a@>>i>b
The w and w regions will be precisely the same as in the
last problem, and the relation between them will be
dw U—a
du a (u—b) (uw —c) ey (29)
The 2 boundary is easily seen to be
it 0=0
1891.] Fluid Motions in two dimensions. 197
where the point marked corresponds to the point from which
the stream comes and the point marked 1 is the origin in the
Q, plane.
Now this polygon has right angles at 1, —1,c, b, an angle 0
at a and an angle 27 at © and it can be conformably represented
upon the half-plane w by means of the relation
dQ _ A
du = /{(u®7—1) (w—b) (u—c)} (w—a)
In the general case the imtegration will require elliptic
functions and I do not propose to proceed with it.
13. In the case of symmetry it is clear that there is no
flow across the line perpendicular to the lamina at its middle
point so that this line may be treated as a real boundary and
the z-region will be
A E
PNG
D
and the image of this in the line AB. It is clear that the
motion takes place as if the boundary were ABCDE. If now
we reflect this in the line DE we get the same figure as in the
escape of a jet from a rectangular vessel by an orifice in the
middle of the base. This is a particular case of our first problem
and the form of the jet has been worked out by Mr Michell.
The relations between z, w, and wu, are
a!
el é SIRS eee eee (31),
dz _1 v(l=a)+V—~) |
duu J(w — a’) J
and we propose to find the pressure on the side of the vessel
in which the aperture is. If we take the part to the left of
the aperture we shall have for the difference of pressure on the
two sides of this plane
fe dw \* | dz
1 = | eee Rearapgee. 508) 31's, oe Se 2
4p | E (5 )| es Wie ieee (32)
198 Mr Love, On the Theory of Discontinuous [May 4,
where p is the density of the fluid, or
1 ti — Op
| E ~ 2-@-w4+2/1-a)vi- A
NG +0)
Jw — a? “it
= : [ {/(1 — w*) + /(1 — a*)} AC lie uw’) du
a (VA -—@)4+/0A— wv’)} fw a*)” 2
ue ie V(1—u*) du
aNG@e=a@) u ”
The value of the integral is
wT /1
Returning now to the problem of the lamina we see that
the pressure on it is
PI (Ll — @)[@iicc. cocci seecccens ene (33),
and in the same case the velocity of the stream at infinity in
the direction from which it comes is
GV A/G = @?)} secon needs ce eee (34),
the breadth of the stream is
d= ll+Wl=a)j\le 2.4. %....0eee (35),
and the breadth of the lamina is
= Tr 2
(—7r ae —— ir —2 sina) eee (36).
Now let the stream flow from o with a given velocity V in
a canal of breadth d, and impinge symmetrically and directly on
a pier of breadth /. Then if a quantity a be determined from the
equation
1l-a+V/1-@) (1 a sin” a)
ae ee Rs
d (1 —a’*) ‘
the pressure on the pier will be
(b= a) (bh V/a— a}
(or (1—a)+ Va — a’) (7 — 2sina)} ©
By writing a= cosa we find the convenient form
pr Vi
l 2a
ae F 1 Tt) PP ORCS EPO Re eGEe ESS e G6 8.8 8 fie
7 tan a+ - (37),
1891.] Fluid Motions in two dimensions. 199
and the mean pressure is
pm V* (sec’a — sec a) (1 + sin a)’
am (1 —cos a) + 2asina
tai eet mentite (38),
where a is determined by (37).
14. When the sides of the canal are distant, / is small com-
pared with d, and if we take //d=e we shall have
2 iE a
e=—+5at+5, 0+ ecgtee Siciocs nee: (39),
so that
2 4
“a= — €— = (=) SE ECBEIV wweanscaeens (40).
As a first approximation taking @ small, we have for the mean
pressure
pa V"(4a*) pw V*
ae a ore _—
7 +20 ae
2
the same as for a lamina held in an infinite stream. Going to a
second approximation we have merely to retain the term 2¢ in
the expansion of (1 + sin 2)’ and this gives for the mean pressure
so that the effect of the sides of the canal is to increase the mean
pressure by
and the fraction of itself by which the mean pressure is increased
is about 4e, in which it is to be remembered that ¢ is the ratio,
breadth of pier to breadth of canal.
_ 15. Prosiem (v). Stream flowing past an obliquely projecting
pier.
Suppose the z boundary consists of parts of two straight lines
one of them infinite in one direction and terminated at the point
marked Q, and the other finite and inclined to the first at a given
angle m—a, and that the fluid is on the side of the boundary
200 Mr Love, On the Theory of Discontinuous [May 4,
within the angle 7 —a and comes from © in the direction indi-
cated. The figure in the z plane is
WV
@
There being but one bounding stream-line, the figure in the w
plane is
w
so that there is no necessity to transform to a new w plane. We
take then the point w=0 to be the corner, w= 1 the extremity
of the broken line and w= the point to which the stream goes.
The figure in the 0 plane is
d=0
if SE
0= —(r-~a)
D
and this can be conformably represented on the half-plane w by
means of the relation
dQ, _ A
dw /(w—1).w
1891.] Fluid Motions in two dimensions. 201
so that
tog {p [tt YA=™)]49._ 159
erat tc Ey an) Se oe
and since O = 0 when w= 1, and the argument of dz increases by
(7 — a) as w goes through zero, we find
dz _[1+/(1—w)]!-4l" g
1-— V1 —w)
dw
16. With our choice of constants the length of the part
between the points corresponding to w=0 and w=1 is
1(/14+/d—w)|1-4 |
} Fe CES =a dw,
putting w=sin’@ and tan $0 =a, a/a7 =n, we transform this into
pp (1 = x”)
8 | TT EC ee 46),
Jo qd + a’) da ( )
and when @ is an exact submultiple of 7, n is an integer and we
shall be able to evaluate the integral.
The pressure on the part between the same two points is
i | dw\* | dz
4 wey (pets eas
zP ie i ( ] ) i, dw,
or, making the same transformations as before,
rl Ga —_ G7") (1 am x”) % ere
4p |. (+a%) DB QRS E (47).
The above might be applied to find the pressure on the rudder
of a ship when turned obliquely to the length of the ship. The
average pressure, the centre of pressure, and the moment of the
fluid pressures can all be expressed by means of definite integrals
of similar form to the above. If it were worth while tables might
be constructed giving the values of these quantities for any given
inclination of the plane of the rudder to the longitudinal plane of
the ship. As however the motion here considered is in two
dimensions it is unlikely that the formule would yield any result
of use in Navigation.
(4) On thin rotating isotropic disks. By C. Curer, M.A.,
Fellow of King’s College.
The following solution might be shortened by assuming certain
results from a previous paper in the Transactions’. As the
subject appears, however, to be one of considerable practical
_! Vol. xtv. p. 328 et seq.
202 Mr Chree, On thin rotating isotropic disks. [May 4,
importance, I have on the advice of Professor Pearson made the
proof complete in itself.
The term disk is here restricted to mean a thin plate of which
a section parallel to the faces is bounded by a circle or two con-
centric circles. The disk is supposed of uniform density p and
of an isotropic material, for which m and m are the elastic con-
stants in the notation of Thomson and Tait’s Vatural Philosophy.
Taking the axis of the disk for axis of z with the origin O in
the central plane—or plane bisecting the thickness of the disk—
we see from the symmetry that as the disk rotates about its axis
with uniform angular velocity » the displacement at every point
is in the plane through that point and the axis, having for its
components w parallel to the axis and w along the perpendicular
r on the axis directed outwards. At any point 7, z in the disk
the strains are as follows:
Normal strains. Tangential or shearing strain.
SGP aa dé Aelia alatlan J, ] ae
5 2 /W au aw.
ongitudinal —— parallel Oz —— + —— in the plane of zr.
J dz ; dz adr P
1 db
radial a along 7,
u :
transverse — perpendicular to r and z.
*
It is obvious from the symmetry that the two other tangential
strains must vanish.
The expression for the dilatation 6 is
du w dw
ir ems FF wieesha ieee tne tee (1).
The stress system is as follows:
[* =(m—n)6+2n - parallel to oz,
Normal } _ du
stresses | 7r =(m—n) 6+ 2n—— along 7,
dr
$¢=(m—n) d+ 2nu/r perpendicular to Oz and to r;
Shearing {-— : du ~ a Ae
112 = }L ee | 1 .
stress He ae plane 27
The other two shearing stresses must vanish from the symmetry.
1891.] Mr Chree, On thin rotating isotropic disks. 203
We may suppose the disk at rest, acted on by a “centrifugal
force” w*pr per unit of volume. Thus the internal equations are
the following two'
drr drz .tr—$¢., 2
ee | 1 Re ate 2
a ae ae ORR (2),
drz tz . Qzz fs. :
Sopa S| pee oe (3).
Let 2/7 denote the thickness of the disk, a the radius of its outer
a that of its inner cylindrical surface, or edge. Then supposing
the disk exposed to no surface forces, the solution ought to satisfy
the following surface conditions*—
over the flat vom a z L fz = 0 See ee (4),
re SOR (5),
over the edges r=aand r=d’, 2 4 | Sea ee (6),
= Ov eeeelevecssees (7).
Substituting for the stresses their expressions in terms of the
strains and using (1), we easily transform (2) and (8) respectively
into
dé d du dw ae
(meen) omg |e (Te ef an Oph en 8)
dé d du dw
(m+n)1 ae ae & = 4) 4) ee eo Seer (9).
Differentiating (8) with respect to 7 and (9) with respect to z,
then adding and dividing out by (m+ n)7, we get
dé i 1ddé dé _ go 2w"p
dr? rdr aa dz m+n
Of this a particular solution is
d6=—ow pr +2(m-+n).
A complementary solution in ascending powers of r and z with
arbitrary constants can be obtained, as I have shown in a previous
paper’, Of this we require for our present purpose only the
constant term and that of the second degree, or
§=A4+0(2-kr).
1 Pearson’s Elastical Researches of Barré de Saint-Venant, foot-note, p. 79, or
Ibbetson’s Mathematical Theory of Perfectly Elastic Solids, p. 239.
2 Ibbetson’s Mathematical Theory......1. ¢., or Todhunter and Pearson’s History
of Elasticity, Vol. 1. Art. 614.
3 Transactions, Vol. xiv. p. 328 et seq.
204 Mr Chree, On thin rotating rsotropic disks. [May 4,
Putting the right-hand side of (10) zero, it is easily verified
that this is a solution. We thus have
§8=A+C (2 —4$2") —$o'’pr'/(mt n) «0.0.00 (11).
Noticing that
a {,(du_dwy) _,d8_ dtd (, du
dr )' & ~ar/\ dz dz Al dr]
we may transform (9) into
dw ildw dw m ds 2M ny ee
dr r ar a? a Nan tre ee tee eeee (12),
Of this a particular solution is
w=— ali C2.
3n
A complementary solution is easily obtained in ascending powers
of r and z. Of this we require for our present purpose only the
terms of the first and third degrees, which separately satisfy (12)
when the right-hand side is zero, Thus we get
gycus t100
w=a,2+ €, (22° — 32r*) — Sn
where a, and e, are new constants.
Employing (1) and (11), we in like manner easily transform (8)
into
d’u Idu wu , du mdd_ w pr
de 'rdr r 4 dz ndr we
2
m w'p
= pane Serr 14
: ( mM M+ =) ( )
A particular solution is
Dae (ome),
n m+n
For the complementary solution we require only the terms in odd
powers of r and z up to the third degree. Terms in negative
powers of z are of course inadmissible, and 7” is the only negative
power of 7 which satisfies the differential equation. Thus the
complementary solution is
D ) ; An
B= |) ae (2° — 2zr*) + €(42*r — 1°),
where D, a, e«, € are new constants whose coefficients separately
satisfy (14) when its right-hand side is 0. Thus for our complete
solution of (14) we get
m wp
D
u=—-+ar+e (2 — Qzr*) + € (42’r — r*) + : (c —
n We +
) 7.15).
1891.] Mr Chree, On thin rotating isotropic disks. 205
The constants in (11), (15) and (15) are not all arbitrary, being
connected through the identity (1). From it we find
a,=A — 2a,
lm+n +
7 hace ame he
e—u
Thus the solution we have arrived at is
= A+ C0 (2 — 47") — he’pr?/(m +n),
1/m+n , |
33 pias! Beye: 223 — 357°
C2 + a #2 C 8¢)(2 327 ),| (16),
m
3n
w=(A —2a)2-
D " ay fm 2) :
Ree fers a. | _
where all the constants are independent.
To determine the constants we have the surface conditions
(A(T). 4¢
From (4) and the expression for % in terms of the strains we
easily find
(m+n) A —4na+ P {(m +n) C — 16nf}
i lin—n j
2 ( ‘ — — 8, — — 2 — 7
Sp ye 5 (38m +n) C Dan 7 P| = 0--.(17).
Since this holds for all values of 7 between a’ and a, the constant
part and the coefficient of 7° must separately vanish. Thus we
get
(m+n) A —4nat+l {(m +n) C—16nf} =0...... (18),
k 1 Aan) eet :
Sn — g (Bm +n) C— a wp =0 Seles views (19).
From (5) and the expression for 72 in terms of the strains we
at once obtain
es 67 Mee) 1 OA | ei (20).
The equations (18), (19) and (20) are satisfied by
_m+rn
An 7
7 ie Uo (21).
~— 2m(m+ n) |
a *,
ae ~ 32 mn -
WN vw \
\
204 Mr Chree, On thin rotating tsotropic disks. [May 4,
Substit\ting these values in the expressions for the strains
and stressus we find for all values of r and z
zz ='0,
du dw :
anes Baan | eee eee 22
dz r dr p Se
and so rz = 0
Since 72 is everywhere zero, the condition (6) over the edges is
exactly satisfied. The only surface condition left is (7), but this
we cannot exactly satisfy, unless n =n, by means of the present
solution. For, substituting the above values of the arbitrary con-
stants, we obtain from the expression for 7 in terms of the strains
pg =4(3m—n) A —Qna?D
16m °° 4m (m + n)
rr;=q = Similar expression, replacing a@ by @’ ............ (24).
ee is obvious we cannot make these stresses vanish for all values
of z.
If the thickness 27 of the disk be of the same order of mag-
nitude as the radius a this failure renders the present method
inapplicable; but when J/a is small it is easy to obtain a solution
which according to Saint-Venant and other eminent authorities
must be very approximately exact except in the immediate neigh-
bourhood of the edges.
The principle this solution is based on is that of statically
equivalent systems of loading. According to this principle when
a surface of an elastic solid has a small dimension—such as the
thickness of a thin disk—all systems of surface forces which in
their distribution along the small dimension are statically equi-
valent produce, except in the immediate neighbourhood of their
points of application, practically identical strains and stresses.
We may thus for practical purposes replace any system of surface
_ forces over the small dimension by any statically equivalent
system.
For a discussion of this principle and illustrations of its
application, the reader is referred to Saint-Venant’s Théorie de
DL Elasticité...de Clebsch, p. 174 et seq. and p. 727 et seq., also to
i ae Elastical Researches of Barré de Saint-Venant, Arts. 8
and 9,
In my previous treatment of this problem’, which was limited
to a complete disk, I determined the constant A so as to make
' Transactions, Vol. x1v. pp. 334—5, § 76, first two cases; and Quarterly Journal,
Vol, xx. 1889, pp. 24—28.
1891.] Mr Chree, On thin rotating isotropic disks. 207
y= Vanish for a given value of z, viz. either z=0orz=+l. In
either case we are left with a system of unequilibrated normal
forces along each generator of the edge. On this ground Professor
Pearson has recently’ expressed his opinion that my solution
cannot be regarded as “final” even for a thin disk. As the
normal forces in question are of the order of the square of the
thickness of the disk, I am not altogether sure what weight may
be attached to this criticism. In deference however to Professor
Pearson’s opinion, and to what I believe the view Saint-Venant
would have taken, I propose the following method of solution
which removes at least this objection.
It consists in determining A and D from the equations
This still leaves normal stresses of the order of the square of the
thickness over the edges, but the forces along each generator of an
edge form a system in statical equilibrium. Thus according to
the principle of statically equivalent systems, the solution we
shall obtain—which must be strictly limited to thin disks—gives
expressions for the strains and stresses which can differ sensibly
from those supplied by the complete solution only in the im-
‘mediate neighbourhood of the edges.
For the case of a complete disk D must vanish and A is to be
determined by (25).
For the annular disk we find from (25) and (26)
7 a Mm—N 47
~ 8m (3m—n) ACB e As Gm (m ma, pl
7m—-Nn , 4
~ 32mn ” P*
Also from equations (21) we have C' and € determined explicitly,
and a found in terms of A. Thus all the constants of our solution
are determined. For a complete disk we have only to put D=0,
and a’ = 0 in the expression for A in (27).
The physical results attainable from the solution will perhaps
be rendered more practically serviceable by replacing the m, n of
our previous work by Young’s modulus # and Poisson’s ratio 7.
To express the values obtained above for the arbitrary constants
in terms of # and 7 we require the relations
m=ZE/{(1—2m)(L+}J
1 Nature, 1891, p. 488.
eyo}
12
208 Mr Chree, On thin rotating isotropic disks. [May 4,
Substituting the expressions found for the arbitrary constants in
terms of E and n in (16), we find for the strains in an annular
disk
"o(1—2 ; ,
Aen 7) {(3+)(a@+a*)-2(1 +) 1"}
4E
yale Foyt fies 3
ES pS 1) (p _ 304)......(29),
im OP (B+ nla +a”)z—2(1 + 9) r°z}
1
gee Tey 2) nce OE
w= CP =m 4 me ta) r= A=) + 14 MB+m}
a oe nd +7)r (P —32’)...... (31).
From these strains we find for the stresses
a ae . wa”
a ae (3+) \« +q?-Pr— =
:
PA vee (P — 32)......(32),
6
P| wa”) :
a=A Pe ae + (8m T 2 | dui tayd’s spol eilae oe era (33).
For a complete Hea we have
Pp (ih
ae ”) {((3 + n) a’ —2(1+n) 7°}
Cl 29) re
+50 are — 32°)......(34),
(3+) a’z—-2(1+)7r*z
1
- op"
op alta cp
SET [oy FOP (35)
ao . 2 3
= on i —M(B+)ar—(1+n)"}
vi =e 7B 7(l +)r (0 — 82*)......(36),
5 F @ E+ P 2 E
=P 3 +m \(a? =r") + Py ee — Bei (37),
reese Py Lilges ee SEI) PR Ree eo" (38).
1891.] Mr Chree, On thin rotating isotropic disks. 209
For both the annular and the complete disks 2 and 72 are by (22)
everywhere zero. The expressions for the strains and stresses in a
complete disk are correctly deduced from those in an annular
disk by leaving out all terms containing a”. Allowing for the
change of notation, the solution for the displacements in a com-
plete disk differs from my previous one’ only by terms in 77 in
u, 2? in w and [? in 6. Thus it only adds to the strains given by
the previous solution certain constant terms of order /’, and in no
respect modifies the conclusions derivable from that solution as
to the mode in which the strains and stresses alter with the
variables 7, 2.
- The expressions (32) and (53) for the stresses in an annular
disk when terms in 7? and 2 are neglected agree with those
which Professor Ewing’? quotes as obtained by Grossmann’,
They likewise agree with those found by Clerk Maxwell* when
the error in the sign of his equation (59) pointed out by
Mr J. T. Nicolson’ is corrected. The expression (31) for the
radial displacement when terms of order /* are neglected is iden-
tical with that given implicitly or explicitly by Maxwell, and by
Grossmann putting his V,=0, and to the same degree of approxi-
mation (30) coincides with the value for the longitudinal displace-
ment to which Maxwell’s theory would lead if fully worked out.
I shall thus for brevity speak of the expressions our solution
supplies both for the complete and annular disks when terms of
order [ are neglected as constituting the Maawell solution.
The conclusion we are led to is that the methods of Maxwell
and Grossmann—which seem practically identical—while involving
inconsistencies® and certainly inconclusive from a strict theoretical
standpoint, perhaps even “paradoxical” as Professor Pearson’
states, yet lead to results which if the present investigation can
be trusted are sufficiently exact for practical purposes so long as
the disk is very thin.
From (33) it is obvious that $$ is everywhere greater than 7
in an annular disk. The same result follows from (38) for a com-
plete disk, except at the axis where the two stresses are equal.
1 Quarterly Journal of Pure and Applied Mathematics, Vol. xx111. 1889, Equa-
tions (129), p. 28.
2 Nature, 1891, p. 462.
3 Verhandlungen des Vereins zur Befirderung des Gewerbfleisses, Berlin, 1883,
pp. 216—226.
+ Transactions of the Royal Society of Edinburgh, Vol. xx. Part 1., 1853, pp.
111—112; or Scientific Papers, Vol. 1. p. 61. For corrections to Maxwell’s second
equation (57) see Todhunter and Pearson’s History of Elasticity, Vol. 1. ft.-note,
. 827.
ris Nature, 1891, p. 514.
6 They lead to results inconsistent with one or both of the original assumptions,
viz. that 72 is everywhere zero, and that Zz if not also zero is independent of r and 2,
7 Nature, 1891, p. 488,
210 Mr Chree, On thin rotating isetropic disks. [May 4,
Also 2 and 72 vanish at every point, thus both in the annular
and in the complete disk 4 is everywhere the stress-difference
and w/7 the greatest strain. Both quantities for any given value
of 7 are greatest when z=0, and for any given value of z are
greatest when r=0 for the complete disk, or r=a’ for the
annular. They are thus according to the solution greatest in
the central plane, at the centre of a complete disk and at the
inner edge of an annular.
According, however, to the principle of statically equivalent
surface forces our solution does not strictly apply for values of r
which differ from a or from a’ by quantities which do not exceed
several multiples of /. In other words, it possibly may give values
for the strains and stresses over the edges differing from the true »
values by terms of the order /’. Thus in determining the greatest
values of the stress-difference or greatest strain, which occur
at or immediately adjacent to the inner edge in an annular
disk, we are not warranted in retaining terms of this order of
small quantities. I thus propose in determining these quantities
to neglect terms of this order as being of doubtful accuracy, at
least in an annular disk, and of insignificant magnitude im any
thin disk. It should be noticed, however, that our complete
solution gives at all radial distances larger values for the strains
and stresses in the central plane than when terms in [’ are neg-
lected. Thus it would certainly only be prudent to regard the
values we are about to find from the Maxwell solution for the
maximum stress-difference and greatest strain as minima, which
in all probability are exceeded in any actual case. In a thin
disk, however, the true values can exceed these only by small
terms, of order (//a)’ at least.
Neglecting then terms in /? and z’, we find for the maximum
stress-difference S and the largest value of the greatest strain 5—
S, =d0'pa' (3 +'n).. se (39),
Hs = (1 = 9) 8, 0s: ee (40),
In an annular disk, S, = #3, = }w’p {a* (3+ )+a"(1—7)}...(41).
Since S,= Hs, the maximum stress-ditference and greatest strain
theories lead to identically the same result for the so-called
“tendency to rupture ”—i.e. approach to limit of linear elasticity
—in the annular disk. In the complete disk the maximum stress-
difference theory assigns for all possible values of , except 0,
a lower limit than the other for the safe velocity of rotation.
In a complete disk,
Supposing », and w, the limiting safe angular velocities in a
complete and in an annular disk of the same material and external
radius, then putting in succession S, = S, and 3, =3,, we find
;
1891.] Mr Chree, On thin rotating isotropic disks. 211
On the maximum stress-difference theory
w/o, =2(1+5—75) pens (42),
On the greatest strain theory
eas laa let
2) ae = ny
@,/@, =2 ‘TVS (CES ee ee
When (a’/a)’ is negligible, or there is only a very small axial hole,
these give respectively
w,/@, = /2 for all values of »,
o,/o, = /2/(1 — n), or nearly 1°633 for n =°25.
