2].
In like manner, if we multiply the forces [8708, 8709, 8710], depending on the resistance
of the air, by the elements of their directions Sr, 66, d-ss.s'mJ, and take the sum of these
products, they produce, in [87 12^, line 2], the same terms as in [8712 line 3J; therefore
the equation [8712] expresses truly the fundamental equation of the motion of the falling
body, arising from the principle of virtual velocities.
[8713]
X. V. <§. 15.] FALLING FROM A GREAT HEIGHT. 577
-'<S)-»(f)-'<f)
dt dt ^ dd
The differential symbol 6, refers to the co-ordinates r, ^, ^ ; and x, y, z,
are functions of these quantities. If we substitute, in [8712], the values
of X, y, z [8704, &c.], we shall have, by neglecting terms of the order a^*,
^ ( dds r. • o ^ <^^ O <^* ) \ ,
+,2.5^.^a/^-2an.sinAcosA^-faS.^l f 2
i dt^ dt ' dt !^ \^^^ ^8714]
. „ ^ ( . „. ddv , ^ . , , du 2an.sin.24 ds , ct • q^ ''^ > I
' ( dt^ dt r dt ' dt ) I
* (4002) The complete variation of — F", considered as a function of a?, y, z, is the
same as the expression in [8712 line 2], which must therefore be equal to — 8V.
Substituting this in [8712], it becomes,
This may be reduced to the form [8714], in the same manner as [325] is deduced from