The former result was given by Professor Ewing in Nature, and
he also directed attention to the great diminution it represents
in the strength of a disk due to the removal of a small axial
core. The effect is even more striking on the greatest strain
theory for ordinary values of 7.
Since the striking character of this result may arouse doubts
in some minds as to the validity of any investigation which leads
to it, I would point out that it is not an isolated fact. The
/ removal of a central spherical core of any radius however small
from a sphere rotating about a diameter has, as I have shown
in a previous paper’, a precisely similar effect, increasing very
largely the greatest values both of the stress-difference and great-
est strain. The same result also follows the removal of a thin
axial core from a rotating right circular cylinder whose length
is constrained to remain constant”.
In discussing the nature of the strains and stresses we may
for most purposes leave out of account in the first place terms of
order /* or 1’, regarding them in the light of small corrections to
the principal terms.
According to the Maxwell solution, every originally plane
section parallel to the faces of a complete or annular disk ap-
proaches at every point the central section, z=0, and assumes
the form of a paraboloid of revolution about the axis of rotation.
In this respect the phenomena are precisely similar to those
presented aby a flat oblate spheroid rotating about its axis of
symmetry *.
The latus rectum of the paraboloid into which is transformed
1 Transactions, Vol. xiv. pp. 467—83. See Tables IV. and VIII. and their
discussion.
2 Transactions, Vol. xtv. p. 339.
3 Transactions, Vol. xv. pp. 10—13.
VOL. VII. PT. IV. LZ
212 Mr Chree, On thin rotating isotropic disks. [May 4,
an originally plane section at distance z from the central plane is
in both the complete and annular disks
2E ~ {n(1+ ) w’pz}.
It thus depends merely on the angular velocity, the nature of the
material and the original distance from the central plane. Its
reciprocal, and so the curvature at the vertex of the paraboloid,
varies directly as the square of the angular velocity and as the
distance from the central plane.
The amount by which the axial point on an originally plane
section parallel to the faces—of course an imaginary point in an
annular disk—approaches the central plane varies as the square
of the radius of the complete disk. The corresponding approach
in an annular disk varies as the sum of the squares of the radii
of its edges, and is greater than in a complete disk of the
same external radius. The magnitudes of the reductions in the
thickness, 2/, of the disks will Pe seen from the following data—
(x inner edge ow ply (3 +) a’+(1—7) a*},
‘ 2
In annular disk
[at outer edge 2 *ely {(38 +) a*+(1—7) a},
at axis = at (3+) a,
In complete disk
at outer edge =, 5 = o'pln(1 — m) a’.
The terms in z/’ and 2 in (30) and (35) cut out when z= +1],
so that as regards the preceding results as to the change of thick-
ness there is an exact agreement between the Maxwell solution
and the more complete solution.
It will be noticed that the reduction in thickness at the inner
edge of an annular disk equals the reduction at the axis of a
complete disk equal in radius to its outer edge, together with the
reduction at the rim of a complete disk equal in radius to its
inner edge, the thicknesses, materials and angular velocities being
the same in each case. Also the reduction in thickness at the
outer edge of an annular disk exceeds what it would be if the
disk were complete by the reduction at the axis of a complete
disk equal in radius to its inner edge.
The longitudinal strain dw/dz parallel to the axis is a com-
pression at every point, unless 7 =0, both in the complete and
annular disks, and diminishes numerically as 7 increases. For
ordinary materials it is a quantity of the same order of magnitude
as the radial and transverse strains, but it vanishes throughout if
n= 0.
_
1891.] Mr Chree, On thin rotating isotropic disks. 213
The terms in zl? and 2 in w would indicate that the longi-
tudinal compression is somewhat greater near the central plane
and somewhat less near the faces of the disk than according to
the Maxwell solution.
For a first approximation confining our attention to the Max-
well solution in (31) and (36), we see that every point in the
disk, whether complete or annular, increases its distance from the
axis, and the transverse strain w/r is thus everywhere an extension.
In the complete disk the radial strain is an extension inside
and a compression outside of a cylindrical surface co-axial with
the disk, and of radius 7,, given by
1 =AJSE(B + AEM) -reeeeereeeeees (44).
This gives a value for 7, less than a for all possible values of 7
except 0. Any annulus of the disk increases or diminishes in radial
thickness according as it lies inside or outside this surface.
In the annular disk the increases da and oa’ in the radii of
“a edges, and d(a—a’) in the radial thickness, a—a’, are given
7
2
a= {(l—n)@4+(38+7)a" 2.0.00... (45),
2 /
aa’ = iS 9)? 9). 2 (46),
2 ¢ /
d(a-—a’)= a fa-a')y—n(at+ay}...(47).
Thus the radial thickness is increased or diminished according as
@/a@< or > (1 — 4/9) = (1 4 4/9). «sneer... (48).
For the ratio of the radii in the annular disk whose radial thick-
ness is unaltered, we find
7 = 0, i/a=1;
ae. aja ="3,
n = °36, a /a = °25,
n='5, w/a = "1716 approx.
The radial strain given by the Maxwell solution in (31) is
2
2s = i POLITE BIBL Puy ds
where f(r) =(1—7)(8+)(@+a7)—3(1—7*) 7°
—(1l+)(84+)@a°?r™...(50).
Since J (a) == 2y {A -—) a +3B4+n) a",
and f(@) =— 2m (l—n)a*+(3+n) a),
17—2
214 Mr Chree, On thin rotating isotropic disks. [May 4,
we see that the radial strain is a compression at both edges for
all ratios of a’: a, and for all possible values of » except 0. For
n = 0,
A= ~ (7? — a*)(a* — 97°), 3... sane (51),
and so the radial strain is for all ratios of a’:a@ an extension
everywhere except at the edges, where it vanishes.
For all other values of 7 there are always portions of the
disk immediately adjacent to the edges wherein the radial strain
is a compression, and it is easily proved that the radial strain is
everywhere a compression when
a’/a>/(1 = )(3 +9) + {88 (1 + 9) + 4 (2 + m)}..-(52).
An idea of the nature of the radial strain under various conditions
for the value ‘25 of » may be derived from the following table :
TABLE I.
du
Sign of aie and loci where it vanishes for » = ‘25.
| du |
dr |
Value of a’/a 0, a+ oe
rja="l 130 927 Ps)
2 2 264 912 |!
3 3 409 882 i
+ 4 597 806 1
> 426 Everywhere a compression.
From the Maxwell solution in (32) and (37) it is obvious that the
radial stress is a traction for all possible values of 7 at all points
not on the edge or edges of the disk.
The terms in rl? and rz* in (31)—(33) and (86)—(38) show
that the complete solution gives algebraically greater values for
the radial and transverse displacements, strains and stresses at
all axial distances in the central plane than the Maxwell solution.
It will be noticed, however, that the mean of each of these
1891.] Mr Chree, On thin rotating isotropic disks. 215
quantities taken between the limits —/ and +1 of z is the same
as when terms of order 7’ are neglected. Thus the Maxwell-
Grossmann method of solution leads to values for all the radial
and transverse strains and stresses which are identical with those
which the present solution supplies for mean values taken through-
out the thickness of the disk. It also as we have seen when
fully worked out supplies the same value for w at the plane
surfaces of the disk, and so the same mean value for the longitu-
dinal compression throughout the thickness.
In consequence of the second relation (22) the mutual inclina-
tions of all material lines in the disk remain unchanged. Thus
what were originally cylindrical surfaces co-axial with the original
edges cut orthogonally the surfaces into which have been trans-
formed what were originally planes parallel to the faces; i.e. they
become orthogonal to what are practically a series of paraboloids,
whose common axis is that about which the rotation takes place.
This perhaps will convey the clearest idea of how material lines
originally perpendicular to the faces become under rotation con-
cave to the axis of the disk.
May 18, 1891.
PROFESSOR LIVEING, VICE-PRESIDENT, IN THE CHAIR.
The following Communications were made to the Society:
(1) On Parasitic Mollusca. By A. H. Cooke, M.A., King’s
y. §
College.
[Received July 18, 1891.]
Various grades of parasitism occur among the Mollusca, from
the true parasite, living and nourishing itself on the tissues and
secretions of its host, to simple cases of commensalism. Some
authors have divided these forms into endo- and ecto-parasites,
according as they live inside or outside of their host. Such a
division, however, is hardly tenable. Certain forms are indif-
ferently endo- and ecto-parasitical, while others are ecto-parasitic
in the young form, and become endo-parasitic in the adult. It
will be convenient therefore, simply to group the ditferent forms
according to the home on which they find a lodgement.
; On Celenterata. (a) Sponges. Vulsella and Crenatula almost
invariably occur in large masses of irregular shape, boring into
sponges. (b) Corals. These form a favourite home of many
species, amongst which are several forms of Coralliophila, Rhizo-
chilus, Leptoconcha, and Sistrum. The common Magilus, from the
Red Sea and Indian Ocean, in the young form is shaped like a
216 Mr Cooke, On Parasitic Mollusea. [May 18,
small Buccinum. As the coral (Meandrina) to which it attaches
itself grows, it develops at the mouth a long calcareous tube, the
aperture of which keeps pace with the growth of the coral,
and prevents the mollusc from being entombed. The animal
lives at the free end of the tube, and is thus continually shifting
its position, while the space it abandons becomes completely
closed by calcareous matter. Certain species of Ovula inhabit
(orgonae, assuming the colour, yellow or red, of their host, and,
in certain cases, developing for prehensile purposes a pointed
extension of the two extremities of the shell. Pedicularia in-
habits the common WMelithaea rubra of the Mediterranean, and
another species has been noticed by Graeffe’ on J. ochracea in
Fiji.
; On Echinodermata. (a) Crinoidea. Stilina comatulicola lives
on Comatula mediterranea, fixed to the outer skin, which it pene-
trates by a very long proboscis; the shell is quite transparent’.
A curious case of a fossil parasite has been noticed by Roberts®.
A Calytraea-shaped shell named Platyceras always occurred on
the ventral side of a crinoid, encompassed by the arms. For
some time this was thought to afford conclusive proof of the
rapacity and carnivorous habits of the echinoderm, which had
died in the act of seizing its prey. - Subsequent investigations,
however, showed that in all the cases noticed (about 150) the
Platyceras covered the anal opening of the crinoid in such a way
that the mouth of the mollusc must have been directly over the
orifice of the anus. (b) Asteroidea. The comparatively soft tex-
ture of the skin of the starfishes renders them a favourite home
of various parasites. The brothers Sarasin noticed* a species of
Stilifer encysted on the rays of Linckia multiformis. Each shell
was enveloped up to the apex, which just projected from a
hole at the top of the cyst. The proboscis was long, and at its
base was a kind of false mantle, which appeared to possess a
pumping action. On the under side of the rays of the same
starfish occurred a capuliform molluse (Thyca ectoconcha), fur-
pished with a muscular plate, whose cuticular surface was indented
in such a way as to grip the skin of the Linckia. This plate
was furnished with a hole, through which the pharynx pro-
jected into the texture of the starfish, acting as a proboscis and
apparently furnished with a kind of pumping or sucking action.
Adams and Reeve’ describe Pileopsis astericola as living ‘on the
tubercle of a starfish, and Stilifer astericola, from the coast of ~
1 Described as a Cypraea, but no doubt an Ovula or Pedicularia: C. B. Bakt.
Par. v. 543.
2 Von Graff, Z. Wiss. Zool. xxv. 124.
3 Proc. Amer, Phil. Soc, xxv. 231.
4 Ergeb. naturw. Forsch. Ceylon, abstr. in J. R. MW. S. (2) vr. 412.
5 Voyage of the Samarang, Moll. p. 69, Pl. x1. f. 1; p. 47, Pl. xvi. f. 5.
1891.] Mr Cooke, On Parasitic Mollusca. 217
Borneo, as ‘living in the body of a starfish! In the British
Museum there is a specimen of Pileopsis crystallina in situ on
the ray of a starfish. On the brittle starfishes (Ophiwridae) occur
several species of Stiliferina. (c) Echinoidea. Various species of
Stilifer occur on the ventral spines of echinoderms, and are some-
times found imbedded in the spines themselves. St. Turtoni
occurs on the British coasts on several species of Hchini, and
Montacuta substriata frequents Spatangus purpureus and certain
species of Amplidetus, Cidaris and Brissus. Lepton parasiticum
has been described from Kerguelen I. on a Hemuaster, and a new
genus, Robillardia, has recently been established? for a Hyalinia-
shaped shell, parasitic on an Hchinus trom Mauritius. (d) Holo-
thuroidea. The ‘sea-cucumbers’ afford lodgement to a variety
of curious forms, some of which have experienced such modi-
fications that their generic position is by no means established.
Entoconcha occurs fixed by its buccal end to the blood-vessels of
certain Synaptae in the Mediterranean and the Philippines. /nto-
colax has been dredged from 180 fath. in Behring’s Straits, attached
by its head to certain anterior muscles of a Myriotrochus*. A
curious case of parasitism is described by Voeltzkow® as occurring
on a Synapta found between tide-marks on the I. of Zanzibar.
In the oesophagus of the Synapta was found a small bivalve
(Entovalva), the animal of which was very large for its shell, and
almost entirely enveloped the valves by the mantle. As many as
five specimens occurred on a single Synapta. In the gut of the
same holothurian lived a small univalve, not creeping freely, but
fixed to a portion of the stomach wall by a very long proboscis
which pierced through it into the body cavity. This proboscis
was nearly three times as long as the animal, and the forward
portion of it was set with sharp thorns, no doubt to make it to
retain its hold and resist evacuation. Various species of Hulima
have been noticed in every part of the world, from Norway to
the Philippines, both imside and outside Holothurians*. Stilifer
also occurs on this section of echinoderms’.
On Annelida. Cochliolepas parasiticus has been noticed under
the scales of Acoetes lwpina (a kind of ‘ sea-mouse’) in Charleston
Harbour’.
On Crustacea. A mussel, 2in. long, has been found’ living
under the carapace of the common shore-crab (Carcinus maenas),
but this is not so much a case of parasitism as of involuntary
1 KE. A. Smith, dnn. Mag. Nat. Hist. 1889 (i), 270.
2 Journ. de Conch. (8) xxrx. 101.
3 Zool. Jahrb. Abth. f. Syst. v. 619.
+ See especially Semper, Animal Life, Ed. 1, p. 351.
5 Gould, Moll. of U.S. expl. exped. 1852, p. 207 (St. acicula, from Fiji).
6 Stimpson, Proc. Bost. Soc. N. H. v1., 1858, p. 308.
? Pidgeon, Nature, xxx1x. p. 127.
218 Mr Cooke, On Parasitic Mollusca. [May 18,
habitat, the mussel no doubt having become involved im the
branchiae of the crab in the larval form.
On Mollusca. A species of Odostomia (pallida Mont.) is found
on our own coasts on the ‘ears’ of Pecten maaimus, and also’ on the
operculum of Turritella communis. At Panama the present writer
found Crepidula (2 sp.) plentiful on the opercula of the great
Strombus galea and of Cerithium trroratum. Amalthea is very
commonly found in Conus, Turbo, and other large-sized shells, but
this is probably not a case of parasitism, but simply of con-
venience of habitat, just as young oysters are frequently seen on
the carapace and even on the legs of large crabs.
On Tunicata. Lamellaria is said to deposit its eggs on an
Ascidian (Leptoclinum), and the common Modiolaria marmorata
lives in colonies imbedded in the tegument of Asczdia mentula
and other simple Ascidians.
Special points of interest with regard to parasitic mollusca
relate to (1) Colowr. This is in most cases absent, the shell being
of a uniform hyaline or milky white. This may be due, in the
case of the endo-parasitic forms, to absence of light, and possibly,
in those living outside their host, to some deficiency in the
nutritive material. A colourless shell is not necessarily pro-
tective, for though a transparent shell might evade detection,
a milk-white hue would probably be conspicuous. (2) Modifica-
tions of structwre. These are in many cases considerable. Hnto-
concha and Entocolaz have no respiratory or circulatory organs
and no nervous system; Thyca and certain Stilifert possess a
curious suctorial apparatus; the foot in many cases has aborted,
since the necessity for locomotion is reduced to a minimum’, and
its place is supplied by an enormous development of the proboscis,
which enables the creature to provide itself with nutriment with-
out shifting its position. Special provision for holding on is
noticed in certain cases, reminding us of similar provision in
human parasites. Kyes are frequently, but not always wanting,
even in endo-parasitic forms. A specially interesting modification
of structure occurs in (3) the Radula. In most cases (Hulima,
Stilifer, Odostomia, Entoconcha, Entocolaux, Magilus, Coralliophila,
Leptoconcha) it is absent altogether. In Ovula and Pedicularia,
genera which are in all other respects closely allied to Cypraea,
the radula exhibits marked differences from the typical radula of
the Cypraeidae. The formula (3°1°3) remains the same, but the
laterals are greatly produced and become fimbriated, sometimes
at the extremity only, sometimes along the whole length. A
1 Smart, Jowrnal of Conch. v. 152.
* Semper notices a case where a Eulima whose habitat is the stomach of a Holo-
thurian retains the foot unmodified, while a species occurring on the outer skin, but
provided with a long proboscis, has lost its foot altogether.
1891.] Mr Cooke, On Parasitic Mollusca. 219
very similar modification occurs in the radula of Sistrum spectrum
Reeve, a species which is known to live parasitically on one of
the branching corals. Here the laterals differ from those of the
typical Purpwridae in being very long and curved at the ex-
tremity. The general effect of these modifications appears to
be the production of a radula rather of the type of the vegetable-
feeding Trochidae, which may perhaps be regarded as a.link in
the chain of gradually degraded forms which eventually terminate
in the absence of the organ altogether. The softer the food, the
less necessity there is for strong teeth to tear it; the teeth either
become smaller and more numerous, or else longer and more
slender, and eventually pass away altogether. It is curious, how-
ever, that the same modified form of radula should appear in
species of Ovula (e.g. ovwm), and that the same absence of radula
should occur in species of Hulima (e.g. polita) known not to be
parasitical. This fact perhaps points back to a time when the
ancestral forms of each group were parasitical and whose radulae
were modified or wanting, the modification or absence of that
organ being continued in some of their non-parasitical descendants.
(2) Exhibition of models of double supernumerary appendages
in Insects: also of a mechanical method of demonstrating the
system upon which the Symmetry of such appendages is usually
arranged. By W. Bateson, M.A., St John’s College.
(3) On the nature of the excretory processes in Marine Polyzoa.
By S. F. Harmer, M.A., King’s College.
[Abstract: reprinted from the Cambridge University Reporter, May 26, 1891.]
This communication was the result of an occupation of a
University table at the Zoological Station at Naples during the
Easter Vacation of 1891.
Observations were made on the manner in which various
artificial pigments were excreted in Bugula and in Flustra, on
the lines adopted by Kowalevsky (Biolog. Centralblatt, 1x., 1889—
1890, pp. 33 etc.) for other Invertebrates. The general result of
the experiments was to show that excretion is not performed by
organs comparable with nephridia, but that this process is carried
on by free mesoderm cells, and to some extent by the connective
tissue and by the walls of the alimentar y canal. Evidence was
obtained to show that the periodic loss of the alimentary canals
leading to the formation of the “brown bodies” may be regarded
as, to some extent, an excretory process.
220 Mr Brown, On the part of the parallactic [June 1,
June 1, 1891.
Pror. G. H. DARWIN, PRESIDENT, IN THE CHAIR.
The following communications were made to the Society:
(1) On the part of the parallactie class of inequalities in the
moon’s motion, which is a function of the ratio of the mean motions
of the sun and moon. By Ernest W. Brown, M.A., Fellow of
Christ’s College.
In a paper to be published shortly, a solution will be given by
approximation in series, of the equations for this class of in-
equalities. In Vol. 1. of the American Journal of Mathematics,
Mr G. W. Hill has shown that by using rectangular instead of
polar co-ordinates, the inequalities depending only on the mean
motions of the sun and moon can be found to a high degree of
accuracy with comparatively little trouble; and that the series to
be obtained may be rendered more convergent by developing in
terms of n’/(n—n’)=m’, instead of n’/n=m as has been done
in most of the previous theories. He further shows that by
developing in terms of m’‘/(1—4m’)=yp a still greater degree of
convergency is obtained. The results are expressed in rectangular
co-ordinates and on that account are not convenient for obtaining
the algebraical expressions of the longitude and parallax of the
moon. But these transformations will still have force when we
change to polar co-ordinates, so that by putting in Delaunay’s
series for this class of inequalities m= m’'/(1+m’)=p/(1+ 4p)
and expanding in powers of m’ or w we should get a better ap-
proximation to the truth. This is not necessary in the Variation
Inequality which Delaunay has found with sufficient accuracy for
practical purposes. But in the Parallactic Inequality he stops at
(a/a').m', the numerical multiplier of which is roughly 55113;
the term expressed in seconds is 0/"38. By substituting
m=im'/(1 +m’)
and developing in terms of m’, this multiplier is reduced to about
one half its former value; but since m’ is nearly one-twelfth
greater than m the accuracy is not increased, though the new
series has greater convergency.
On using Hill’s method with rectangular co-ordinates for the
parallactic class of inequalities (i.e. those dependent on the ratio
of the mean distances of the moon and sun) a factor whose value
is 1/(1—4m’/—...) appears, and to the expansion of this factor in
powers of m’ is “due the slow convergence of the series giving the
.
1891.] class of inequalities in the moon's motion. 221
coefficient of the parallactic inequality. Delaunay’s series for this
ee is*
ae 93 6887 73 4 137197 pt 4 4628333 5 4 63106813 76
= [y mm + 23m? + 888i m m* + 4828333 n° + 83106813 mn
a ie
"86 071
+ LON sp oe LSS I
‘O” 38
where the coefficients expressed in seconds of are are written
below. In this expression put m=m'/(1+m’) and expand in
powers of m’, subtract from the result the expansion of
(43m’ + 2m”)/(1 — 4m’)
which are the first two terms found by rectangular co-ordinates)
iy g
in powers of m’ and we obtain instead of Delaunay’s series the
expression
Deira 34°33 1189 4°43
a :
ina 82 Dian l2\ [6 __ IN 2298 Ade 8423 y)/4 273317»
> [4b m’ + 2m™)/(1 — 4m’) — $44? — 8423 4 4 2238317 m
—118"°25 —11"°47 +1783 + ()"-37 = 0716
— 3140657 »,,/6 38320080247,,,/7
sane me + sae ee Mm |.
+ 002 =i 30)
Newcomb’* suggests that the last term of Delaunay’s expression
is wrong and this expression which is deduced from Delaunay’s
would seem to support that view. I hope however to verify the
term. Leaving this term out of consideration the increased con-
vergency of the series is manifest. The first three terms give
nearly the whole value of the coefficient. In the paper the other
coefficients of the periodic inequalities of this class will be dealt
with in a similar manner, and expansions will be given for the
whole class of inequalities, the factor 1/(1 — 4m’ — mi being intro-
duced also into the higher powers of m’. It will then not be
necessary to go further than m’a/a’ to get the values of the
coefficients correct to one hundredth of a second of arc; and, for
this degree of accuracy, by the methods given the approximations,
either for algebraical or numerical results, are not long.
(2) On Pascal’s Hexagram. By H. W. RicuMonp.
The author applies Cremona’s method of deriving the hexa-
gram by projection of the lines on a nodal cubic surface from the
node. By use of a new form of the equation to this surface the
equations of the lines are obtained in a perfectly symmetrical
form, and their properties thence developed.
1 Mémoires de Vv Académie des Sciences, Tome xxix. p. 847.
2 Astronomical Papers for use of American Ephemiris, Vol. t. pt. u. p. 71.
222 Mr Chree, On some experiments on [June 1,
(3) A Linkage igi describing Lemniscates and other Inverses
of Conic Sections. By R. 8S. COLE.
(4) Some experiments on liquid electrodes in vacuum tubes.
By C. CureE, M.A., Fellow of King’s College.
The experiments discussed in this paper were undertaken
at the suggestion of Professor J. J. Thomson, in order to throw
further light on the nature of the electric discharge in a vacuum
tube with liquid electrodes. To render the work intelligible it
is necessary to give a brief sketch of the general nature of the
discharge at low gaseous pressures, and to mention certain results
of previous observers and certain of their theoretical conclusions.
At low pressures the phenomena at the cathode—or electrode
to which the positive current travels in the tube—are the most
striking, so that it is the most convenient point of departure.
The phenomena ordinarily observed between the electrodes when
not too near together are as follows :—
(A) a thin luminous envelope covering the cathode,
(B) a much thicker, but still except at very low preseam
short dark space,
(C) a bright, usually blue, space of considerable length,
(D) a second dark space,
(E) a more or less luminous interval extending to the anode.
There is unfortunately no universally accepted English termi-
nology for these spaces. The following table shows some of the
terms most commonly used in English, and also the ordinary
German terminology.
TABLE I.
Gee Solaire SPOTTISWwooDE | ORDINARY
: AND Movuton | USAGE
|
(A) | Lichtsaum Narrow layer |
| | |
(B) dunkle Kathodenraum Dark space Crookes’ space | Crookes’ space
Glimmlicht
(C) | \helle Kathodenschicht | Glow proper
(Glimmlichtstrahlen |
Negative glow | |
Negative haze |
Negative dark
space Faraday space
dunkle Trennungsraum | Dark interval
|_—- r
|
(E) positive Licht | Positive light | Positive light |
1891.] liquid electrodes in vacuum tubes. 223
(A), (B) and (C) are regarded as forming the negative light
or discharge, and (D) as separating the negative and positive
discharges. (A) is very inconspicuous, if actually existent, at
gaseous pressures exceeding 1 or 2 mm. of mercury, but at very
low pressures it is fairly bright though very thin.
(B) is also insignificant so long as the pressure exceeds a few
mm. of mercury, but at very low exhaustions it has been ob-
served to exceed a length of 2cm. Its length has been shown
not to depend much on the material of the cathode when metallic.
It is also usually but little dependent on the strength of the
current. It varies to a considerable extent with the nature
of the gas, being according to Professor Crookes’ decidedly longer
in hydrogen and shorter in carbonic acid gas than in air. In
any one gas it is supposed to increase in length as the pressure
is reduced, so that its magnitude gives a useful if not very exact
indication of the degree of exhaustion®*. The following table
gives some of the measurements of Crookes*—altered to mm.—
and Puluj* for air vacua.
TABLE II.
Crookes. Puluj.
| Se a a ee
Pressure in mm. Length of (B) in Pressureinmm. Length of (B) in
of Hg. mm. of Hg. mm.
313 6°5 1°46 2°5
123 8°5 66 45
‘078 12:0 "30 78
042 15°0 ‘24 9°5
‘020 25 ‘16 140
‘06 22°0
For a given length of (B) the pressures found by Crookes
are very considerably less than those found by Puluj, a dis-
crepancy ascribable perhaps to differences between their tubes
and cathodes but due probably in greater measure to the un-
certainties attending the determination of such low pressures.
The conditions of Puluj’s experiments resembled more closely
those of the present paper than did Crookes’, so the former's
results seem a priort the best for comparison with those de-
scribed here. It must, however, be remembered that at the
lowest pressures in my experiments, the gas in the tube was
doubtless in great measure, if not almost exclusively, vapour
from the liquid electrodes. Though not absolutely black, (B) in
1 Phil. Trans. 1879, pp. 138—9.
2 Phil. Trans. 1879, p. 137.
3 Phil. Trans. 1879, pp. 158—9.
4 Sitzungsberichte Math. Nat. Classe der k. Akad., Bd. uxxx1., Abth. 1. Wien,
1880, p. 874. :
224 Mr Chree, On some experiments on [June 1,
general seems so to the eye, and the surface separating it from (C)
is usually sharply defined.
(C) has its brightest side next the cathode, and it is by some
writers divided into a brighter portion, the negative glow, and
a less bright portion, the negative haze. (C) appears almost as
soon as the pressure is sufficiently reduced to allow the dis-
charge to pass. It increases in length’ as the pressure is re-
duced within at least certain limits. The transition from (C)
to (D) is usually so gradual no exact line of separation can be
drawn. (D) is not always visible. Its presence depends to a
considerable extent on the strength of the discharge. At low
pressures an increase in the strength of discharge tends to make
(D) contract *. At high pressures when the discharge first passes
(E) consists of a succession of twig-like independent discharges,
which, as the pressure is lowered, transform into what seems to
the eye a tolerably uniform column whose colour in most gases,
especially in air, is bright red. As the exhaustion proceeds the
colour becomes less bright, and the apparent uniformity of the
column tends to disappear. At moderate pressures such as 1 or
2mm. of mercury with a regular source of current, (E) consists
in general of a succession of similar units, striae, each having a
sharply defined luminous head on the side next the cathode, and
gradually fading on the side next the anode into a seemingly
non-luminous portion. At very low pressures the striae are
sometimes very faintly defined, if existent, and (E) appears as
a hazy luminosity, generally of a blue tint.
Taking the simplest case, viz. a long, uniform, straight cylin-
drical tube with flat metal electrodes, whose planes are per-
pendicular to the axis of the tube, and whose diameters are
not very small, each stria in the positive column has in ana-
tomical language an opisthocoelous form, the convex surface being
that of the luminous head. This surface is, however, in general
of smaller curvature than a sphere of diameter equal to that
of the tube. Whether the positive column be striated or not,
the surface separating it from (D) has its convexity on the side
next the cathode. This surface is usually sharply defined when
(D) appears at all.
_ There is another phenomenon whose relation to the discharge
is somewhat doubtful, viz. the phosphorescence observed in good
vacua. The more brilliant phosphorescent phenomena are
beautifully shown by many well-known experiments of Professor
Crookes. He believes the “molecular streams” or “radiant
matter ’—German “ Kathodenstrahlen”—whose incidence on the
* After completely covering a wire cathode it expands as the current increases.
See Hittorf, Wied. Ann. 20, 1883, p- 746,
° Hittorf, 1. c. pp. 736—7.
1891.] liquid electrodes in vacuum tubes. 225
glass of the tube sets up this luminosity, to be gas molecules
charged at the cathode and projected with great velocity at
right angles to its surface. According to Goldstein and others
the direction of projection varies to some extent from the
normal. Professor Crookes* appears to have originally thought
these molecular streams peculiar to very low vacua and indicative
of a fourth state of matter, bearing to the gaseous state some-
what the same relation as it bears to the liquid. Messrs Spottis-
woode and Moulton” have, however, shown that phosphorescence
can be produced at quite high pressures provided the intensity
of the negative discharge be sufficiently increased.
These latter observers’ have treated with great fulness other
less conspicuous phosphorescent effects. They found that under
certain conditions a portion of the wall of a vacuum tube touched
by the finger or having an earthed conductor in its neighbour-
hood acts as a sort of secondary cathode, setting up phosphor-
escence on the opposite side of the tube. They also found that
at very low pressures the positive discharge when in the form
of a hazy luminous column, occupying in general only a portion
of the tube’s cross-section, creates a sort of demand for negative
electricity which may be supplied by a discharge proceeding from
the walls of the tube in directions at right angles to its length
and creating phosphorescence. Further when the positive column
is cut at an angle by the wall of the tube, as at a sharp bend,
phosphorescence appears whose position is as if it were due to
the impact of molecules travelling along the positive column
in the direction from cathode to anode. This latter phenomenon
has been more exhaustively treated by Goldstein*. He found
that at very low pressures in a tube bent at right angles any
number of times, there is at every bend between the cathode
and anode where the positive column extends, a patch of phos-
phorescence, situated as if due to rays travelling from the cathode
to the anode. Supposing an electrode at A in a tube AD at
right angles to a tube BDL, and that the latter tube is closed
at the end B, while a third tube at right angles to it leads
from £ to an electrode C, then according to Goldstein, if I in-
terpret him correctly, there is phosphorescence at # whether
the cathode be A or CU, but none at B unless C is cathode.
Goldstein’ also describes the production of phosphorescence
at positions which could not be reached by molecular streams
travelling in straight lines from the cathode, which he apparently
1 Phil. Trans. 1879, pp. 142, 143, 163, 164, ete.
2 Phil. Trans. 1880, pp. 582—6, etc.
3 1. ¢., pp. 602—6, 616—7, 620, ete.
4 Wied. Ann. 12, 1881, pp. 104—49, and figs. 16 and 17, Taf. 1.
5 Wied. Ann, 15, 1882, pp. 246—254,
226 Mr Chree, On some experiments on [June 1,
ascribes to a species of diffuse reflexion at a bend in a narrow
part of the tube. Perhaps this is immediately connected with
another phenomenon he observed’, viz. that if the vacuum tube
be of variable section then a place where in passing from cathode
to anode there is a sudden large increase in the diameter takes
upon itself the functions of a secondary cathode, dominating the
character of the striae on the side next the anode and emitting
molecular streams along the axis of the tube.
The view that phosphorescence is due to the impact of finely
divided matter torn off the cathode by the current has been
maintained by Puluj* and others, but it seems obviously in-
applicable to the phosphorescence proceeding from a secondary
cathode, or to that called in by the positive column. Even as
concerns the ordinary phenomena some experiments devised by
Crookes* seem very adverse to Puluj’s view. It is, however,
unquestionable that cathodes of most substances have matter
torn off by the discharge, and that it is frequently largely de-
posited on those portions of the tube where the ordinary phos-
phorescence is most conspicuous. Messrs Spottiswoode and
Moulton* have pointed out other resemblances between the action
of the molecular streams and that of matter such as lamp-black
actually projected from the cathode.
As regards the function of the molecular streams in the
discharge no final results have been obtained, though much
speculation exists. Messrs Spottiswoode and Moulton give as
the result of their investigations:—“‘At present we have come
to no definite conclusion..., but we cannot say that we are
aware of anything that conclusively shows that they (the mole-
cular streams) have any definite electrical function to perform
in the discharge,” |. c. p.650. “The most attractive hypothesis re-
lating to their functions is that they officiate at the birth of the
discharge and enable it to get into the gaseous medium...,”
Lae} pobils
From the remainder of their cautiously worded remarks, I
believe Messrs Spottiswoode and Moulton would regard the
function ascribed to the molecular streams by this “attractive
hypothesis” to be that of carriers of a convective discharge out
into the gas. A slight modification would however adapt the
hypothesis to the very suggestive views of Messrs E, Wiedemann
and H. Ebert®.
1 Wied. Ann. 11, 1880, p. 836, and Ann. 12, 1881, p. 276. .
Wiener, Sitzungsberichte, Bd. yxxxx1., Abth. 11., 1880, pp. 864—923, see specially
p. 873.
* Journal of Electrical Engineers, Vol. xx., Feb. 1891, pp. 29—36.
+ Phil. Trans. 1880, pp. 582 and 649—650.
° Wied. Ann. 35, 1888, pp. 217 and 258—9,
1891.] liquid electrodes in vacuum tubes. 227
These observers found that the illumination of the cathode
surface by ultra-violet light, which in general has at high gaseous
pressures a remarkable effect in facilitating the discharge, ceases
to have any certain effect at low pressures when the presence
of phosphorescence shows the existence of molecular streams.
Their explanation is that ultra-violet light promotes the dis-
charge by setting up at the cathode surface, apparently in the
ether, vibrations of great rapidity such as would be produced
by heating the cathode to a high temperature—in general a
most efficacious way of promoting discharge—, but that the mole-
cular streams are the consequence or the concomitant of the
production of such rapid vibrations by some independent cause
which they do not specify. Thus the function which the ultra-
violet light performs at high pressures, is already fully provided
for at low pressures.
The point I more specially wish to bring out is this. At
high pressures whatever tends to increase the violence of the
negative discharge—e.g. an air spark in the circuit on the cathode
side of the tube—tends to set up the molecular streams. There
seems, however, strong grounds for believing that the production
of these streams actually facilitates the discharge, rendering it
less violent and disruptive than it otherwise would be, especially
at very low pressures.
As regards the nature of the discharge itself various views
are entertained, to some of which reference is necessary to explain
what follows. In the opinion of Professors J. J. Thomson’,
Schuster* and others the ordinary vacuum tube discharge is
in a way electrolytic. The molecules of gas become dissociated
and the atoms act as carriers of positive and negative electricity.
The heating of the cathode or anode, as the case may be, or the
presence of a heated wire, influences which have been shown by
Hittorf*, Elster and Geitel* and others to promote discharge
in a wonderful way, operate on this theory by dissociating the
gas and so furnishing atoms ready to respond at once to the
directive action of the external source of electricity. At a
luminous part of the discharge there is a production of heat
owing to the coming together of oppositely charged atoms, at a
dark part such coalitions are rare. Other writers, e.g. Lehmann’
and EK, Wiedemann’, regard the discharge as usually of two co-
1 Phil. Mag. Vol. xv., 1883, pp. 427—434; Aug, 1890, pp. 129—140; March
1891, pp. 149—171, etc.
2 Proceedings of the Royal Society, Vol. xxxvut., 1884, pp. 317—339, and Vol.
xuit., 1887, pp. 371—9.
3 Wied. Ann. 21, 1884, pp. 106—139.
4 Wied. Ann. 37, 1889, pp. 315—329, and Ann. 38, 1889, pp. 27—39, ete.
5 Molekularphysik, Bd. 11. pp., 220 et seq.
6 Wied. Ann. 35, 1888, p. 256.
MO VIL, PT. IV. 18
228 Mr Chree, On some experiments on [June 1,
existent but more or less independent parts, a dark probably
convective discharge independent of chemical action, and a lu-
minous discharge. Lehmann’s idea is something of the following
character. The luminous discharge is essentially disruptive and
intermittent whatever be the nature of the source of electricity.
Hittorf’ and Homén” it is true, employing for the source a large
number of cells, have imagined they got a steady luminous dis-
charge through the tube like the current in a metallic conductor.
But their reasons for this view such as the silence of a telephone
in the circuit are, according to Lehmann *, not conclusive, because
the steadiness of the current outside the tube does not necessarily
prevent its being intermittent inside. He regards the electrodes
as charging and discharging like condensers, requiring a certain
potential depending on the density, temperature, etc. of the gas
before the luminous discharge is possible. Simultaneously, how-
ever, the electrodes leak imto the tube, the discharge being
carried off probably convectively without any necessary lumi-
nosity. An increased brilliancy in the tube only implies an
increased quantity of electricity passing at each individual lu-
minous discharge, and thus it accompanies whatever raises the
capacity of the electrodes or affords an obstacle to rapid dis-
charge. This explains the action of an air spark in the cireuit
outside the tube. A diminished brilliancy may mean the passage
of a smaller current, or it may indicate an increased rapidity in
the succession of luminous discharges without any alteration in
the current outside the tube, or it may indicate the operation
of some agency facilitating the convective discharge. It is to
cne or both of the latter causes that Lehmann would ascribe
the effects of heating a cathode or exposing it to ultra-violet
light. It is unquestionable that in many such cases the diminu-
tion in the luminosity of the tube is very striking. Lehmann *
does not regard the luminous discharges at the anode and cathode
as having any necessary relation in the rapidity of their suc-
cession. He regards a red colour as merely indicating a strong,
a blue colour a weak discharge. Thus if, as usual, the positive
discharge is red and the negative blue, the difference is to be
accounted for either by the generally smaller cross-section of the
positive column, or by the interval between successive discharges
being longer at the anode than at the cathode.
A good many writers while recognizing a convective discharge
ascribe it to the action of dust particles®, These may exist in
1 Wied. Ann. 20, 1883, pp. 705—712, ete.
2 Wied. Ann. 38, 1889, pp. 172 et seq.
3 Molekularphysik, Bd. 11. pp. 234—7.
4 Molekularphysik, Bd. 1. p. 257.
° See Lenard and Wolf, Wied. Ann. 37, 1889, pp. 443—456.
1891.] liquid electrodes in vacuum tubes. 229
the original gas or be derived from the cathode by the dis-
integration which has been shown to accompany the passage of
the discharge, and in many cases at least to follow the incidence of
ultra-violet light.
Whatever the nature of the discharge may be it certainly does
not follow Ohm’s law. Hittorfand Homén, using a current which .
outside the vacuum tube seemed steady and continuous, found
under certain limitations—depending on the spread of the ne-
gative glow over the cathode—that the potential difference at
low pressures between the electrodes was nearly independent of
the strength of the current. At very low pressures the fall of
potential took place in great measure quite close to the cathode
surface. Hittorf’ found in the positive part of the discharge a
more or less regular fall of potential, and this fall per unit length
of tube was much greater than that in the non-luminous Faraday
space. Thus here at least non-luminosity seems to indicate
diminished resistance.
As one of the liquids I tried was mercury some points con-
nected with discharge through Hg. vapour claim special notice.
Mercury vapour being usually considered mon-atomic, it is clear
that discharge through it presents some peculiarities on the
electrolytic theory of discharge, there being no very obvious
means of separating the gas into carriers of positive and negative
electricity.
This peculiarity drew to it the attention of Professor
Schuster? who states as the result of very careful experiments:
—“T find that if the mercury vapour is sufficiently free from
air, the discharge through it shows no negative glow, no dark
spaces, and no stratifications.’ He also found the discharge to
present almost exactly the same features at both electrodes. The
introduction of a very slight trace of air set up the Crookes’
dark space at once. Experiments in agreement with Schuster’s
are described by Natterer®. Professor Crookes* has, however,
arrived at results diametrically opposed to those of Professor
Schuster. After taking the greatest pains to prevent the pre-
sence of any foreign gas, he found a distinct Crookes’ space and
at least traces of a dark Faraday space. He employed aluminium
electrodes, while Professor Schuster preferred them of platinum
with only a small surface exposed.
Elster and Geitel’ examining the effect of the presence of
a heated wire in a tube on the contained gas, found mercury
1 Wied. Ann. 20, 1883, pp. 726 et seq.
2 Proceedings of the Royal Society, Vol, xxxvu1., 1884, p. 319.
3 Wied. Ann. 38, 1889, p. 669.
4 Journal of Electrical Engineers, Vol. xx., Feb. 1891, pp. 44—6.
® Wied. Ann, 37, 1889, pp. 319 and 327.
18—2
230 Mr Chree, On some experiments on [June 1,
vapour to possess the rare, if not unique, property of showing
no electrification. This of course fits in extremely well with
the electrolytic theory, which these writers seem to favour.
Gassiot' more than thirty years ago made some interesting
experiments on mercury. His apparatus was so constructed that
he could have for his electrodes either metal wires or the Hg.
surface itself. Describing the appearance of the discharge as
the mercury rose in the tube, he says, “as soon as the mercury
ascends above the negative wire, a beautiful lambent bluish
white vapour appears to arise, while a deep red stratum becomes
visible on the surface of the mercury,’ p. 4. This red glow
was only sometimes apparent. In one place Gassiot suggests it
may be due to impurities in the mercury, in another he considers
it analogous to the glow on a platinum wire electrode, but he
seems to have arrived at no final conclusion. A mercury surface,
he says, when cathode is all covered with a luminous white
glow, but when anode only the extreme point is luminous. He
records numerous observations on striae in tubes in which
mercury was present, but it is not always easy to follow the
exact conditions of the experiment. He mentions that by re-
ducing the temperature of a vacuum tube containing mercury to
— 102° F. or raising it to 600° F. he caused the striae to disappear.
He also on several occasions got rid of striae at ordinary tem-
peratures by using for cathode the surface of the mercury itself.
Thus on p. 7, Phil. Trans. 1858, he says, “...1mmediately it
(the mercury) covers the negative wire the stratifications dis-
appear, and the interior of the globe is filled with bluish light.”
I think one may fairly conclude from his experiments that the
nature of the electrodes and especially of the cathode exercises
an important influence on the phenomena observed in the dis-
charge through mercury vapour.
In the experiments now to be described I used the secondary
current from an induction coil, varying the primary current ac-
cording to circumstances. When the resistance in the vacuum
tube circuit is sufficiently large only the direct extra current
passes. The appearance of the discharge shows at once whether
this is the case.
The tubes which I employed were constructed by Professor
Thomson’s assistant, Mr Everett, who rendered me valuable assist-
ance in the course of the experiments. As the first tube had
only a brief existence it will suffice to describe the second which
resembled it in all essential points. The diagrammatic sketch,
fig. 1, p. 231, will explain the general character. The nature
of the electrodes varied but their position in the tube was fairly
' Phil. Trans. 1858, pp. 1—16, and 1859, pp. 137—160.
1891.] liquid electrodes in vacuum tubes. 231
constant. For one of the vertical tubes, say AZ, the distance
of D above the electrode was 170 mm. of which the lowest
100 mm. was of uniform diameter, 13°5 mm. externally. Above
this the tube narrowed for some 12 mm. and then continued of
A\
}
Fig. 1. JB
}
uniform external diameter 8 mm. up to # at 250 mm. above
the electrode. The horizontal tube DF had an external diameter
of 6 mm. and a total length of 160 mm. Thus the total distance
between the electrodes was about 50 cm. These measurements
were not made with any great exactness and the surfaces of the
electrodes were not kept at a perfectly constant level. The
tube CHP led to the mercury pump, CH being vertical, and
HP parallel to DF. This tube was of about the same diameter
as DE.
Supposing the tube AD to have a liquid electrode, this was
connected with the exterior by a fine wire whose top was about
1 em. below the liquid surface, and which passed down air-tight
through the bottom of the tube.
The point to which Professor Thomson originally directed my
attention was the question whether there was any change in-
troduced in the liquid surface by the discharge which altered
the subsequent character of the luminous appearances. As my
results on this point are intelligible only when the form of the
discharge has been described, their discussion is postponed to a
later part of the paper.
There were three sets of experiments on mercury. In the
first both electrodes were mercury surfaces; in the second one
was mercury, and the other an uncovered platinum wire ex-
tending some 2 cm. up the axis of one vertical tube ; in the
third set the platinum wire was replaced by a flat horizontal
232 Mr Chree, On some experiments on [June 1,
plate of aluminium. The gas originally in the tubes was always
air, and a mercury pump was used.
In all three cases the following phenomena were observed
in the Hg. tube—i.e. the branch containing a mercury electrode.
At fairly high pressures when the spark first passed freely the
positive discharge was bright red, as is usual in air, As the
pressure diminished the colour became whiter, and there ap-
peared numerous striae, a distinct Faraday space, and a whiteish
blue negative glow. Before the striae became conspicuous the
discharge left an Hg. anode from its extreme summit, but as
the exhaustion proceeded it left from an increasing area, till
finally the whole surface became luminous.
As the pressure was diminished there appeared what seemed
to be a Crookes’ space over an Hg. cathode. It was not so clearly
defined as that space usually is, but the surface separating it from
the negative glow was at first tolerably distinct. This surface
was convex like the Hg. surface itself and had a very similar
curvature. This Crookes’ space was, however, by no means very
dark, being in general of a distinctly red aspect, and on occasions
almost fiery in appearance. The introduction of a minute air
bubble into the tube made the boundary of this space appear
much more distinct. As the pressure was further reduced the
Crookes’ space increased in length and the top of the Faraday
space moved up the tube. This went on until the Faraday
space extended to 80 or 90 mm. from an Hg. cathode. At this
stage the stratification in the tube containing an Hg. anode
was fairly distinct, the length of a stria being about 1 cm. and
its luminous portion being nearly pure white. After this stage
there appeared a change in the character of the phenomena most
conveniently dealt with in the separate discussion of the different
sets of electrodes.
Electrodes both Hg. surfaces.
When the stage just referred to was reached in this case,
the variety in the appearances at different parts of the discharge
tended to disappear, the whole assuming a more or less uniform
white colour. The Faraday space seemed on some occasions
certainly absent, and the Crookes’ space became, to say the
least of it, exceedingly indistinct. The difficulty of coming to
a decision as to the existence of these spaces was much in-
creased by the deposition of matter on the walls of the tube
containing the cathode. The nature of this deposit was the
same as that described in the next case. All I can say with
certainty is, that the red-black space—assumed to represent
Crookes’ space—did not continue to expand continually as the
1891.] liquid electrodes in vacuum tubes. 233
exhaustion proceeded. It seemed when longest to reach as far as
lem. above the cathode, becoming less distinct as it expanded.
At the lowest pressures reached, however, while the 4 or 5 mm. of
the tube immediately above the cathode were perhaps somewhat
darker than the average, the luminosity had certainly attained a
maximum within not more than 7 or 8 mm. of the Hg. surface.
The stage at which the Faraday space became indistinct depended
to some extent on the nature of the break in the induction coil.
With a rapid break and quiet spark the luminosity fell off and
the Faraday space remained longer distinct.
Electrodes Hg. surface and Pt. wire.
With electrodes so different in form the variations in the
phenomena cannot be entirely ascribed to difference of material.
According to Goldstein’ an alteration in the size of the anode
does not affect the striae in a stratified discharge, but a dimi-
nution in the size of the cathode while leaving unaltered the
length of the individual striae increases the distance of the head
of the positive discharge from the cathode. E. Wiedemann *
found that the relative facility of discharge between points and
between plates changes with the pressure. Thus at pressures
over 1 mm. he found the discharge to pass between points and
not between plates at the same distance apart, whereas at lower
pressures it passed most easily between the plates. Lehmann *
lays down some apparently general laws as to the effects of
making blunter a cathode, but he does not always seem con-
sistent on this point, and I have some doubts as to how far he
bases his views on experiment. So many secondary influences
are at work, such as the size of the tube and the distance of
the electrodes, especially when comparable with their transverse
dimensions or with the length of Crookes’ space, that one would
hardly expect @ prior any simple general law to apply.
At the lower pressures the platinum wire rapidly became
red hot and the deposit on the tube around it became very
thick, so that it was impossible to see anything of the Crookes’
space. If it did exist it must have been considerably less than
the Crookes’ space in the Hg. tube when that was last seen
distinctly. The comparison instituted was thus between the
distances from thé cathodes of the further extremity of the
Faraday space, ie. the head of the positive discharge. The
following are some of the data obtained, the simultaneous lengths
of the Crookes’ space at the Hg. cathode being given when
noted :
1 Wied. Ann. 12, 1881, p. 275.
2 Wied. Ann, 20, 1883, pp. 795-7.
3 Molekularphysik, Bd. 11. pp. 277—8, and Wied. Ann. 22, 1884, pp. 320—1.
bo
vs
frag
Mr Chree, On some experiments on [June 1,
TABLE IIT.
Hg. Cathode Pt. Cathode
ae a = (ee ae
Length of Distance above cathode Distance above cathode
Crookes’ of head of positive of head of positive
space. discharge. discharge.
30 13
35 14
+ 60 20
5 65 20
7 15 20
The distances are in mm., and are all measured from the
upper extremity of the cathode. The observations recorded in
the same horizontal line were taken in immediate succession,
the make and break regulator of the mduction coil remaining
untouched. The uniformity of the pressure and of the make
and break was tested by reversing the current twice, which
showed whether the appearances at the surface which was first
cathode had altered during the observations. The experiments
extended over a considerable interval and the tubes were cleaned
out and refilled more than once, thus the difference between
the cathodes shown by the table, which was confirmed by nu-
merous observations in which accurate measurements were not
taken may, I think, be fully relied on. The table shows that
the distance above the cathode of the head of the positive
column was on an average about thrice as much when the
cathode was mercury as when it was platinum. The difference
was greater the lower the pressure. At the highest pressures
when the Faraday space first became distinct no exact measure-
ments were recorded, but the difference though then not so
striking as in the table was still conspicuous. As the exhaustion
proceeded very slowly and both tubes were at intervals heated
up by a burner, the difference can hardly be attributed to any
great extent to a difference in the gaseous contents of the two
vertical tubes.
The difference may be due to the difference in the material
of the electrodes or to the dissimilarity of their sizes and shapes.
According to Goldstein the effect of the difference in size of the
cathodes should have tended in the opposite direction. For
reasons explained in treating of the next case, I am inclined to
attribute the difference in considerable measure to the difference
in shape.
1891.] liquid electrodes in vacuum tubes. 235
Appearance of Discharge.
The tube containing the Pt. electrode always showed a redder
tint than that containing the Hg. electrode under similar con-
ditions.
At a certain stage of exhaustion the former tube was some-
times the redder even when the Hg. surface was the anode.
At the lowest pressures, however, the difference between the
colour of the tubes was inconspicuous, both being prevailingly
white. At the highest pressures the spark left the tip of the
Pt. anode and the extreme summit of the Hg. anode, but as
the exhaustion proceeded it gradually extended down the Pt.
anode and spread over the surface of the Hg. anode. Eventually
the positive column covered the whole of the Hg. surface, but
within 1 or 2 mm. up the tube it had contracted somewhat in
diameter, leaving an annular dark space between it and the
glass. The Faraday space became gradually indistinct over the
Hg. cathode, and the same phenomenon appeared over the Pt.
cathode but at a lower pressure. Thus at one stage of the ex-
haustion the appearances in the two tubes when containing
the cathode were widely different. At the lowest pressures
reached, both tubes, so far as clearly seen, were very similar in
appearance, and the phenomena agreed with those observed when
both electrodes were of mercury.
Phosphorescence.
For clearness let us suppose, as was actually the case, A the
Hg., B the Pt. electrode. In the tube BG the phosphorescence
first appeared at the level of the upper portion of the wire, and
gradually spread both up and down as the exhaustion proceeded,
reaching the summit G of that tube. In the tube AZ the
phosphorescence hardly appeared within 2 cm. of the Hg. surface.
In the lower parts of both tubes phosphorescence was most
brilliant at about the positions of the Faraday spaces at the
lowest pressures, and so at a much higher level in AZ than in
BG. Also the phosphorescence at # was much more intense
than that at G, though the latter was very well marked.
At the lowest pressures reached a faint nebulosity, presumably
the positive column, extended to a considerable distance along the
tube CHP. It was difficult to detect when the platinum was
cathode, but when the mercury was cathode it could be traced
almost as far as the pump.
There then appeared at H a patch of phosphorescence on the
convex side of the bend. This was very faint when the platinum
was cathode, but when the mercury was cathode it was fairly
236 Mr Chree, On some experiments on [June 1,
bright. This is clearly a variety of the phenomena observed by
Goldstein and by Spottiswoode and Moulton, but it seems worth
noticing as the tube CHP did not lead to an anode. The Hg.
column in the pump did not lead to earth, and further the
luminosity in HP decreased as the distance from H increased,
which it would hardly have done if the pump had acted as anode.
Deposit on the tubes.
In the tube BG containing the platinum there were two,
generally distinct, principal areas of deposit on the glass. When
the platinum had served for some time as cathode there was a
dense black deposit from a little above the level of the top of
the wire downwards, and a second less dense deposit separated
in general from the lower by an almost perfectly clean and
sharply defined area only about 1 mm. broad but extending
right round the tube. Roughly speaking, the upper deposit ex-
tended to about the highest level attained by the extremity of
the Faraday space, but there were traces of it further up the
tube. Thus the portions of the walls of this tube where the
deposit was thickest were precisely those where the phosphor-
escence was strongest ere the darkening of the glass reduced
its brightness. At the same time phosphorescence was con-
spicuous on the narrow patch of clean glass separating the two
principal areas of deposit, and also in the upper portion of the
tube #’G where no deposit was seen.
In the tube AFH the permanent deposit extended in general
from 20 or 30 to 80 or 90 mm. above the Hg. surface, none ap-
pearing in the neighbourhood of #. It was nowhere so thick as
that on the other tube. Looking at it in strong light one could
see small drops of mercury scattered about, but its exact com-
position was not determined. There were other deposits of a
more temporary character near the Hg. surface. One had the
appearance of dew spreading up the tube for a few millimetres
when the Hg. was made cathode, and gradually creeping down
when the current was stopped, taking only a short time to dis-
appear. It was seen only at low pressures. In some cases
there was a sort of white deposit separated from the Hg. surface
by a clear space of a few millimetres, which increased apparently
as the exhaustion proceeded. On readmitting air to the tube
this last form of deposit in great measure disappeared. What
has been called above the permanent deposit bad a blackish
colour and was but little atfected by the readmission of air.
These deposits by obscuring the portions of the tube where the
Faraday and Crookes’ spaces had to be looked for, increased very
much the difficulty of reaching final conclusions as to the existence
of these spaces at the lowest pressures.
1891.] liquid electrodes in vacuunr tubes. 237
Electrodes Hg. and Al.
The aluminium electrode was a flat circular plate of about
two-thirds the internal diameter of the tube. The following ob-
servations were taken on several occasions:
TABLE IV.
Hg. Cathode Al. Cathode
tt a a. c 25 1
Length of Distance Distance Length of Distance Distance
Crookes’ above above Crookes’ above above
space. cathode of cathode of space. cathode of cathode of
top of head of top of ~ head of
negative positive negative positive
glow. discharge. glow. discharge.
9) 18 If
8 20 5 12
5) 11 22 7 13
1 15 23 12°5 16°5
75 24 18
25 17
30 20
15 40 32°5
2 40 30
40 25
8 80 6 70
The distances are in millimetres, and were all measured from
the centre of the cathode surface. At the higher pressures the
upper limit of the negative glow was pretty distinct and so has
been recorded above. Observations in the same horizontal line
were taken in rapid succession as in the case of Table m1. The
ratio of the mean distance from the cathode to the end of the
positive column when the cathode was mercury to the corre-
sponding mean distance when the cathode was aluminium is as
1:37 : 1, and so is very much less than the ratio found when the
solid electrode was platinum. Since no such striking difference
between the metals platinum and aluminium as electrodes seems
to have been noticed by previous observers, the inference would
seem to be that we are here concerned with the size and shape
rather than with the material of the cathode.
Owing to the appearance of a slight crack in the tube with
the Al. electrode its base had to be immersed in mercury con-
tained in a paraffin cup. Thus the length of the Crookes’ space
when short could not be accurately observed without lowering
the cup. This was not done in the cases recorded in the table,
to avoid the risk of leakage, variation of current, etc. but on
238 Mr Chree, On some experiments on [June 1,
other occasions such observations were taken and it was found
that from the highest pressures where it was visible down to
the lowest in the above table, Crookes’ space appeared slightly
but distinctly longer over an Hg. cathode than over an Al.
cathode.
Appearance of the discharge.
Under similar conditions the Al. tube was always redder
than the Hg. tube, though both were at the lowest pressures
mainly white. When the pressure was reduced to a certain
stage, the Faraday space over the Hg. cathode became more
and more indistinct till it seemed to vanish. The Faraday
space over the Al. cathode was at this stage unmistakeable,
but at a lower pressure it too eventually became undistinguish-
able. ‘The way in which the Faraday space over the Hg. cathode
disappeared was rather remarkable.
The pressure reached a point at which the appearance in the
tube was unstable. There might be an unmistakeable Faraday
space and negative glow, and then a sudden transition to a stage
in which distinct striae reached down the tube to near the Hg.
surface. During this time the discharge in the Al. tube showed
no fluctuation. As the pressure was carried lower the striae
became less and less distinct until the Hg. tube whether the
Hg. were cathode or anode seemed an almost uniform white.
The lowest 5 or 6 mm. in the tube appeared sometimes per-
ceptibly, sometimes doubtfully darker than the rest. Distinct
striae eventually ceased to appear even in the Al. tube, but it
showed to the end an unmistakeable Crookes’ space which how-
ever was becoming increasingly indistinct at the lowest pressures
reached.
The greatest length reached by the Crookes’ space over the
Al. cathode was 14 mm. At this stage the tube contaiming
the Hg. cathode was as bright as anywhere within 7 or 8 mm.
of the mercury surface.
Phosphorescence.
When the aluminium was cathode phosphorescence extended |
at the lowest pressure throughout the whole of the tube BG,
whereas when the mercury was cathode phosphorescence was
not observed within some 2 cm. of its surface. The phosphor-
escence at the top # of the Hg. tube was always brighter than
that at G at the same exhaustion. At the best exhaustion with
the Hg. cathode there was distinct luminosity along the tube
CHP as far as the pump, a distance of 42 cm. from H, and a
patch of phosphorescence was distinctly visible at H. At the
1891.] liquid electrodes in vacuum tubes. 239
same exhaustion with the Al. cathode a faint luminosity was
seen to some distance past H, and a very faint phosphorescence
could be made out at the bend.
Deposit on the tubes.
In the Hg. tube there were the same deposits as before, and
a further one was observed under the following circumstances.
On several occasions on examining the Hg. surface by day-
light it was found to present a yellow metallic appearance. The
conditions preceding its first appearance were as follows. An
air-bubble which had remained under the mercury was driven
up whilst the discharge was passing. Mercury splashed up the
walls of the tube and adhering to some extent presented a
concave surface. Some observations were taken with the surface
in this state, and next day it was noticed that the surface was
yellowish, and that there were traces of a yellow deposit not
only up the tube AF but also at intervals along DF and even
for some distance down FB.
On another occasion when the tube had just been carefully
cleaned and dried and fresh mercury introduced, I noticed after
passing the spark both ways for some time that the Hg. surface
though retaining its ordinary convexity had a decidedly yellow
appearance, and that the tube near it was slightly yellow. Air
was allowed to leak in, and the tube being left for some days
appeared when next examined quite clean, while the Hg. surface
had its usual colour. When however the tube was again ex-
hausted and left for some days, the Hg. surface and a few
millimetres at the base of the tube were found yellow as before.
However, on passing the spark the phenomena had their normal
character—which was not the case when the mercury surface
was concave—and on re-examining the tube the yellow colour
was found to have entirely disappeared and it was not observed
again. With the exception of the yellow patches above men-
tioned and a narrow dark ring sometimes observed on the glass
near the head of the positive column, the tube with the Al.
electrode showed no distinct deposit. The origin of the yellow
colour was not discovered. Its appearance in the tube FB sug-
gests but does not prove a capacity in the discharge to transport
particles from a cathode surface round corners. The transporting
agency might of course have been vapour rising from the Hg.
surface and condensing on cooler portions of the tube.
Effect of sudden alteration of the Hg. surface.
The experiments on this point were those first carried out.
Both electrodes were then of mercury, and the tube CHP instead
240 Mr Chree, On some experiments on [June 1,
of being fused to the pump as it was during the other experi-
ments, was connected to it by some thick-walled india-rubber
tubing. The mode of altering the surface was simply by shaking
or vigorously tapping the tube. With the head of the positive
column from 9 to 30 mm. above the cathode, the stage at which
the Faraday space was most distinct, no certain change in its
position was observed to follow the alteration of the surface.
The only stage of exhaustion at which a distinct effect of any
kind was observed was that where the Faraday space seems to
be in the act of disappearing. On first starting the current,
especially with a slow break and noisy spark, there appeared
more or less uniform whiteness in the cathode tube and in-
distinct striae in the anode tube. After the discharge had
passed a short time, the colour tended to fade out of a portion
of the cathode tube. roughly speaking, between 30 and 90 mm.
over the Hg. surface, and phosphorescence became much more
conspicuous, especially in this part of the tube; also the striae
in the anode tube became more distinct. A shaking or sharp
tapping of the tube instantly restored the more or less uniform
white colour throughout the cathode tube and tended to obliterate
the striae in the other tube. The effect lasted only a short time,
the discharge gradually reverting to the appearance it presented
before the disturbance. ‘This phenomenon invariably presented
itself under the conditions stated. The only explanation that
occurs to me—suggested by the views of Messrs E. Wiedemann
and H. Ebert—is that, at least at certain stages of exhaustion,
the condition at the cathode surface which leads to the pro-
jection of molecular streams takes some time for its full develop-
ment, and that on its development the successive discharges
follow one another more rapidly and consist each of a smaller
quantity of electricity. The shaking of the tube and consequent
distribution of fresh mercury over the cathode surface restores
the original conditions, which are less favourable to the production
of molecular streams.
Electrodes H,SO, and Al.
In the next set of experiments the electrode in A was some
pure sulphuric acid, the aluminium plate electrode in the other
tube being retained.
Some experiments with sulphuric acid electrodes have been
described by Paalzow*. He gives an interesting account of the
electrolysis of the acid, and of spectroscopic observations on the
discharge. He observed the positive discharge to start from the
line of separation of the fluid surface and the wall of the tube.
' Wied. Ann. 7, 1879, pp. 130—135.
1891,] liquid electrodes in vacuum tubes. 241
As I hardly follow his description of the appearances at the
cathode I give his own words: “ Von der negativen Fliissigkeits-
oberflache selbst erhebt sich in einigem Abstande von derselben
ein schwach conischer Lichtring, aihnlich wie die Flamme eines
ringformigen Brenners,” p. 131. With increasing exhaustion :
“um so mehr verlingert sich dieser negative Lichtcylinder, und
um so griésser wird sein Abstand von der Fliissigkeitsober-
fliche,” p. 132. At very low pressures he observed the pheno-
mena to be much the same at both electrodes.
In this case my observations commenced as soon as the
pressure was sufficiently reduced for the discharge to become
visible in a dim light. This invariably occurred when the alu-
minium was cathode, and the luminosity took the form of a
thin purplish negative glow. At somewhat lower pressures the
Al. tube was fairly luminous whether it contained the cathode
or anode, the H,SO, surface when cathode showing a thin blue
glow. The horizontal tube and a small portion of the tube AH
below D then showed a red spark discharge. The greater portion
of the latter tube remained however dark, except that at in-
tervals red twig-like discharges passed down it. At this stage
the spark in the tube DF was most twig-like and of least diameter
at that end which was nearest the cathode, whether Al. or H,SO,.
The pressure had to be further reduced to a considerable extent
before luminosity could be detected at the H,SO, surface when
anode. When the discharge was first clearly seen at an H,SO,
anode it took the form observed at pretty low pressures with
an Hg. anode. Throughout the greater portion of the tube the
positive column appeared as a solid cylinder of considerably less
diameter than the interior of the tube, but near its base this
cylinder increased in diameter so as just to fill the tube on
reaching the liquid surface.
At this stage the appearance in the tube AH over an H,SO,
eathode took the following form. A column of very small
diameter, usually bright red in colour, extended down the axis of
the tube to within a short distance of the liquid surface. The
end of this column was sometimes truncated, but frequently it
presented a sharp point like that of a pencil. Over the liquid
surface, completely occupying the cross-section of the tube, there
existed the ordinary blue negative glow. At first this was sepa-
rated by several millimetres of a dark, presumably Faraday, space
from the red column, but as the exhaustion proceeded while the
head of the column retired from the H,SO, surface the negative
glow overtook it. With progressing exhaustion the point and a
gradually increasing length of the red column were immersed in
the blue glow. The difference in colour rendered the red column
conspicuous through the blue glow, but I am unable to say
a)
242 Mr Chree, On some experiments on [June 1,
b
whether the two were actually in contact. Possibly they may
have been separated by a thin hollow conical dark space.
At this stage the tube containing an Al. cathode showed a
positive column whose diameter was several times greater than
that of the column just described. Further, the head of the
positive column over an Al. cathode had the usual convexity, and
was separated by a distinct Faraday space from the negative glow.
As the exhaustion proceeded an ordinary Crookes’ space appeared
over an Al. cathode, and the other phenomena seemed of the usual
type. Over an H,SO, cathode, however, the phenomena seemed
of an exceptional kind.
The liquid surface was of course concave, the depression of
the vertex below the rim being about 1? mm. Thus on looking
sideways at the tube one saw a curved dark line answering to
a vertical section of the liquid surface. Between this and the
horizontal plane through the rim of the surface there appeared
a blue tint. This might have proceeded from a luminous film
over the liquid surface or merely from the reflexion of the glow
further up the tube. Now supposing a Crookes’ space to exist,
the surface separating it from the negative glow would by analogy
from other cases be expected to resemble in form the liquid
surface. Thus so long as the thickness of this space was less than
the sagitta of the liquid meniscus, one would have expected the
blue negative glow to pass near the axis of the tube into the
blue of the liquid surface, while close to each of the tube walls
one would look for a dark space shaped like a right-angled
triangle, with a curved hypothenuse answering to the surface
of the negative glow. Let us call this type (a). An idea of
it will be easily derived from (b), fig. 2, by supposing ACB non- -
existent. The spaces HAD and KBE are to be supposed black,
while below DABE and above HABK blue prevails for some
distance. DASE represents the liquid rim cutting the surface
HABK of the negative glow in the straight line AB. At lower
pressures one would have expected to see the length AB
gradually diminish, till finally there appeared over the whole
1891.] liquid electrodes in vacuum tubes. 243
liquid surface a black space, which would appear to the eye to be
bounded below by a straight line and above by the curve of
intersection of the negative glow by a vertical plane. Let us
call this type (d). These types were actually observed, but in
addition two other types were seen, viz. those represented by (b)
and (c), fig. 2.
In both, DABL represents the rim of the liquid surface as
seen by an observer's eye at the same level. In (b) the small
areas HAD, KBE had a black appearance, but were in general
not conspicuous. The negative glow extended from HABK for
some distance up the tube as in type (a). But it now showed a
sort of tuft ACB of a much whiter and less translucent blue than
the rest, with more or less distinct curvilinear boundaries which
seemed to cross at C and gradually fade away beyond it. In (c)
there was no trace of any black spaces, and the tube appeared
prevailingly blue all the way up with the exception of an almost
pure white tuft ACS. It had a tolerably distinct outline except
at the top C where the colour passed gradually into blue. It was
sometimes more conical and sometimes more depressed seemingly
than in the figure, but exact measurements were not attempted.
A slight unsteadiness in the type (c) happened to catch the
eye when looking at the liquid surface while putting on the
current. Further examination led to the following results.
_ After keeping the pump at work for some time and then
suddenly starting the discharge with the H,SO, as cathode one
saw a sort of white cloud instantly gather in the tube, separated
apparently from the liquid surface by from 2 to 4 mm. of a
dark interval. The cloud, however, immediately stretched down
the tube and transformed into the appearance (c). This trans-
formation was on several occasions so distinctly seen that the
observer could hardly be mistaken; but it was not always seen
under these conditions. The whole thing happened so fast that
sometimes all one could say was that some rapid change had
occurred. When the current was simply stopped and renewed
without intervening exhaustion, the tuft appeared at once in the
position shown in the figure.
I am not aware of previous observations on the form of the
negative glow near concave cathodes forming surfaces of revolu-
tion, and it would obviously be very difficult to see what actually
exists within the rim of such a cathode. Professor Crookes, how-
ever, in Plate 14, Phil. Trans. 1879, gives some beautiful coloured
illustrations which show the negative glow and the Crookes’ space
near a concave cylindrical cathode. I would more especially draw
attention to his illustrations b and ¢ fig. 11, which show a great
concentration of negative glow near the plane of symmetry
through the axis of the cathode. In Crookes’ fig. b there is a
VOL. VII. PT. IV. 19
244 Mr Chree, On some experiments on [June 1,
regular tuft, and in his figure c the appearance of a bundle of rays
projecting into a dark Faraday space. In fact if in these figures
all to the left of the plane through the rim of the cathode were
removed there would be a considerable resemblance to the pheno-
mena illustrated by my (0) and (c) fig. 2. The principal difference
is that with Crookes’ electrode there appears to have been a
Faraday space limiting a distinct and, except in the plane of
symmetry, thin negative glow, whereas with the H,SO, cathode
no such distinct Faraday space was seen.
I would also call attention to Crookes’ fig. 2, p. 643, Phal.
Trans., 1879, as showing a concentration of negative glow in
positions opposite the hollows of a corrugated cathode.
The resemblance of the boundaries of ACB in (6) and (c)
fig. 2 to caustic surfaces unquestionably suggests that we may
have here to do with some species of emission from the surface
separating the liquid and gas, each element of the surface acting
as a source more or less independent of its neighbours. If, as
seems to be the case with the molecular streams}, the direction of
emission from the elements near the rim deviates more than else-
where from the normal to the surface, one can easily see that there
would be a crossing of the trajectories near the axis of the tube,
even close to the liquid surface.
The sudden change noticed when the type (c) fig. 2 appeared,
might be accounted for by the discharge leaving at first from only
a portion of the liquid surface, or possibly it may be connected
with the rise of the small bubbles accompanying the electrolysis
which take some short time to reach the surface.
The following table gives results from several series of observa-
tions, all data in a horizontal line being taken in immediate
succession with a constant rate of make and break. All the dis-
tances are in millimetres. Except at the highest pressures even
the approximate position of the upper end of the negative glow
could not be fixed. Such an entry as “7 +” in the column headed
“Distance...glow” means that the glow reached to some unde-
termined height above the point of the red column. In the case
of the H,SO, cathode all the distances were measured from the
rim as the most convenient starting-point. It ought to be remem-
bered, however, in instituting any comparisons that in the axis of
the tube the true liquid surface was some 1? mm. below the rim.
The negative glow in type (d) had a considerably greater curva-
ture than the liquid surface, so that its distance from that surface,
measured parallel to the axis of the tube, was least in the axis,
1 See Goldstein, Wied. Ann. 15, 1882, pp. 254—277, specially pp. 274—5.
1891.] liquid electrodes in vacuum tubes. 245
TABLE V.
H,SO, cathode Al. cathode
=... Ga ae ee
Crookes’ space Distance Distance Length Distance Distance
: above above of above above
Type Height rim of rim of Crookes’ cathode cathode
above top of head of space. of top of of head of
Tim. negative positive negative positive
glow. discharge. glow. discharge.
+ 5 jis
6 6 17
7+ 7 9 19
7+ 7 22
13 10 17 29
(a) 10 + 10 2:5
13+ 13 4 40
15+ 15 23 38
(b) absent 45
(c) absent 5
(d) 1°5 7+ 17 5 dubious
(dq) 15 ‘pe 18 7 50
(d) 2 24 + 24 8 dubious
(d) ao 30 + 30 8 dubious
None of the distances admitted of very accurate observation
because the positions of the various parts of the discharge, especially
the head of the positive column over an Al. cathode, were seldom
quite stationary. At the lower pressures in the table the blue -
light surrounding the red column over an H,SO, cathode extended
up the tube AZ to above the level of the horizontal tube. When,
however, the interrupter was adjusted to give a slower and more
noisy spark the red column sometimes completely disappeared.
It also sometimes faded out when the discharge was kept passing
for some time. In either case there appeared from about 20 to
40 mm. above the H,SO, cathode a considerably darker blue than
elsewhere verging towards black. This darker band always accom-
panied the types (6) and (c) of discharge with which the red
column was not observed. Once or twice this column instead of
being red was white. In general its outline appeared straight and
regular, but on at least one occasion it showed numerous short
horizontal projections like hairs.
The colour of the positive column at the higher pressures and
until the Crookes’ space over the Al. cathode attained a length of
1 or 2mm. was always red. At lower pressures when red its
colour was much fainter, and at the lowest pressures it generally
tended to white or even blue. The striae were most conspicuous
when the colour was white, but they were seldom very distinct.
At the lowest pressures only faint phosphorescence was observed in
19—2
246 Mr G. H. Bryan, Nete on a Problem [June 1,
the tube over an Al. cathode, and none was noticed over an H.SO,
cathode. There was also no deposit on the walls of the tube.
The joint absence of these two phenomena ‘is rather striking, but
the exhaustion was not, I think, carried so low as in the previous
experiments.
At the lowest pressure attained, with a Crookes’ space of from
9 to 10 mm. in length over an Al. cathode, the appearance of the
discharge became unsettled. There was nowhere any very bright
colour, but throughout the greater part of the tube over an H,SO,
cathode the colour along the axis was unmistakeably red. There
was a red column somewhat resembling that seen at higher
pressures, terminating in a sharp point about 40 mm. over the
H,SO; surface. Immediately below this the tube appeared blue
throughout the entire cross section, but a little lower down there
appeared a red column of about half the diameter of the tube. Its
base was curved, and its convexity was directed towards the
H,SO, surface, from which it was separated by an interval of only
2to4mm. The colour of this interval seemed to vary from black
to faint blue. Both these red columns had a blue haze between
them and the walls of the tube. This stage seems to answer to
that observed with a mercury electrode previous to the discharge
assuming a nearly uniform appearance throughout.
My best thanks are due to Professor Thomson for putting at
my disposal the necessary apparatus, and for numerous suggestions
during the course of the experiments, which were performed in the
Cavendish Laboratory.
(5) Note on a Problem in the Linear Conduction of Heat. By
G. H. Bryan, M.A., St Peter’s College.
THE problem of conduction of heat in a bar one end of which
is subject to radiation while the other end is at an infinite distance
away, has been treated by Mr Hobson in his paper “On a Radiation
Problem” published in the Proceedings, Vol. VL, page 184 The
author there finds the expressions in the form of definite integrals,
representing the temperature due to the given distribution of heat
in the medium at the extremity of the bar and to the initial
distribution of heat in the bar respectively.
The expression which Mr Hobson obtains for the second part of
the temperature is, however, open to several objections. It appears
to fail entirely if the initial temperature is anywhere discontinuous
or if any sources, doublets, or other singularities are initially present
in the bar,
Moreover, from the integral obtained, it is shown that the
initial distribution is equivalent to a certain distribution of lines
of sources and sinks in a rod extending to infinity in both directions,
1891.] in the Linear Conduction of Heat. 247
but this interpretation is also liable to objection. For the distri-
bution of sources and sinks corresponding to the initial temperature
in any single element of the rod is not in itself equivalent to that
element. Hence Mr Hobson’s-solution gives no idea of the part
played by the initial temperatures of the separate elements of the
rod, and in fact it does not even give a correct result if the effect
of the initial temperature in any part of the rod is considered
apart from that in the rest.
The problem is best solved by starting with a single instan-
taneous source at one point of the rod, since from this the more
general solution can be obtained by integration.
Consider then the effect of an instantaneous source Q of heat
generated at the point 2’ at the time 0. The temperature due to
such a source if there were no boundary would be
Qeitimongciaty
2 akt > kt
Let this expression be denoted by v,, and let the temperature
when the boundary is taken into account be v where
v=, +2,.
If the external medium be at temperature zero, the boundary
condition gives, when «= 0,
dv, - d Q (a — 2’)
Hence =o hv=- (=. - h) 2 lt cS aaa ae
SPN 0 eg ee ee
=O feck | Dt vy exp — 47, When a= 0.
But v, must be the temperature due to a series of images on
the negative side of the origin, hence the conditions of the problem
will be satisfied by taking
PL a (a + 2’)?
dlc =I Diet +i} «PD ake
_(d Q (27+ 2’)
= = + h) ea ull exp
_(%_ Q (x + a’)
F; e h) jako. || Akt
2hQ _ (e+e'p
Sake OF
248 Mr G. H. Bryan, On Conduction of Heat. [June 1, 1891.
Therefore by integration
at Qube see Vt seal uae
(oe
ay 2 /akt ao Akt 2 2 /arkt .
Akt
e~ exp —
))
In the last integral put €=a+z. Also add v,, thus the whole
temperature assumes the form
a _(@e-#y (w+ a
V=7,+2,= yaa? erie x} Abt
Ope (c+z2+2'/
— 2h 02 Jaki exp— ar dz.
The first line represents the temperature due to the given source
and an equal source at the geometrical image, the second line
represents the temperature due to a line of sinks extending from
the geometrical image to infinity in the negative direction. None
of the i images of the source lie on the positive side of the origin.
For the temperature due to the initial distribution v= ¢ (a) we
write d(«’) dz’ for Q, and deduce by integration,
\3 je De i] (a—2'? (c+a’P ; :
=o Jeni), (OP fg toe — “Fe | 6 @) da
~9 oe oe exp (— hz). exp— (a +2 4a!) da'ds.
This solution holds good whether ¢ (w’) be finite and continuous
or not.
[Note by Mr Hobson.—Mr Bryan’s formula is undoubtedly an
improvement on the one I gave in the paper referred to, It should
be observed that his formula may be obtained from mine by
integration by parts.]
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PROCEEDINGS
OF THE
Cambridge Philosophical Society.
October 26, 1891.
ANNUAL GENERAL MEETING.
Proressor G. H. DARWIN, PRESIDENT, IN THE CHAIR.
The following Fellows were elected Officers and new Members
of Council for the ensuing year:
President :
Prof. G. H. Darwin.
Vice-Presidents :
Prof. Hughes, Prof. Thomson, Mr J. W. Clark.
Treasurer :
Mr R. T. Glazebrook.
Secretaries :
Mr J. Larmor, Mr S. F. Harmer, Mr E. W. Hobson.
New Members of Council :
Mr H. F. Newall, Mr C. T. Heycock, Mr A. E. H. Love.
The following Communications were made to the Society :
(1) On the Absorption of Energy by the Secondary of « Trans-
former. By Prot. THOMSON.
won: VIL. PT, V. 20
val
\)
250 Mr Sedgwick, on a Peripatus from Natal. [Nov. 9,
(2) Onan experiment of Sir Humphry Davy’s. By G. F.C.
SEARLE, M.A., Peterhouse.
Two copper wires are passed up through holes about 5 centi-
metres apart in the bottom of a flat trough, their ends being level
with the surface of the trough. Mercury is then poured into the
trough to a depth of about 4 millimetres. On sending a powerful
current through the mercury by means of the two wires the
mercury in the immediate neighbourhood of the electrodes was
elevated into a small cone 2 or 3 millimetres in height.
(3) Some notes on Clark’s Cells. By R. 'T. GLAZEBROOK, M.A.,
Trinity College, and 8S. SKINNER, M.A., Christ’s College.
The paper relates to the causes of the variation of electromotive
force of the cells. In addition to the causes indicated by Lord
Rayleigh the authors find that the state of amalgamation of the
zinc pole may cause a fall in force if the zine does not shew a
bright surface. This is worked out by means of a testing cell into
which the faulty zines are transplanted. The result is confirmed
by Swinburne’s experiments on zinc rods in zinc sulphate solution.
To correct this fault previous amalgamation in the presence of
dilute sulphuric acid is recommended, or immersion of the zinc in
the paste. Dr Hopkinson’s method of testing cells by tapping
was shewn.
(4) Illustrations of a Method of Measuring Ionic Velocities.
By W. C. D. WueruaM, B.A., Trinity College.
(5) On Gold-Tin Alloys. By A. P. Laurie, M.A., King’s
College.
November 9, 1891.
Dr GASKELL IN THE CHAIR.
The following Communications were made to the Society:
(1) Note on a Peripatus from Natal. By A. Sepewick, M.A.,
Trinity College.
[Received November 10, 1891.]
Last spring I received from Mr P. 8S. Sutherland a single
specimen of Peripatus, which was found by Mr J. F. Quickett in
the Botanic Gardens of Pietermaritzburg, Natal. The specimen
possessed 22 pairs of claw-bearing legs, and presented all the
characters of P. Moseleyi, as described in my monograph of the
genus,
1891.] Mi Bateson, On Variations in colour of cocoons. 251
The specimen measured 5} centimetres in length. The ventral
surface was of the usual light colour. The ground colour of the
dorsal surface was dark green, with a sufficient number of yellow
papillze to give a yellow tinge to the specimen. There was a light
yellow band, from which the green was almost entirely absent, just
dorsal to the insertion of the legs.
The specimen was a female, and the last leg was without a
white papilla.
The genital opening was subterminal, and behind the legs of
the last pair.
The uterus was full of embryos, at a stage of development
more advanced than in a Cape specimen killed at the same time of
year.
The number of legs of the embryos was, as in the adult, 22
pairs.
The specimen is of interest, as being the first recorded from
Natal.
(2) On Variations in the Colour of Cocoons (Saturnia carpini
and Eriogaster lanestris), with reference to recent theories of Pro-
tective Coloration. By W. Bareson, M.A., St. John’s College,
[Abstract ; received November 11,1891. Reprinted from the Cambridge
University Reporter, November 24, 1891.]
The cocoons of several moths, eg. the Emperor and Small
Egger, vary in colour from dark brown to white. It is believed by
some that these colours have a protective value as a means of con-
cealment, and it has been stated by Poulton and others that when
spun on leaves which will turn brown, or in dark surroundings, the
cocoons are dark, while they are white if spun on white paper. To
account for this phenomenon “the existence of a complex nervous
circle” has been assumed. The present experiments shewed that
it is true that larvee left to spin on their food-leaves produce dark
cocoons, and also that if they are taken out and put in white paper
the cocoons are white. But it was found that larve similarly
taken out and made to spin in dark substances also spun white
cocoons, and indeed that starvation, or merely interference at the
time of spinning, may lead to the production of a white cocoon. On
the contrary, if white paper is put amongst the food, so that the
larvee can, of their own choice, walk into it and spin, the cocoons
are generally dark. It was noticed in several cases that larve
which had been shut up evacuated a quantity of dark juice having
the natural tint of the cocoon, and the suggestion was hazarded
that absence of colour in the cocoon perhaps results from the loss
or retention of this juice, which may be of the nature of me-
conium,
20-—2
252 Miss R. Alcock, On the Digestive [Nov. 9,
(3) Ewhibition of Phylloxera vastatrix. By A. E. SHIPLEY,
M.A., Christ’s College.
PROFESSOR HUGHES IN THE CHAIR.
(4) The Digestive Processes of Ammocetes. By Miss R.
ALCOCK (communicated by Dr GASKELL).
[Received December 3, 1891.]
In all the higher vertebrates digestion is carried on by means
of the secretion of specialised glands localised in certain definite
portions of the alimentary canal or of glandular masses which are
formed as appendages of the same. The formation of a peptic fer-
ment is confined to the glands of the stomach, of a tryptic ferment
to the pancreas and of diastatic ferments to the salivary glands and
pancreas. Passing to lower forms we find in Amphibia that the
formation of peptic ferment is not restricted to the stemach, but
extends oralwards into the cesophagus, which is even more active
as an organ for digestion than the stomach. Then in Fishes this
tendency to diffuseness in the position of the proteid-digesting
glands is even more pronounced, and it is remarkable, as Kruken-
berg has pointed out, how their position varies even in nearly
allied families. Sometimes a pancreas is present, sometimes
absent; in some the appendices pylorice are well developed, in
others they do not exist, and in some they function as digestive
glands, whilst frequently they seem merely to act as organs of
absorption. In some cases the so-called pancreas does not func-
tion in proteid digestion, and in many fishes certain cells forming
part of the liver secrete a tryptic ferment, which enters the ali-
mentary canal by means of the bile-duct. The pepsin-forming
glands also vary in position, sometimes extending oralwards and
sometimes into the upper part of the intestine.
These observations of Krukenberg lead to the conclusion that,
as far as digestive organs are concerned, the lower we descend in
the scale of evolution of the vertebrates, the greater is the ten-
dency towards a decrease of specialisation in function and a diffuse-
ness in the position of the secreting cells. If this is the case, the
study of the digestive processes in the lowest vertebrates ought to
shew this absence of concentration of the secreting tissues in a still
more pronounced manner. With this object Dr Gaskell proposed
to me to find out how the digestive processes were carried on in
the Ammoccetes, as no physiological observations have been made
on the digestion of this primitive vertebrate by Krukenberg or any
other observer.
——_——
1891.] Processes of Ammoceetes. 253
We may consider for this purpose that the alimentary canal
consists of three portions, lst the pharynx, 2nd the narrow tube or
anterior intestine which leads from the pharynx to the mid-gut
and terminates posteriorly at the entrance of the bile-duct where
the sudden enlargement of the alimentary tube marks the begin-
ning of the 3rd portion, the intestine proper. The glandular
appendages in connection with these parts are, (1) the so-called
thyroid gland, with its duct opening into the pharynx, and (2)
the liver with its duct opening into the intestine.
In order to test for the presence of digestive ferments, I made
extracts of the epithelium lining the pharynx, the liver, the intes-
tine and the thyroid in ‘2 °/, HCl. or in glycerine, using in each
experiment the organs of two or more Ammoccetes of Petromyzon
Planeri. I may here mention that I can confirm Krukenberg’s
experiments on the digestive ferments in fishes and invertebrates
as to the temperature at which they are most active. I find my
extracts are very much more powerful at 38° to 40°C. than at -15°
to 20° C., many authors giving the lower temperature as that at
which the digestive extracts of fishes and cold-blooded animals
generally are most active. I found that all parts of the alimentary
canal, with the exception of the thyroid gland, were capable of
digesting fibrin in a ‘2°/, HCl. medium with greater or less
rapidity. I used carmine-stained fibrin after Griitzner’s method,
and always used as control experiment a ‘2°/, HCl. solution alone
and an extract of some tissue of the animal which was inactive in
digestion.
The results of these experiments may be summed up as
follows :—The extract of the liver is always the most powerful ;—
thus in one case about 1 c.c. fibrin was entirely digested in from
4 to ? hr. by an extract made from the livers of two Ammoceetes,
‘The epithelium of the pharynx comes next in activity, thus in the
case mentioned the epithelium of the pharynx of the same two
animals digested the same amount of fibrin in about Id hrs. Next
in order of activity comes the extract of the intestine; this always
gives decided evidence of digestive power, but if carefully cleaned
out with a soft brush before the extract is made, so as to remove
as far as possible the secretion from the liver, then its power of
digestion is very far behind that of the pharynx or liver, although
in all cases digestive activity is still manifest. Finally the extract
of the thyroid is always inactive.
So far I have not succeeded in obtaining any results in a 1°/,
sodium carbonate medium, and conclude that tryptic ferment is
absent. In relation to this it is interesting that in a young
Selachian Krukenberg found that a tryptic ferment was entirely
wanting, and he suggests, for this and other reasons, that in primi-
tive vertebrates the digestion was rather peptic than tryptic.
254 Miss R. Alcock, On the Digestive [ Nov. 9,
I conclude then from these experiments that :—
1. The proteid digestive ferment in the Ammoceetes is of the
nature of pepsin rather than trypsin.
2. This ferment is very diffuse in position, being found in all
parts of the alimentary tract.
3. It is found mainly in the anterior part of the tract,
especially in the respiratory portion of the pharynx and in the
liver.
4. The so-called thyroid gland has nothing whatever to do
with the digestion of proteids.
Perhaps the most novel and important feature of this series of
experiments is the evidence of the importance of the pharyngeal
cavity for proteid digestion in this primitive form of vertebrate ;
and when we come to examine the histological structure of the
alimentary tract we find that glandular secreting structures are
more conspicuous in this part of the alimentary canal than in the
intestine proper. In the pharynx the epithelium lining the body-
wall and the adjacent branchial surfaces is undoubtedly different
from the single layer of flattened epithelium cells which cover the
lamellar folds of the branchize themselves. It is usually described
as consisting of several layers; but very conspicuous amongst the
small epithelium cells are numerous glandular looking cells, to
which I have never found any reference in descriptions of this
region. ‘These cells are arranged in groups of five or six together,
and correspond in height to the whole thickness of the epithelium ;
they are somewhat swollen in the middle and are covered super-
ficially by the small epithelium cells with the exception of a small
space above the centre of the group where their tips converge to-
gether at the surface. In some preparations a collection of deeply
stained granules can be seen at the outer ends of these cells, and
in others a stringy mucus-like secretion is issuing from them. It
is striking how on the branchial folds of the anterior wall of the
first gill-pouch the epithelium containing these cells predominates,
only the most internal folds appearing to have retained their respi-
ratory function. Clearly the abundance and the evident activity of
these pharyngeal glands is a sufficient histological reason why the
extract of the pharynx is so active in digestion. On the other
hand the epithelium of the narrow anterior intestine consists of a
single layer of tall columnar cells with a distinct cuticular border
and uniformly ciliated ; there is no histological evidence here of the
presence of any secreting cells. The epithelium of the rest of the
intestine is very uniform, and similar in character to that of the
anterior intestine, though only the anterior dilated portion is
ciliated ; and the ciliation even here is not uniform, but occurs in
patches. The whole structure of this region suggests an organ of
absorption rather than of digestion.
——
1891.] Processes of Ammoceetes. 255
The only glandular organ in connection with the intestine is
the liver, with its large gall-bladder ; in structure it is typically a
tubular liver throughout, and there is no evidence of any differen-
tiation of function in the cells.
Round the entrance of the bile-duct into the intestine are a
few small glandular follicles which have received the name of
pancreas, but in most specimens they are too few and too small for
it to be possible that they could play any important part in diges-
tion, at all events during the Ammoccetes-stage. After transfor-
mation the liver becomes completely separated from the alimentary
canal, the duct becomes obliterated, and the liver itself undergoes
fatty degeneration. At the same time the small glands in the
wall of the intestine which have been called the pancreas increase
in number and form quite a prominent ring. If these glands are
pancreatic in function, it suggests itself that possibly the tryptic
digestion may become more important than the peptic as the
animal advances to the higher stage of evolution in the adult
Petromyzon. I hope to be able to work out the digestive processes
of the adult animal as soon as I can obtain material.
Finding the presence of a peptic ferment to be so general in all
parts of the alimentary canal, the question arises if it is present
also in other tissues of the animal. I have already mentioned that
ne sign of digestive activity can be obtained from extracts of the
so-called thyroid, extracts of the central nervous system are also
quite inactive, and extracts of muscle shew only the faintest signs
of activity after many hours. Then I thought it might be of
interest to test for peptic ferment in the skin as being an epithelial
structure continuous with the lining of the pharynx and on account
of its active secretion; and I found that a -2°/, HCl. extract of
the skins of two Ammoccetes digested lec. fibrin in a little over
lhr. The skin consists of several layers of cells, the most super-
ficial being those which secrete. They are peculiar in having a
very thick cuticular border, whose striated appearance is due to
the presence of fine pores through which the secretion exudes,
These cells are full of granules and when treated with methylene-
blue in the living condition, as observed by Mr Hardy, the granules
at the base of the cell stain blue, whilst those towards the surface
appear rose-coloured in artificial light; this reaction he has every
reason to believe indicates the presence of a zymogen.
As to the significance of this secretion, it does not seem
probable that it could function in the digestion of food. I think
it must have some important function, and considering the habits
of the animal, which lies buried in the mud, Mr Hardy has sug-
gested that this secretion of the skin may act as a protection
against the attacks of bacteria and other organisms, which might
otherwise be injurious.
256 Mr Hardy, On the Reaction of certain [Nov. 9,
(5) On the Reaction of certain Cell-Granules with Methylene-
Blue. By W. B. Harpy, B.A., Gonville and Caius College.
[Received November 30, 1891.]
In 1878, Prof. Ehrlich pointed out the fact that the granule-
containing cells of the body, whether found free in the body fluids
or elsewhere, could be distinguished from one another by the
character of the reaction of their granules with different aniline
dyes*. He distinguished five classes of granules characterised by
staining with acid, basic, or neutral dyes, or inditferently with acid
or basic dyes (amphophil). The present communication deals with
a further subdivision of the basophil granules into two groups,
characterised by very distinct colour-reactions with the basic dye
methylene-blue, the one class of granules staining a deep blue, the
other a bright rose.
The distinction of tint depends, for some reason not at present
at all obvious, on illumination with yellow artificial light. Under
these circumstances the colour-contrast is one of extraordinary
brilhancy. With daylight, or with gaslight after it has been
filtered through neutral-tint glass, the rose colour either appears a
blue like that shewn by the blue-staining granules, or is dulled to
a blue-violet tint. The explanation of this change may be found
in the fact that the yellow gaslight is relatively richer in red rays
(or poorer in blue rays) than is daylight, or the phenomena may be
of a more complex nature. Be this as it may, the abrupt transi-
tion trom bright rose to bright blue produced by directing the
mirror from the gas flame to the window is a striking feature of
this colour-reaction.
The discrimination of basophil granules into rose-staining or
blue-staining varieties may depend upon the dichroic nature of
methylene-blue. Thin films produced by running a minute quan-
tity of a strong solution under a coverslip appear rose with arti-
ticial light, while more dilute solutions appear blue.
Methylene-blue is a salt having the composition of a chloride,
the base being a pigment of the aniline series, and it has already
been noticed that alkalies produce a rose or reddish modification
possibly by decomposing this salt and liberating the pigment-base.
This suggests that the rose tint may not be solely a physical
phenomenon but may depend upon a chemical action of the
granule substance on the dye, or a chemical change produced by
the osmosis of the pigment through the cell protoplasm to the
granule.
That the reaction does not depend simply upon the thickness
* The various papers dealing with this subject have been republished by
Dr Ehrlich in pamphlet form under the title ‘‘ Farbenanalytische Untersuchungen
zur Histologie und Klinik des Blutes.” Berlin, 1891.
1891.] Cell Granules with Methylene-Blue. 257
of the film of methylene-blue is shewn by the fact that the
rose-tint may be developed by granules of sizes varying from mere
points to spherules 2 to 4 in diameter. Nor does it depend upon
the solidity or fluidity of the staining substance; for the contents of
large vacuoles in the ectoderm cells of Daphnia frequently stain
an intense rose, the rest of the cell appearing blue. Lastly the
rose tint may be produced neither in the cell nor in the animal,
but (in the case of Daphnia) by the action on the dye of a sub-
stance poured by the ectoderm cells into the surrounding water.
The hypothesis that the reaction is really of a chemical nature
is favoured (1) by the general fact that the imbibition of dyes by
the fresh unfixed cells is determined by the chemical nature of
those dyes—whether the pigment be basic, or acid ; and (2) by the
peculiar method of imbibition of dyes by fresh cells. If still living
cells, such as basophil blood corpuscles, are treated with either an
organic fluid or normal salt solution, in which a small quantity of
methylene-blue is dissolved it is noticed that imbibition of the dye
is coincident with the onset of death. So long as the cell remains
fully alive it resists infiltration by the pigment, and the granules
remain uncoloured. This condition may, especially with eosinophil
cells, last for hours. With the first onset ot death the dye makes
its way through the protoplasm and the granules become coloured.
Later, when rigor mortis has become thoroughly established, the
nucleus and cell body absorb the dye, and appear blue. In other
words the first imbibition of the dye occurs at a period when the
complex cell protoplasm is commencing to disintegrate, and when
therefore profound chemical changes are taking place.
In order to determine whether granules are of the rose or blue
staining varieties it is necessary to apply the stain in some rela-
tively innocuous Huid to the living cells; and subsequent treat-
ment with fixing reagents entirely obliterates the reaction. This is
because all fixing agents with which I have experimented have
some action on methylene-blue. Thus corrosive sublimate pro-
duces a rose-coloured modification, and converts blue staining into
violet or rose. Ammonium picrate produces a violet tint except
in the case of very intensely blue granules. Osmic acid converts
the rose into a blue tint.
Rose-staining basophil granules have been found by me in free
basophil cells of Astacus, and of Vertebrates, in the ectoderm of
Daphnia, and of the Ammoccete larva of Lampreys, and in the
alveolar cells of salivary glands.
The last two instances are of a specially suggestive nature, as
affording instances of cells containing at the same time blue and
rose-staining granules. In the cells lining the alveoli of the sub-
maxillary gland of a rabbit I have seen, after treatment with dilute
methylene-blue in normal salt solution, a zone of rose-coloured
258 Reaction of certain Cell-Granules with Methylene-Blue. [Nov. 9,
granules surrounding the lumen and extending about half-way
towards the basement membrane, while outside this there was a
zone of blue-staining granules. This suggests that the rose-stain-
ing condition is a final stage in the elaboration of the constituents
of the granules of these cells.
A still more instructive example is found in the ectoderm cells
of the Ammoceete larva which I examined at the request of Dr
Gaskell. These cells are each overlaid by a thick cuticle per-
forated by coarse canaliculi which lead from the cell protoplasm to
the external surface of the animal. Miss Alcock has shewn that
these cells, under appropriate stimuli, discharge on to the general
surface a viscid substance which has the power of rapidly digesting
fibrin in an acid medium. If these cells are treated with methy-
lene-blue we find (1) that the extruded secretion gives the rose
reaction, (2) that the pores in the cuticle may appear as rose-
coloured rods, owing to their being filled with the secretion, and
(5) that the cells themselves are occupied by rose-coloured granules
which lie in the half of the cell next to the cuticular border, and
by blue-coloured granules which occupy their deeper portions.
In the ectoderm of Daphnia rose-coloured granules are scanty,
while, under certain circumstances to be detailed elsewhere, the
cells may include a number of large vacuoles, the contents of
which give a brilliant rose reaction. In connection with the
presence of these vacuoles we find that Daphnia possesses the
power of extruding on to its surface, through cuticular pores, a
substance which swells up to form a jelly in water, and stains
brilliant rose. This particular case will receive more detailed
description on some future occasion. For the present I will only
say that the secretion is used by the animal to prevent parasitic
vegetable or animal growths obtaining a foothold on the shell.
I have never yet found a blood or lymph cell with both blue
and rose-staining granules. It may be regarded as probable that
blue-staining granules are absent from wandering cells. The cells
with rose-staining granules have a remarkable distribution. In
Astacus, as I have noted elsewhere*, they occur normally lodged in
the spaces of a peculiar tissue which forms an adventitia to some
of the arteries. They are only discharged into the blood as a
result of special stimuli. In Vertebrates they occur to a marked
extent in the peculiar adventitia of the blood-vessels of the spleen.
It is noticeable that I have so far failed to find rose-staining
granules in endoderm cells, though I have examined the lining cells
of the alimentary canal and of its glands in very diverse animal types.
The cells of the excretory organ (end-sac of Daphnia) contain
granules which have a remarkable affinity for methylene-blue and
stain a deep opaque blue.
* Journal of Physiology, 1892.
1891.] Mr Macdonald, Self-Induction of two Parallel Conductors. 259
November 23, 1891.
Proressor LEWIS IN THE CHAIR.
S. Skinner, M.A., Christ’s College, was elected a Fellow of the
Society.
The following Communications were made:
(1) The Self-Induction of Two Parallel Conductors. By H.
M. Macpona.pD, B.A., Clare College.
(A bstract.)
In Article 685, Maxwell’s Electricity and Magnetism, Vol. 11.,
: ; Be haan
the expression 4 (yu -+ mw) + 2, log aq’ © given for the self-in-
duction per unit length of two parallel infinite cylindrical con-
ductors, the radii of their sections being a, a’, the distance between
their axes b, w, w’ their magnetic permeabilities, and pw, that of
the surrounding medium. Lord Rayleigh remarked in the Phil.
Mag. 1886, that this expression only holds when w=p’=yp,. To
solve the general problem for steady currents it is necessary
to know the conditions satisfied by the components of vector
potential at the bounding surface of two media of different
magnetic permeabilities ; they are found to be
a Py |
LOE 1 oy see
pe On pi’ On
Transforming the equations by the relation
x+ecy=ctand(E+ 0),
taking 7 =a, 7 =—( as the bounding surfaces of the conductors,
we have
+ Aro z9 Bo ae
Medico cst.
oH, vn, ke =
og* = 0, ye 4 to Nese B,
OH’ cm Anrp w’ 4 eae
oF" + On? “ain tasce 7=—B8 to Ss
a — HH, and yee zilellly when 7 =a,
On pf, On
H,= H’ an cau? peg a oe when 7 =— 8,
aver. On
F and G being ike
260 Mr Macdonald, On the Self-Induction [Nov. 23,
Solving these for H, H, and H’ and determining LZ from them,
we obtain
L=}(u+p')+2u (a+)
(142) en (a—B) 4 (1+) en 8-2) + ( He) gon (a+ 98) s (1-He) e~méSetA) _ ge-ma+6)
“u u a Me
a]
=
~b4
(2 +H) en(a+B) _ (2 z= He) (a 2 Hs) e—n(atB)
be be rr
When yp’ =n, we have the case of the a conductor iron, the other
being copper, and then
b?
Ga
h?
L=}(ut+m)+ 2y,log 7 + ay
Ko lor =
a’ M+ By ay
the repulsive force between the conductors being
Sue (1-4 = )
b e+p, Pa
Taking the case of conductors of equal section, the following
table shews how the variable part of the coefficient of induction
varies with their distance apart.
b | ee B-Po;, P Increase L—50°5 L-50°5
~ aa’ +p, °b?—-a*| per cent. Maxwell from above
2a 1-38629 ‘282007 20°3 2°7725 3°3365
| 3a 2°19722 ‘11776 5°3 4:3944 46299
| 4a 2°77258 06326 2-2 55451 5°6716
| 5a 3°21887 039829 1-2 64377 6°5173
6a 358351 027583 “¢ 771670 72221
7a | 389164 020211 D 7°7832 78237
8a | 4:15888 ‘015936 “3 83177 8-3496
9a 4°39425 012131 "2 87885 8-8127
10a 460517 ‘009851 “2 9-2103 9-2300
The first column gives the distances between the axes of the
conductors, the second the values of half the variable term in
Maxwell’s formula, the third half the term which has to be
added to it, the fourth the increase per cent. of the variable part
due to the term neglected by Maxwell, the fifth and sixth the
values of the variable parts in both cases; w, being taken unity
and 4=100. The table shews that the term neglected is con-
siderable when the conductors are close to one another, and
decreases rapidly at first as b increases, afterwards slowly.
7.
1891] of two Purallel Conductors. 261
Again taking the conductors touching one another, the follow-
ing table gives the maximum values of the correction as the
radius of the iron conductor increases.
a b log My | ase b° | Increase L-50°5 |L-—50°5 from
aa’ |utph, ~b?—-a?) per cent. Maxwell |above formula
a 2a’ | 1:38629 *282007 20°3 2°7025 3°3365
2a’ 3a’ | 150407 | 576147 38°3 30081 4°1604
3a’ | 4a’ | 1:67397 810307 | 48:0 3°3479 4:9685
da’ | 5a’ | 1-83257 | 1001419 | 546 36651 56679
5a' | Ga’ | 197407 | 1-162144 | 58-8 3-9481 62724
6a’ | 7a’ | 2-10005 | 1300593 | 61-9 42001 68012
Ta’ | 8a’ | 2-21297 | 1:422097 64-2 44259 72701
8a’ | 9a’ | 2-31447 | 1-530317 66:1 4-6289 76895
9a’ | 10a’ | 2-40794 | 1627843 67°6 4-8159 8-0716
10a’ | lla’ | 2-49320 | 1:716587 68-8 49864 8-4195
_
The first column expresses the radius of the iron conductor in
terms of that of the other; the remaining columns are as in the
first table.
The expression for the force can by suitably choosing a, a’, b
so that b is somewhat greater than a/2, be made to change sign,
so that the force between the conductors would then be attractive.
In the case of two iron conductors
b° b*
L=pt+ 2p, [toe oa +A log @— a) =a")
lie (b° = a =o a”)
(b° at a”)(b* = a”)
: b? (b° = a es a”)?
+2 log (— a*)(b*— a?) (e— Fs ab} (be ay a°b*) Sh et. >
+ 207 log
St Pres:
+ My
In this series the coefficients of the powers of \ rapidly diminish.
It is perhaps worthy of remark that when p» is 100 or greater,
the part depending on the size of the conductors and their
distance apart is only slightly affected by the value of yp, as >
differs but little from unity.
where
Mr Orr, On the Contact Relations of certuin [Noy. 23,
bo
o>
bo
(2) The Effect of Flaws on the Strength of Materials. By J.
Larmor, M.A., St John’s College.
The effect of an air-bubble of spherical or cylindrical form in
increasing the strains in its neighbourhood was examined; and it
was suggested that the results might be of practical service in
drawing ‘general conclusions as to the influence of local relaxations
of stiffness of other kinds. In particular, it is shewn by the aid
of the hydrodynamical form of St Venant’s analysis, that a cavity
of the form of a narrow circular cylinder, lying parallel to the
axis of a shaft under torsion, will double the shear at a certain
point of its circumference; and the effect of a spherical cavity
will not usually be very different. For a cylindrical cavity of
elliptic section, the shear may be increased in the ratio of the
sum of its two axes to the smaller of them, this ratio becoming
infinite in accordance with known theoretical principles for the
case of a narrow slit. It is assumed in the analysis that the
distance of the cavity from the surface of the shaft is considerable
compared with its diameter, so that the influence of that boundary
may be left out of account in an approximate solution’.
The results will however also give the effect of a groove of
semicircular or semi-elliptic section, running down the surface of
the shaft, provided the curvature of the surface is small compared
with the curvature of the groove.
(3) The Contact Relations of certain Systems of Circles and
Conics. By W. MF. Orr, B.A., St John’s College.
(Abstract.)
The following theorem is first established:—If four circles
X, Y, P, Q, in a plane or on a sphere, are such that a circle can
be drawn through one of each pair of intersections of X with P,
X with Q, Y with P,and Y with Q respectively, (and therefore
another circle through the remaining four intersections of the
same circles,) the eight circles which touch X, Y and P, and the
eight which touch X, Y and Q, can be arranged in sixteen groups
of four circles, each group consisting of two touching X, Y and P,
and two touching X, Y and Q, such that each group has two
common tangential circles besides X and Y.
A similar result is of course true for groups of circles touching
P, Q and X, and P, Q and Y respectively.
The above relation of condition is tr iply satisfied by the four
circles that form any Hart-group of circles touching three others
(restricting the title Hart-group to the eight groups that are
analogous to the inscribed and escribed circles of a triangle).
1 See Phil. Mag., Jan. 1892.
1891.] Systems of Circles and Conies. 263
Hence, taking as a particular case the inscribed and _ escribed
circles of a plane or spherical triangle, the following result is
obtained:—If we describe circles touching three by three the
inscribed and escribed circles of a plane or spherical triangle, we
obtain four sets of four circles, exclusive of the sides of the
original triangle and its Hart-circle ; each set forms a Hart-group
and in addition we can obtain twenty-four groups of four circles by
taking two out of any one set and two out of any other such that
each group is touched by two circles besides the two circles they
have been constructed to touch in common.
Any group whatever of four circles of the eight that touch
any three given circles satisfy the above relation of condition,
some singly, some doubly, and some trip!y; and by taking all
such groups of four, and describing circles touching them three
by three the following result is obtained:—Eight circles can be
described to touch three given circles; these eight form fifty-six
groups of three; to touch any three we can describe a set of four
circles exclusive of the original three, and one which with them
forms a group of four circles touching four others; and we can
form a thousand and eight groups of four circles, two out of one
set and two out of another, such that each group is touched by
two common circles besides the two they have been constructed
to touch in common.
These theorems are then extended to co-vertical cones which
are either circular or have double contact with a given one, and by
projection to conics having double contact with a given one.
In the last case one of the results obtained is:—If four
straight lines X, Y, P, Q are such that through the intersections
of X with P, X with Q, Y with P and Y with Q there can be
described a conic having double contact with a given one ¢, then
the four conics touching X, Y and P and the four touching XY, Y
and Q, all having double contact with ¢, can be arranged in eight
groups each consisting of two touching X, Y and P, and two
touching X, Y and Q, such that each group, besides touching X
and Y, touch in common two tangent conics having double contact
with ¢; and similarly for conics touching P, Q and X, and P, Q
and Y respectively.
The enunciation of the reciprocal theorem is obvious.
Another result is as follows:—Four conics can be described
baving double contact with a given one ¢, and touching three
given lines or passing through three given points; to touch any
three of these four conics sixteen conics can be described having
double contact with ¢, exclusive of the original lines or points
and four Hart-conics; there are thus four sets of sixteen conics. In
addition to the groups of four touched by four conics having double
contact with @ that can be formed by taking four out of the
264 Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23,
same set, there are thirty-two groups of four conics formed by
taking two out of any one set and two out of any other, such
that each group is touched by two conics having double contact
with the given one ¢, besides the two they have been constructed
to touch in common, and as the four sets can be taken in pairs in
six ways, there are thus one hundred and ninety-two such groups
in all.
Some other theorems are obtained of a more complicated
character.
The method of proof is purely geometrical and the first pro-
position, though not proved as shortly as it might have been, is
made to depend mainly on a property of Bicircular Quartics
given by Mr C. M. Jessop in the Quarterly Journal of Mathematics,
Vol. xx1tI., which is equally true for Sphero-Quartics, and which
for the case of two circles may be enunciated a little differently
as follows:—If X, Y are any two circles of the same family
touching two given circles A and B, in a plane or on a sphere,
and P, Q are any two circles of the other family touching A and
B, then a circle can be drawn through one of each pair of the
intersections of X with P, X with Q, Y with P, and Y with Q, and
of course another circle through the remaining four intersections
of the same pairs.
(4) On Liquid Jets under Gravity. By H. J. SHARPE, M.A.,
St John’s College.
1. The motion (Fig. 1), which is in two dimensions, is sup-
posed to be symmetrical with regard to a’Ox which is the axis
of the vessel and jet. BHF is the semi-outline of the vessel,
FJ of the jet. AF is the semi-orifice, which is small compared
with the dimensions of the vessel and the depth OA of the
liquid. Gravity acts parallel to #’Oz. OF is the surface of the
liquid, which is maintained steady. AF is supposed to be so
small that it may be considered either as the are of a circle
1891.] Mr Sharpe, On Liquid Jets under Gravity. 265
with centre O in the surface of the liquid, or as a small straight
line perpendicular to Ox. For simplicity we shall take OA the
radius of the circle (or the depth of the liquid) as unity.
If g be the acceleration of gravity referred to this unit it
will be convenient to put
tog... FR ld ade (1).
We shall take O as the origin of Cartesian and Polar Coordinates
x, y, 7, @ and we shall put
ST et (Ss Bee <2) ee eee eee (2).
Let x be the stream function to the right of AF’; u, v the
velocities parallel to Ox, Oy.
Further let oA MES” olf an oo See SRO rence (3),
where p is a large number.
On the right of AF we will take
dx _ re eee IRIE yA
a u =ar* cos 0 + Xe,’e cos pny |
1 —pnx’
os. seal
348s — 2 sin 40 + Xc,’e rg
where c,’ is an arbitrary constant. = indicates summation with
regard to n for all integral values from 1 to «.
Therefore along AF, we have on the right of it at every point,
nearly
u=a+Xc,’ cos pry (5)
ee ee eee :
[Of course AF is really half the small segment of a circle.
The equations (4) and (5) are only approximate (the more so
the larger p be taken) but it will be pointed out afterwards
(Art. 5) how their exact forms can be found—forms which would
be suitable for all values of p—but as these are rather complicated
it is better to begin with the simplest case first. ]
Let be the stream function on the left of AF, and
on the left of AF we will take
oY u=S(a,e"" cos my) + Se," cos pny +A
dy (6)
pra’
e”” sin my) — Xc,e”"” sin pny
— a= S(a
™m
where a,, c, and A are arbitrary constants, and S indicates
summation with regard to m for a finite number of values of
m, the largest of which is supposed to be small compared with p.
VOL. VII. PT. V. : 21
266 Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23,
Therefore along AF we have on the left of it at every point
u=S(a,, cos my) + Xe, cos pry + A\ (7)
vy =—S(a,sinmy)—Xc,sinpny )
But as the motion must be continuous through AF, the w’s
and v’s in (5) and (7) must be the same. Therefore we get
— S(a,, sin my) +4ay=% (c,/+¢,)sinpny ) 0” }
These must hold from y=0 to y=7/p. But if we expand
the left-hand sides of (8) by Fourier’s Theorem, we can get c, and
c,’ as functions of n.
2. Doing so, we shall get
¢’ +¢,= “(3 rinses =) 6 i (9).
sf p MN
UT sae Se
P .
c, —c,=—2S/a,,. a in P a” Sinan (10).
nar — —
Also 7 ar (a, x sin =) Bybee. (11).
mar p
Also in order that the second of the equations (8) may hold
at the limit y = 7/p, we must have
-8 (a, sin ™™) oT =0
p/ 2p
We shall now prove an important property of c, and ¢,’.
As 1 is the least value of n, and as by hypothesis m/p is always
a small fraction, we may always (if need be) safely expand the
fractions in (9) and (10) in ascending powers of m*/p*n’.
ee
Doing so in (9), we shall get
2
of $6,228 fa, sin 8 (4 ME be
p nT
But by (12) the first and last term here disappear, so that
(c,/+c¢,) 18 always a small quantity at most of the order 1/p*.
Similarly from (10) we see that (c,’—c,) and therefore ¢, and c,’
are always small quantities at most of the order 1/p*. We say
‘always’—even when n =1 when they are largest.
1891.] Mr Sharpe, On Liquid Jets under Gravity. 267
3. From (6) the equation to BHF is
8 (t= 2” sinm 4S Sn CPR! in om + Ar
=: y) +> pny + Ay
= § (Se sin mn), =2 ee (14).
m p p
If y=b is the equation to the asymptote to BEF
Ab=8 (2 sin™) = ae (15),
iu 7
so that if b be finite, A is a small quantity of the order 1/p.
Looking now at (6) we see that if OZ is to be the surface
of the liquid, uw and v must, when 2’ =—1, be small quantities
at most of the order 1/p. A and the & term already satisfy
that condition. In the S term m has several values, enough
to satisfy conditions (11), (12) and (15). Suppose the particular
m in (6) to be the smallest of these values, and suppose m = log p,
then when 2’ = —1, the S term also satisfies the surface condition,
and the more accurately the larger p is, since log p/p diminishes,
as p increases.
If FJ is to be a jet we must have, since AF is small, at every
point of the jet, nearly
w+ v= gr.
But we see at once from (1) and (4) and Art. 2 that this con-
dition is fulfilled, the error being of the order 1/p’.
4. To get some idea of the maximum value of this error, we
see from (4), since at F' we have nearly
u=a-—c,
that — 2c,’/a is a fair measure of this maximum.
From (10) and (13)
C, = SR a see ee (16).
From (11) and (12) we have nearly, since A is a small
quantity,
—a+S(a,)=0, and 5 S (ma,,) = 0.
If for instance we take 8 and 9 for the two values of m, then
p will be about 2981 and the maximum error about
+ 0000143.
268 Mr Sharpe, On Liquid Jets under Gravity. [Nov. 25,
If we took a sufficient number of values of m to satisfy, in
addition to previous conditions, the condition S(m’*a,,)=0, we
see from (16) that the maximum error would be of the 3rd order
of smallness, and so on for higher orders.
5. Suppose p instead of being large, were somewhat smaller,
we should then proceed thus.
NA x
From F (fig. 2) draw FN perpendicular to Az.
Let VF = 7/p’ where
In equations (4) to (8) &c. put p’ for p, and consider (4) to
apply to the right, and (6) to the left of NF. (8) also would have
to hold from y=0 to y=7/p’. Of course from (17) p’ could be
expressed as accurately as desired in terms of p.
6. From (4) the equation to the outline FJ of the jet is
2a 3 . 3 Cy —p et Bee _ 2a . 30
es Sin) + Fo pty = ee ooh LOR
As the & term is of the order 1/p’, we see that in all solutions
obtained by the present method the shape of the jet is nearly
independent of the shape of the vessel, and is dependent only
on the angle which the orifice subtends at O.
Upon this point light may be thrown by the following
Article.
nN
7. Since writing the preceding, I have examined somewhat
carefully equation (14) which gives the outline of the vessel—
in the case where m has the values 8 and 9. In this case a,,
and a, are determinate. I find that in (15) 6 is not perfectly
arbitrary, but appears to have limits in order that the curve
BEF may be continuous. I have tried to take it as large as
possible. I have actually taken it =27/9 or about ‘6981, but
1891.] Mr Sharpe, On Liquid Jets under Gravity. 269
whether this is the largest admissible value of b (for m = 8 and 9)
I am not sure. The result is that I get a curve something of
this shape (fig. 3) for BEF. .
y
One noticeable feature is the existence of a long spout or pipe-
like portion GF’ before we reach the orifice #’, which may perhaps
explain the result noticed in Art. 6.
I am afraid this spout exists in all solutions obtained by the
‘present method. There is an interesting point about the curvature
of the outer stream-line in the neighbourhood of F. It will be
found from equations (14) and (18) that there is a sudden change
of curvature at /—that the curvature on the nght of F divided
by curvature on left of F gives a small quantity of the order
1/p, thus corroborating (as far as the approximate character of
the present method will allow) a remark of Kirchhoff found at the
end of Art. 96 of Lamb’s Motion of Fluids.
(5) Theory of Contact- and Thermo-Electricity. By J.
PaRKER, M.A., St John’s College.
In this paper, which treats of an electrified system of metals
situated at rest in a vacuum in an unvarying state, we shall use the
electromagnetic C.G.S. units, so that if Q, Q be the charges of
two small bodies at a distance of 7 centimetres, the electric
,
repulsion between them is as dynes, where e is the constant
87 x 10”.
We first require to know the energy U and entropy ¢ of our
system. The system being necessarily so chosen that its total
270 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
charge is zero, let us suppose that by altering the relative positions
of its parts, every one of its charges becomes zero, and denote the
corresponding values of the energy and entropy by U, and ¢,. If
from any cause, such as the parts of the system not being sufficiently
numerous, this (or any other) operation cannot be directly per-
formed, we may always make use of subsidiary bodies; and the
final result being independent of the subsidiary bodies employed,
we may argue as if they were entirely unnecessary.
For the energy U, we assume Helmholtz’ expression
/
U=U,+ «> oe 0 IEeeaees Reva
= U,+€ {4Q,V, + 405g t+... }4+ QF, + Q,F,+...0+ (1),
where A, B, C,... are the different homogeneous bodies of uniform
temperature which the system contains; Q,, Q,, Q,, ... their
charges; V,, V,,; Vg, --: their potentials; and H,, #5, 830s
quantities which depend respectively on these bodies but not on
their electric states. In what follows, the states of the different
bodies will be completely defined by their temperatures, so that
F will be a function of the temperature depending in form on the
nature of the metal.
To obtain the value of ¢, we make use of Joule’s law that if a
steady current J be flowing in a homogeneous body of uniform
temperature and resistance R, the heat evolved is RJ” ergs per
second. From this we deduce that while a quantity Q is being
transmitted, the heat evolved is RJQ; and therefore, since R is
independent of J, as J diminishes, the heat evolved when a given
finite quantity of electricity is transferred, diminishes in the same
ratio as J. Now the process becomes more and more nearly
reversible as J diminishes, but does not actually become reversible
until J vanishes. Hence if our given system be made to undergo
a reversible operation of any kind in which no part of it is com-
pressed or distorted, and no charge made to pass from one body to
a different body or to a body of the same kind but at a different
temperature, there will be no thermal effect produced and conse-
quently no change of entropy. So long therefore as the charges
are not made to leave the bodies on which they were at first, the
entropy of the system is unaltered by any change in the distribution
of the charges or in the relative positions of the bodies. We may
therefore put
$= WalQs) + Pa(Qa) + Wo(Qe) + oes
where y,(Q,) only depends on the body A and its charge, w,(Q,)
ouly on B and its charge, ......
1891.] Mr Parker, On Contact- and Thermo-Electricity. 271
Now take a second system identical with the first, forming
with it a compound system whose entropy is 2¢. By a reversible
process, such as we have already described, let a charge be made to
pass from one metallic body A to the other body A, and suppose
that no other charge passes from one body to another. Then, since
by what precedes the entropy of the compound system is unchanged,
it follows that, if q be the final charge of one body A and 2Q, —q
of the other, w,(q)+,(2Q,—4) is independent of q, whatever q
may be. We therefore infer that y,(Q,) = W,(0)+ Q,H,, where
Hf, is independent of Q,.
Next, let us take a system formed of the original system ¢ and
of a second system which also contains a metal body A at the same
temperature as the body A in the first system but different in
form and size and with any charge Q,’. Let H,’ be the quantity
corresponding to H,. Then, by supposing any charge to pass from
one metal A to the other metal A, without the passage of a charge
between any other bodies, we find H,’= H,. Hence H, is inde-
pendent of the size and form of A as well as of its electric state.
Thus finally
6=W,(0)+,(0)+...4+Q,0,4+0,H,+..-
IO oF ORT BW... ...ssdessetnerasones (2):
To complete the expressions for U and ¢, we require an im-
portant identical relation which holds between F and H. Let the
metal A and any other part which is at the same temperature @,
be slowly heated to 6,+d6,, and let the parts of the system be at
the same time slowly moved about so that no charge passes from
one body to another. Then we have
dU = aU, + ed 2% 4 Q,dF,+ 2
dp=dd,+Q,dH, +...
If dW be the work done on the system, dU—dW is the heat
absorbed, and since the operation is reversible, we have
dU —dW=6,d¢,
or dU,—0,d¢,—dW + ed ae are: ...
—Q,0,dH,,—Q,0,dH, —... =0.
Now take a second system identical with the first except that
every charge is reversed in sign, and let it undergo a reversible
272 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
operation similar to that just described. Then since dU,, dd,, dW,
and ed= “= are the same as before, we obtain
=
aU, 0,d$,—dW+ e322 _ Q,aF, —...
“a OU a, ae 0 ¢..di, +... =) (fh
These two relations being true for all values of Q,, Q,, -». it
follows that
dF G3
Bg werdid@y tots (cits (3).
To obtain the theory of the Peltier effect and the correspond-
ing change of potential, let two long wires A, B, of different metals,
be joined together at J, and also connected, as in the figure, with
B A
Tron B J A Tron
COOUDTOD “TODD OK
two plates A, B, which are respectively of the same metals as the
two wires. Parallel and opposite to these two plates place equal
plates of any the same metal, as iron, and connect the iron plates
by long iron wires with each other, or with a large distant mass of
iron in the neutral state, so that the two iron plates are always at
potential zero, Also, to make the calculations simple, let us
suppose the air exhausted about the plates; but, to make the
results general, the junction J must be surrounded by air, and to
prevent the air coming near the plates, it must be enclosed in a
bag and the junction J and the bag kept at a great distance from
the plates. Then when the system is at a uniform temperature 8,
if Q,, Q, be the charges of the plates; V,, V, the potentials; U
and ¢ the energy and entropy of the system, we have
U= U,+€{3Q.Vn+ 3Q.V 3 + QF, (9) + OF, a
p= hp, + Q,H (0) os Q,H,(0)
where U, and ¢, will remain constant in what follows.
If now by slowly moving the plates B nearer together and slowly
separating the plates A, any quantity of electricity g be made to
pass slowly from A to B against the abrupt rise of potential V,—V,
at J, and the temperature of the system be kept constantly equal
1891.] Mr Parker, On Contact- and Thermo-Electricity. 273
to @, there will be no thermal phenomenon in the system except at
J, and the heat absorbed there will be @ times the increase of
entropy, or g@(H,—H,). This may be written qP,,, and we see
ont se, —— PP ,,, P,, = Pot Py,
Again, the work done on the plates Bis — }geV,, and the work
done on the plates A, $geV,; so that the total work done on the
system is —}ge(V,—V,), or —4qD,,, if D,, stand for e(V,—V,),
or the electromotive force of contact. Hence, since the increase of
energy is 4qD,,+ 49 (f; — F,), we have
4D,,+ F,—F,=—4D,, deeas
or Tee PP nO ey Th ,) --- 2000s (5).
Combining equation (5) with the identity On gee we get
dD,
dé
AE ee
— Se Oe a ee 6),
wal 8’) tid od;
= oa)
10 (3 “do dd 0d
=H, —H,,
eB),
and P.=0
The result P -0 has been given on four independent
occasions :—by Prof. J. J. Thomson in his Applications of Dynamics
to Chemistry and Physics; by Maxwell in his small Treatise on
Electricity, where he has abandoned the older assumption that
P=D,; by Duhem; and, lastly, by the present writer.
Next, let the plates A and B be of the same metal in the same
molecular state but at slightly different temperatures 0, 0+d0.
Then if V and V’ be potentials of A and B, Q and Q their charges,
we have
U=U, + {3Q'V'+4QV}+QF (0+ dé) + QF (8),
d= $,+QH (6 + dé) + QH (@).
If therefore we suppose the change of temperature at J to be
so gradual that the heat conducted across the junction while a
small charge q passes slowly from one plate to the other, can be
neglected, we easily find, if = be the ‘specific heat of electricity,
that is, the cvefficient of the Thomson effect, or Sd@ the quantity
274 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
which corresponds to P,, of equation (5),
dF dH
’ af es = s = ce
e(V'—V) + 7, d0=3d0=0—, dé.
dH dF
Hence = 075 = 76° 1.00 (7).
and therefore lee
The result V’— V =0, which asserts that there is no electro-
motive force of contact between two pieces of the same metal at
different temperatures, is of the utmost importance. At first
sight it may seem to be in contradiction to experiment; but on
closer examination, as we shall shew later on, this is found not to
be the case.
Equations (5), (6), and (7) contain the whole theory of the
Peltier and Thomson effects. They enable us to discuss the
properties of themoelectric circuits, and the results thus obtained
include all those of Thomson and others which have been tested by
experiment. After shewing this, we will point out the close
analogy between our theory of thermoelectric circuits and Helm-
holtz’ theory of the galvanic battery. Then we will consider some
experiments bearing on our theory.
In the first place, we obtain, from equations (6) and (7),
Thomson’s result
d a1) ad ae — >
za ae
Next, let two pieces A, A’, of the same kind of metal, be con-
nected by a piece B of a different kind of metal, and let the free
ends of A and A’ be at the same temperature 0, while the junctions
0 La oe Creal 2 6
A B A’
of B with A and A’ are at the different temperatures @,, 0,, re-
spectively. Then since there is no electromotive force of contact
between two pieces of the same metal at different temperatures, it
follows that, in the state of equilibrium, the potential of the free
end of A’ will exceed that of the free end of A by
= (Dz, (0,) — Des (6,)}
This will not generally be zero, and therefore, as experiment also
shews, if a circuit be formed by joining the free ends of A and A’,
equilibrium will be impossible and there will be a current, called
a thermoelectric current.
1891.) Mr Parker, On Contact- and Thermo-EHlectricity. 275
If A be a piece of metal in the same molecular state through-
out whose temperature varies in any gradual way we please from
end to end but has the same value at both ends, the ends will be at
the same potential, and therefore if they be joined so as to form a
circuit, there will be no current produced. This is the result
obtained experimentally by Magnus, who found it impossible to
obtain a current by unequal heating in a homogeneous circuit. If,
however, we take a homogeneous circuit, and by filing make a
junction of a very thick piece and a very thin piece, it was found
by Maxwell that on applying a flame to this junction, a current
is produced.
Now let a thermoelectric circuit be formed of two different
metals A, B, as in the figure, and let the temperature of every
a B
ZG
part of the circuit be kept constant; @ and 6, being the tempera-
tures of the junctions. Then the ‘electromotive force of the
circuit’ is defined to be e times the sum of the abrupt rises of
potential as we travel round the circuit in the direction of the
current. Hence since the electromotive force of contact of two
pieces of the same metal at different temperatures is zero, if the
current be supposed to flow from A to B through the junction of
temperature 0, the electromotive force # will be given by
fe FS | eS en eee rae (9).
For a circuit formed of several metals, we shall have
Pr PR ee aaades Suua! cae odscn tank (9Y’.
This result, which has not been tested directly by experiment,
has been given by Duhem and assumed by Clausius.
When the current is steady, let J be its strength and R the
resistance of the circuit. Then
E=Ri,
since the sum of the abrupt rises of potential at the various
junctions must be exactly balanced by the gradual fall of potential
in the other parts of the circuit.
Confining ourselves, for the sake of simplicity, to the case of
a circuit formed of only two metals, as in the preceding figure,
the heat absorbed in a second at the junctions @, @,, will be
(P—P,)l=1[{D+ F,(0) —F, (9)} —{D, + F (9) — Fi, (4)}]
=I(E+{F,(@)—F,(6,)} —{F. (9) — F, @)i)
276 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
which is exactly equal to the heat evolved in the rest of the
circuit.
We can now give the principal results obtained by Sir W.
Thomson, who avoids entirely the question of the electromotive
forces of contact at the various junctions, either of two different
metals, or of two portions of the same metal at different tempera-
tures.
Combining equation (9) with the results
dD d (=) eo
de) and 7 A pe
we get Thomson’s formulae
6P 6
E=| 7 0=| (3a:
4% 6
iP = 7 dé 9 ig PS “evelecacnreteleiaretaretetete (10)
GE dr =
de a dé (>, > D4)
From the second of these equations, we see that if the cur-
rent tends to cool any junction of two metals in passing through
it, the electromotive force will be increased by raising the tem-
perature of that junction. The electromotive force, as far as it
depends on that junction, will be a maximum when P vanishes.
As @ further increases, P will become negative and the electro-
motive force will diminish. The temperature 7’ at which P
vanishes is called the ‘neutral point’ of the two corresponding
metals, or of the circuit when it is composed of these two metals
only.
Again, when @ and @, are so nearly equal that we may put
6, = 0 —7, where 7 is small, we have
aD PED _
do’ oan?
and therefore for a circuit of two metals
Po gee
B = ae (G)+-
When the @ junction is at the neutral point 7, this gives Thom-
son’s formula
D,=D—-rT
1891.] Mr Parker, On Contact- and Thermo-Electricity. 277
The current is then of the second order only and flows through
the circuit in the same direction whether the other junction be
hotter or colder than 7.
When P does not vanish, we have another of Thomson’s
formulae
For simplicity taking t to be positive, we see that # and P are
of the same sign. The current therefore travels in such a
direction as to absorb heat at the hotter junction of the two
metals.
When the temperatures of the two junctions differ by a finite
amount, it will be seen from (9) that the electromotive force is
generally finite even when one junction is at the neutral point.
. If we suppose that the hot junction is at the neutral point, it is
clear that both the Thomson effects cannot be absent, for then
the heat that is developed in the homogeneous parts of the
circuit would be all conveyed by the current, without assistance,
from the coldest part of the circuit. It was the consideration of
this case that led Sir W. Thomson to the discovery of the
‘specitic heat of electricity.’
If we had accepted the old assumptions that the thermal
effects measure the electromotive forces of contact, or that P = D
and that the electromotive force of contact of two pieces of the
same metal at slightly different temperatures is d6, we should
have had
E=P-P,-{ (&,-3,)d a ee (13).
%
Now this is the very equation that is obtained by writing down
the condition that the sum of the quantities of heat absorbed at
the four junctions is equal to the heat evolved (according to
Joule’s law) in the rest of the circuit. Here then the two as-
sumptions that P= D and that the electromotive force of contact
of two pieces of the same metal is =d@, exactly neutralize each
other ; but this fact, it is clear, does not prove that the assumptions
are correct.
It may be noticed that the assumption P=D cannot agree
with our result P=@ = unless P is proportional to @, which
experiment shews is not usually the case. Again, if ,= 2,=0,
equations (7) shew that F,, H,, F,, H, are constant, and there-
fore, by equation (6), P= D=C@, where C is independent of @.
But if P=D =C@, it does not follow conversely that F’,, H,, F,, H,
are constant, nor that ,==,=0.
278 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
Returning to our own theory, let us take as a first approximation
D=a+b0+ cé,
where a, b, c are constants. Then
P =0(b+ 2cé).
But P vanishes when 6= 7': hence
6+ 2cT =0.
Thus we have
E=D—D,=b(0-96,)+¢(#—8,)
=—2cT (0-—6,)+c(P — 6,7)
= sae a, 4
the formula of Avenarius and Tait, which has been found to agree 5
sufficiently well with experiment.
a, d a ee —
Again, since 0 (5 HF =
we find F,-F,=-a+ iW
which is satisfied by taking
F=k? +k,
where k and k’ are constants.
Hence =e 2k@, or the ‘specific heat of electricity, &, varies
as the absolute temperature. Also a= = 2k, or H = 2k0 + 1.
The ‘ thermoelectric diagram’ of Prof. Tait is greatly simplified
by our results. For if be the standard metal and WM any
other metal, the ‘thermoelectric power’ of M with reference to o,
dD mu
or 79>
be lead, because the ‘specific heat of electricity’ of lead is zero,
or H, constant. Hence in the diagram, the abscissa represents 6,
and the ordinate, Hy, — H,,, where H, is constant.
is equal to Hy,—H,. The standard metal is taken to
Lastly, let a galvanic battery have both poles of the same
metal, and let every part of it be kept at the constant tempe-
rature 6. Let LZ be the heat evolved when we effect in any
way at constant pressure the same chemical change as is produced
by the passage through the battery of unit quantity of electricity
in the direction in which the battery tends to give a current.
1891.] Mr Parker, On Contact- and Thermo-Electricity. 279
Also let us suppose that when the battery is in action, the
chemical changes which take place are reversible, and the Peltier
effects and electromotive force the same as when the current
is infinitesimal. Then the heat absorbed at the junctions on
the passage of unit quantity can be easily proved to be
dE /. dD
06 (just as P=077)
Hence since the heat evolved in the homogeneous parts in the
same time is /, and the total heat evolved L, we have the result
of Helmholtz and Gibbs,
It now only remains to look at the experimental evidence
relating to the contact theory. In so doing we must remember
that an electromotive force of contact is produced not only by
the contact of two conductors, but by the contact of a conductor
and a non-conductor, and also, but less easily, by the contact of
two non-conductors. Again, for the sake of simplicity we shall
often follow the custom of writing B/A for D,,, and whenever
it is necessary to indicate the temperature, we shall use suffixes:
thus B/A, denotes that the two different bodies A, B, are at
the same temperature 6; B,/B,,, that two portions of the same
substance are at the different temperatures 6, 6,.
It has been pointed out by Maxwell that in experiments
like those of Clifton and Pellat, in which two plates Z, C, of
different metals, are employed in the open air, we really measure
the sum 4/Z+ Z/C+ C/A, or D,,4 A/Z—A/C. In like manner,
when we employ two plates of the same metal Z but at different
temperatures 0, 6,, we measure Z/Z,,+A/Z,—A/Z,. Now
hitherto the terms A/Z,—A/Z,, have been always omitted, and
the experiment has been supposed to prove that Z,/Ze, is not
zero. But clearly we cannot assume that
A/Z,—A/Zo, =90, or that A/Z
is independent of the temperature ; and therefore the experiment
does not contradict our result that Z,/Z,, = 0.
Let us assume that the thermal effects measure the electro-
motive forces of contact; and suppose that we repeat the ex-
periments of Clifton or Pellat with two plates Z, C, of different
metals, first at the temperature 6, and then at a slightly different
temperature. From the results obtained, we find the value of
dP, d
a * dé {A/Z— A/C},
280 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23,
which we will call £ (0). Next, if two plates of the same metal 7
be employed but at slightly different temperatures, we obtain
Zet ~ (A/Z) =a known quantity f,(@). Similarly we may get
d
See
20+ 7 (A/C).
We then easily find
dP 70
Tho - Ge 20 =f) 1, O-£,08))
a result in which the influence of the air does not appear.
If we accept the thermo-dynamical theory here developed,
these experiments give
dD,,. d
dé + 76 {A/Z — A/C} =f, (8),
d d
S(AID=f,0), 49 (Al) =f, O)
and therefore
FOUPOALEC =e =F
iP,
~ dé =(2 = Zo)>
which is the very result we obtained by assuming that the thermal
effects measure the electromotive forces of contact. Hence experi-
ments like those of Clifton and Pellat do not help us to decide
which of the two theories is correct.
Again, on the assumption that P=D, it follows from the
experiment which gives D,,+ A/Z—A/C, since P is very small,
that the electromotive forces of contact between the plates and
the air form great part of the phenomenon observed. It might
be supposed that this conclusion could be tested directly by ex-
periment by exhausting the air; but Pellat, by reducing the
pressure to lem. or 2cms. of mercury, found little difference in
the result.
If we take the thermo-dynamical theory, it will follow that
in Pellat’s experiment with two plates of the same metal at
different temperatures, the whole of the phenomenon observed is
due to the contact of the plates and air. In the case of two
plates of different metals at the same temperature, a large part
of the phenomenon observed will also probably be due to the
contact of the metals with the air. Taking into account the
1891.] Mr Parker, On Contact- and Thermo-Electricity. 281
experiments of Brown and Pellat on the effects of reducing the
pressure of the air, we conclude that within attainable limits of
pressure, A/Z, the electromotive force of contact of the metal
and air is nearly independent of the pressure of the air. It will
now be shewn directly from theory not to be unreasonable to
suppose that this may be the case.
If, for shortness, we take A/Z to be a volt, or 10° times the
electromagnetic C.G.S. unit, and suppose the distance, d, of the
: I :
two electric layers to be ax 10! em., the attraction between the
metal and the air per square centimetre will be
erAraY
Sire” Ga dynes,
or about 40,000 x 10° dynes. Hence since the pressure of one
atmo is only about 10° dynes per square centimetre, we can partly
understand why the reduction of the pressure to 1 cm. or 2 cms. of
mercury produces so little effect.
MOL. Vil, PT. V. F 22
COUNCIL FOR 1891—92.
President.
G. H. Darwin, M.A., F.R.S., Plumian Professor.
Vice-Presidents.
T. M°K. Hucues, M.A., F.R.S., Professor of Geology.
J.J. THomson, M.A., F.R.S., Professor of Experimental Physics.
JOHN WILLIS CxaRK, M.A., F.S.A., Trinity College.
Treasurer.
Rk. T. GLAZzEBROOK, M.A., F.R.S., Trinity College.
Secretaries.
J. Larmor, M.A., St John’s College.
S. F. Harmer, M.A., King’s College.
E. W. Hosson, M.A., Christ’s College.
Ordinary Members of the Council.
W. GARDINER, M.A., F.R.S., Clare College.
W. Bateson, M.A., St John’s College.
A. Cay Ley, Sc.D., F.R.S., Sadlerian Professor.
W. J. Lewis, M.A., Professor of Mineralogy.
W. H. GaskELL, M.D., F.R.S., Trinity Hall.
A. Hiuu, M.D., Master of Downing College.
A. 8S. Lea, Se.D., F.R.S., Gonville and Caius College.
A. Harker, M.A., St John’s College.
L. R. Wivperrorceg, M.A., Trinity College.
H. F. Newatz, M.A., Trinity College.
C, T. Heycocx, M.A., King’s College.
A. E. H. Love, M.A., St John’s College.
PROCEEDINGS
OF THE
Cambridge Philosophical Society.
Monday, Feb. 8, 1892.
PROFESSOR DARWIN, PRESIDENT, IN THE CHAIR.
Mr W. Heaps, M.A., Trinity College, was elected a Fellow
of the Society.
The following communication was made to the Society :
Long Rotating Circular Cylinders. By C. Cures, M.A., Fellow
of King’s College.
§1. In Vol. vit, Pt Iv., pp. 201—215 of the Proceedings I found
a solution for a thin elastic solid disk of isotropic material rotating
with uniform angular velocity about a perpendicular to its plane
through its centre. In the present paper the same method is
applied to a long right circular cylinder of isotropic material
rotating about its axis. The cross section of the cylinder when
solid is supposed of radius a, when hollow its outer and inner
boundaries are of radii a and a’ respectively. The axis of the
cylinder is taken for axis of z, the origin being at the middle
point, and the notation for the displacements, strains, stresses, etc.
is the same as in my previous paper, except that the dilatation is
denoted by A.
If Poisson’s ratio, 7, be zero the solution obtained here satisfies
all the internal and all the surface equations, whatever be the
ratio of the length, 2/, of the cylinder to its diameter. But for
other values of 7 it is only true to the same degree of approxi-
mation as Saint-Venant’s solution for beams, and like that solution
can legitimately be applied only when J/a is large. This re-
striction of Saint-Venant’s solution, whether for torsion or flexure,
is not perhaps in general sufficiently recognised, but the best
authorities I believe regard it as sufficiently exact only when the
VOL. VII. PT. VI. 23
284 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8,
length of the beam attains to something like ten times its greatest
diameter. Unless 7 =0 the present solution ought to be simi-
larly restricted, and it should not be applied to the portions of the
cylinder immediately adjacent to its ends. The solution con-
siders solely the action of the “centrifugal force”, taking no
account of gravity or of the action of any forces applied at the
ends by the bearings.
§ 2. The solution satisfies exactly the internal equations—
viz. (2) and (3) lc. p. 203—, and also the conditions at the
curved surface or surfaces. The only condition it fails to satisfy
exactly is that z vanish at every point of the flat ends. In-
stead of this we have to be content with satisfying
a
ik 2Qrr zdr= 0,
a.e. instead of making the normal stress over every element zero
we make the resultant normal stress zero. The solution is thus
based on the principle of statically equivalent load systems—
referred to im my previous paper, pp. 206-7—and so can be
regarded as satisfactory only when the dimension 2a is small
compared to the dimension 21.
The solution is found by starting with the expressions (16),
p. 205, for the displacements, which satisfy the internal equations.
We have then to determine the arbitrary constants by means of
the surface conditions, viz.,
re = 0 =7rr, when r=a and when r=a,
a
7=0=| Qnrrz dr when z=+l.
Ja
§ 3. It is unnecessary to reproduce the algebraical work, as
the solution may easily be verified. In terms of Young’s modulus
E and Poisson’s ratio 7 it is as follows:
3 — 5 (L=!29)(-F%)
SS 2 2\ aa a Pe is LEU LOeN.
u=op(a +a") sae 5) @ pr 8E(1—7)
2, va” (1+n)(3— 2m)
+@ - SE (I =n) ee eceeeccees (1),
eA ek 2 2 12 a
w=—op(a +O eon Sine mine ea thros nae ba ve cetae me tees eae (2),
(1—29)\(8—n) . .1—2%)Q+%) |
Se — wp? 31 —9) izsshoy
3 —2n
oe»)
A=o'p (a? + a”)
a wp (a? a 7”) (1 —a”/r’)
1892.] Mr Chree, On Long Rotating Circular Cylinders. 285
is 5 aT 3—2n 1+ 2n
eee | 2 2 2/2 /,.2 i 2 a ee
$6=ap(a+a*+aa a oe era salina i (ee hie ats (5),
2 =p (a> +a? — 27° BEA, eS ii1. sworliiloa. sist, taf sonore 6
NEE Ta) JF ROTUHLE UC eee Akela se didi via csda. wld Le (7).
For a solid cylinder the displacements and stresses may be
correctly deduced from the above by omitting all the terms con-
taining @*. This solution for a solid cylinder is identical with
one deduced from the case of a rotating spheroid* by supposing
the ratio of the axis of figure to the perpendicular axis to increase
indefinitely.
§ 4. In what follows I shall assume 0 and ‘5 as limiting
values of n+. As regards the latter limit there is a difficulty that
will be best understood by reference to the relations
n/m =1—2n, 38n(1—n/3m)=F
between FZ, 7 and Thomson and Tait’s elastic constants.
If we regard # as finite, we must when »=4 have m
infinite. Thus those terms inthe expressions for the stresses in
terms of the strains which contain m as a factor may remain
finite, though the corresponding strains vanish in the limit when
™=4%. This explains the apparent inconsistency in the express-
ions supplied by our solution for this value of ». The strains in
this case in the solid cylinder take the remarkably simple forms
u=opar/8H, w=—o'pa'z/4H, A=0 ......... (8) ;
ams : . du dw
so that the three principal strains, viz. a u/r and qe” are every-
where constant. The vanishing of A is a necessary consequence
of m being infinite, for this implies that the material is incom-
pressible.
§ 5. The expressions for the strains and stresses in the axis
of a solid cylinder and at the inner surface of a hollow cylinder in
which @/a is infinitely small are, it will be noticed, totally
different; for in the former case terms in a”/7’ simply do not
exist, whereas in the latter case @?/?>=1. There is thus, as in
the thin disk, a discontinuity in passing from a solid to a hollow
cylinder however small a’/a may be. At first sight this appears
absurd, for it may be argued that if matter has a molecular
structure, as is generally supposed, then cavities exist everywhere
between the molecules, and there is no reason why a cylinder
* Quarterly Journal..., vol. xx111., 1889, p. 23, Equation (103).
+ Phil. Mag. September 1891, pp. 235—6.
23—2
286 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8,
apparently solid should not have a cavity or cavities devoid of
molecules occupying the whole or a great portion of its axial
length. Would not then, it might be urged, such a solid cylinder
act according to our solution quite differently from another of the
same material in whose axial line there happened to be numerous
molecules? The following seems a satisfactory explanation of this
difficulty :
A hollow cylinder in the mathematical sense is one in which
rz and 7 vanish over r=a’; but this implies that a’, even when
infinitely small compared to a, is still so great compared to mole-
cular distances, that the action of molecules separated by a
distance of order @ is inappreciable. There is thus no sudden
discontinuity, as our solution seems to imply, but a gradual trans-
ition as a’ increases from being a molecular distance to being a
distance so great that the mathematical conditions for a free
surface are satisfied. This transition stage is not within the
compass of the present mathematical theory; but this is hardly a
matter of practical importance, for the mathematical conditions
are probably satisfied in any existing hollow cylinder as exactly
over its inner as over its outer surface.
For brevity, a’/a=0 will be employed to denote the cylinder
that is hollow in the mathematical sense, but in which a’/a is
extremely small. In the same way a’/a=1 will be employed to
denote a cylinder whose wall thickness a—a’ is extremely small
compared to a, though great compared to molecular distances.
§ 6. Returning to our solution we see that 2 vanishes when
7 = 0, so that all the surface conditions are then exactly satisfied.
The solution in this case is thus complete and applies to circular
cylinders of all shapes, to thin disks as well as to very long
cylinders. When 7 is small z is very small compared to the
greatest stress $4, and even when 7 is 4 the greatest value of z
bears to the greatest value of $¢ a ratio which is 1 in a solid
cylinder and not more than ;1, in a hollow cylinder. When, how-
ever, 7 is $ the greatest value of 2 in a solid cylinder is one half
the greatest value of $3, and so may be by no means a small
stress. Now in the case of the thin disk the stresses which failed
to vanish over the edges were always very small compared to the
largest stresses. Thus, to all appearance, our solution for long
cylinders is not quite so satisfactory as that for thin disks unless
Poisson’s ratio be small.
§ 7. A striking difference between the effects of rotation on
thin disks and on long cylinders in which 7 is not zero, is that
whereas in the former case originally plane sections perpendicular
to the axis of rotation become paraboloidal, in the latter case
1892.] | Mr Chree, On Long Rotating Circular Cylinders. 287
such sections remain plane. Unfortunately this absence of curva-
ture could hardly be observed except at the ends, where our
solution cannot strictly be applied.
The shortening of the cylinder per unit of length is given by
(— 6l/l) = (—w/z) =@"p (a? +0”) n/2E ......... (9),
with a*=0 for the solid cylinder. Though it vanishes with ,
this is in ordinary materials an important alteration. Its magni-
tude in terms of w’pa’/H—a quantity depending on the density,
Young’s modulus and the velocity at the outer surface—is re-
corded in the following table for the value °25 of », for various
values of a’/a :—
TABLE I.
Shortening of cylinder per unit of length, n =°25.
aja= 90 2 4 a) ‘8 t
( —61/l) +(@*pa’/F) ="125 113 145 7 "205 ‘25
The entry under a'/a=0 applies also to the solid cylinder.
Since the shortening varies directly as 7, its amount in terms of
w pa?/E may be at once written down for any other value of ».
Numerical measures of (— 6//l) in two typical cases will be found
in Tables IX. and XI.
§ 8. The other displacements of most interest are the altera-
tions da and &a’ in the radii of the two cylindrical surfaces. The
ratios of these alterations to the original lengths are given in
terms of w’pa’/E by the formulae
(8a/a) + (w’pa?/E) = $ {1 -—n + (34+) a*/a*}...... (10),
(8a’/a’) + (w*pa?/F) =} {[34++ (1-7) a"/a’}...... (11).
Thus the radii of both surfaces are always increased. ‘Taking
w*pa*/E as constant, the following table shews how these altera-
tions of the radii vary with 7 and with a'/a:—
TABLE II.
Value of (Sa/a) + (w*pa*/E).
nadja= 0 2 “4 6 8 1-0
0 25 28 87 52 0-738 1-0
25 eGo ee SITS «48 7075 10
“5 125 16 265 ‘44-685 1-0
288 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8,
TABLE III.
Value of (a’'/a’) + (w*pa’/E).
naja= 0 2 “4 6 8 10
0 “15 ‘76 ‘79 "84 ‘91 10
"25 8125 "82 8425 ‘88 9325 1:0
"5 875 ‘88 895 ‘92 ‘955 10
The numerical results in these two tables are exact*. Both da/a
and 6éa’/a’ are linear in 7; thus results for other values of 7 are
easily and accurately supplied by interpolation. The results
under a’/a=0 apply also to the solid cylinder in Table IL, but
not of course in Table III. The formulae (10) and (11) are
identical with (45) and (46), Proceedings, l.c. p. 213, which give
éa and éa’ for a thin disk. Thus Tables II. and III. apply also to
thin disks.
The large and steady increase in the value of da/a as a’/a
increases is very conspicuous. It is also noteworthy that when
a'/a has a given value, da’/a’ increases but da/a diminishes as 7
increases. In fact by (10) and (11)
Sa/a + 8a’'/a' = (w*pa*/E) (1 +a”/a°),
so that da/a + 6a’/a’ is independent of 7. When a’/a approaches 1
the influence of 7 on the magnitude of the alterations in the radii
tends to disappear. An idea of the numerical magnitude of
da/a and éa’'/a’ will be most easily derived from the special cases
treated in Tables IX. and XI.
For the alteration d6a—6a in the wall thickness we have
the same formula as for a thin disk, viz. (47), p. 218, and this
thickness is increased or diminished by rotation according as
a'ja< or >(l1—/n) +(1 +n);
see Proceedings, l.c. Equation (48), p. 213, and subsequent remarks.
Comparing Tables I., II. and III. it will be seen that (— 61/2)
is by no means negligible compared to da/a and 6éa’/a’ unless
be small. Thus as a// is small, the shortening of the cylinder
should in general be more easily detected than the alterations
in its radii.
§ 9. Since * is zero the principal strains are everywhere
diy. b) Seey ith
dz’ dr rT
* This term as applied to this and following tables means that the numerical
results are as exact as the formulae, and not merely the first figures of a decimal.
1892.] Mr Chree, On Long Rotating Circular Cylinders. 289
dw . : e
tea e is everywhere negative, 2.¢ a
compression. It has the same constant value as w/z, and so
is given in Table I. The transverse strain, u/r, is everywhere
positive, ze. an extension, and is never algebraically less than
The longitudinal strain
the radial strain _ Its greatest value, 8, which is found at
di
the axis of a solid cylinder or the inner surface of a hollow
cylinder, is thus the greatest strain. For a hollow cylinder it
is the quantity éa’/a’ of equation (11) and Table III. In a solid
cylinder it is given by
whence answering to 7 =0, 7 =‘25 and »="5 we obtain
5 + (w*pa®/H)=°375, +2916 and ‘125 respectively.
The radial strain is most conveniently dealt with by means
of the formula
8# (1 - du =
Sm, r ae =) ee oe (13) ;
w*p
where for a solid cylinder
r f(r) =a (8— 5n) — 37° (1 — 2n) (1 +7) ...... (14a),
and for a hollow cylinder
f(r) =—a@a* (1 + 9) (3 — 2m) + (a +.2*) 7? (3 — 5m)
— 3r* (1 — 2n)(14+7)...... (145).
- we need only consider that of /(7).
For the sign of
§ 10. In a solid cylinder f(r) is positive inside and negative
outside the surface
7? =a? (3—5n) +{3(1—2n)(14+7)}......- ee. (15);
but the radius of this surface exceeds a when 7 > 3.
Thus in a solid cylinder when 7 >°3 the radial strain is every-
where an extension; when, however, 7 <°3 there is a cylindrical
surface, viz. (15), outside of which it is a compression. When
n=0 or ‘3 the radial strain vanishes over the surface of the
cylinder, and elsewhere is an extension; but for intermediate
values of 7 the region wherein this strain is a compression has
a small but finite thickness. For a given value of a this thick-
ness has a maximum value of ‘(03775a approximately when » =°2.
290 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8,
§ 11. The variations in the sign of 8 in a hollow cylinder
may be most easily investigated by means of the equation
where f(”) is given by (140), regard being had to the sign of
the surface values of /(7), viz. :
f(@) =— 2na? (3 —n) a + (1 — 8n) a} 2 (17),
f(a) =—2na? \(1—38n) @ + (3-9) a} 0 (18).
Let us denote by a,'/a the least positive root, when real, of
(a? + 1)* (3 — 5)? — 120" (1 + 9)2(1 — 29) (3 — 2) =0...(19),
and by a,’/a the positive root of
a = (39 —1)/(8—7) ..+... chs. ee (20).
Then a,'/a is that value of a’/a for which f(7) has equal roots,
and a,'/a is that value for which f(a)=0. When a/aa,'/a:
The radial strain is everywhere a compression.
Transition case to Class IIL, 7 ='3:
This follows the same laws as Class IL., except that for w/a =0
one of the two surfaces over which f(7) vanishes coincides with
the outer surface of the cylinder, so that there is not, as in sub-
class (i) above, a volume of finite thickness at the outer surface
wherein the radial strain is a compression.
Crass IIL, 3 <1 <(4—J7)/3, ie. 4514 approximately.
Here a,/a is real.
Sub-class (1), w/a u,/a:
The radial strain is everywhere a compression.
Transition case to Class IV.
n =(4—./7)/3; a, /a=a,/a = °3728 approximately.
This follows the same laws as Class III., except that Sub-class
(ii) is not represented, and for a’/a='3728 the radial strain
vanishes over two surfaces both coinciding with the outer surface
of the cylinder. (
292 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8,
Crass IV., (4—./7)/3 <9 $5.
Sub-class (1), a'/a a,'/a:
The radial strain is everywhere a compression.
§ 12. As illustrating Class IL. and the transition to Class IIT.
we shall consider the values ‘25 and ‘3 of 7. The approximate
values of a,'/a in these two cases are respectively 4276 and
‘3728*; ae. the radial strain is everywhere a compression when,
n being ‘25, a’/a exceeds 4276, and when, » being ‘3, a’/a exceeds
‘3728. When a’/a Fe: a 7)
3p = si+R(s R Sloe
ae
T=8.R (57s) sind.
Now we might proceed to square these two and add them
together to find 4f?, and so go on to find the rigorous expression
for 4f/€, which equated to F/C will give the rigorous equation
to the required surface; but the result would be so complex as
to be of little value because not easily intelligible.
I therefore at once proceed to approximation.
1892.] in the neighbourhood of a planet. 317
S being very large compared with M and m, p will be small
compared with r.
Then since r = R’ + p*+ 2Rp cos 8,
R* P ape, to 2 2
ga Lop cos 0 5 ty BOS 6,
Therefore approximately
3 Te , lop P
+(—5f 0088 — 5 Fat pcos 8) cos 6
La eP 9 p 15 p
=~ Ff {1 Seos O—% cos 8 + > cos’ G ;
and
2
qr =4 58 |(1 ~8 costa)
+3 cos (5 cos’ @—3)(1 —83 cos’ @) 5 -
Again
B= — Fh 13 0080+ 5 he —y renstal sin 8
and
a = aes {9 cos? 8(1 — cos’ 0)
— 9 cos 6 (5 cos’ @—1) (1 — cos’ 4) Ft ,
?
whence
ey p |
f= i I tees ge coe ey
Now
G = M+m
p
And
la Ss 2
— = + =) § Rs
We now have to introduce a similar approximation into the
value of F?/C”.
+f + 3 cos’ @ — 12 cos® oF.
252
318 Prof. Darwin, On the perturbation of a comet [Mar. 7,
7 rT —.- «_ == dis
We have R 1+4 Ro 6,
: E* Siodl iaedir p
and therefore Gn (= _) ° (1 +4 R © @) ;
Equating F’/C® to FP/C’, we get
R\* (S\*(S + m\’ :
a = (Gn) = ) {1 + 3 cos’ 0
as 2 Pp) ‘
4 cos @ (1 + 6 cos*@) RS
ap te S\t (S+m\4 2 A\is
p = (Fy) (ren) a 3.
2 cos 0(1+ 6 cos* @) p
| "5 1430 ae
Thus the equation to the surface is approximately
Rk (S\s(S4+m\s eae
aa ea = + =| (Lr ease)
{1 2 Ga & a) cos 6 (1 + 6 cos? a
5 S S -+ m (1 + 3 cos” gyro ;
It is usually the case that m is negligible compared with J,
and that M is also small compared with S, and in this case we
may write the equation witb sutticient accuracy
R_(8
a =
Laplace gives a formula for the radius of the sphere of activity
which is virtually derivable from the above investigation on the
special hypothesis that the three bodies lie in a straight line.
Thus he puts @ equal to zero or 180° and finds,
2
Rt = 410 (a):
Pp
But to find the true mean value of (1 +8 cos® @)"*, we must
estimate it all over the sphere.
is (1 + 8 cos® 6)**,
Now
1 bly. 1 4
= {fa + 3 cos’ @)'* sin 0d0dd =|. (1+ 32°)" dz.
This integral evaluated by quadratures, is found to be equal
to 1:063.
1892.] in the neighbourhood of a planet. 319
Thus the true mean gives
R S\5
> = 1-068 (37)
Laplace makes it
Re rit gti Aad has. S\é
The ratio of the least to the greatest value of p in the formula
suggested in this note is 1149, and Laplace takes the minimum
value of p as the radius of his sphere.
In the case of Jupiter, Laplace’s formula gives p = 054 R, and
my formula gives p= ‘058 R.
It follows that Laplace’s conclusion is sufficiently accurate for
the purpose for which it is intended.
(3) The change of zero of Thermometers. By C. T. Heycock,
M.A., King’s College.
The author described the result of experiments he had made
in conjunction with Mr Neville to overcome the change in zero
which thermometers undergo when heated for a long time. The
method consisted in boiling the thermometer for eighteen days
in baths of either mercury or sulphur, at the end of this time
the zeros were found to be practically fixed unless they were
exposed to higher temperatures than those of the substance in
which they were boiled. The paper was illustrated by a curve
showing that the change in zero was very rapid for the first
few hours, amounting in a special case to 11°C. for 20 hours
heating, but that afterwards the change became almost nil as
the heating was continued.
(4) The Elasticity of Cubic Crystals. By A. E. H. Love,
M.A., St John’s College.
(5) Changes in the dimensions of Elastic Solids due to given
systems of forces. By C. Cures, M.A., Fellow of King’s College.
[Abstract.]
This paper deduces from a general theorem due to Professor
Betti expressions for the mean values of the strains and stresses
in any homogeneous elastic solid acted on by any given system of
bodily and surface forces. Formulae for the mean strains in
isotropic solids acted on only by surface forces were given by
320 Mr Chree, Changes in the dimensions of — (Mar. 7,
Professor Betti, but he does not seem to have considered the
general case, nor to have made applications such as those treated
here. From the formulae for the mean strains the change can be
found in the mean length, taken over the cross section, of any
right cylinder or prism subjected to any given system of forces.
Similarly the change in the whole volume of any elastic solid of
any shape can always be expressed as the sum of a volume and a
surface integral involving only the applied forces and the elastic
constants of the material.
Thus in an isotropic solid acted on by bodily forces whose
components are X, Y, Z per unit volume, and by surface forces
whose components are F’, G, H per unit surface, the change dv in
the volume is given by ;
dkbv = f[{(Xa + Yy+ Zz) dxdydz + f{( Fx + Gy + Hz) ds,
where & denotes the bulk modulus, or m—in in Thomson and
Tait’s notation. It is obvious from the equations of statical equi-
librium that the position of the origin in the above expressions is
immaterial. In any homogeneous aeolotropic solid the change in
volume may be similarly determined, but the expressions under
the integral signs are a little longer.
The several formulae both for isotropic and aeolotropic solids
are applied to a variety of special cases, a few of which will serve
for illustration. The material to which the following results apply
is, unless otherwise stated, assumed isotropic.
When a solid of any shape is suspended from a point, or a
series of points in one horizontal plane, its volume v is greater
than if “gravity” did not act, and the increment 6v due to
“gravity”, represented by g, is given by
v/v = gph/3k,
where p is the density and h the distance of the centre of gravity
below the point, or points, of suspension. On the other hand, if a
body be supported on a smooth plane, or at a series of points in a
horizontal plane, its volume is diminished owing to the action of
gravity, the diminution (— 6v’) being given by
— $v'/v = gph'/3k,
where h’ is the height of the centre of gravity above the plane of
support.
When a right cylinder or prism is suspended with its axis
vertical its length / is increased, and the mean increment 6/ taken
over its cross section is given by
6l/l = gpl/2E,
where # is Young’s modulus. When the cylinder rests on a
smooth horizontal plane with its axis vertical, it shortens under
1892.] Llastic Solids due to given systems of forces. 321
gravity by an amount equal to the above. When the cylinder is
suspended with its axis horizontal, in such a way that bending
does not occur, it shortens, while when supported on a smooth
horizontal plane in that position it lengthens. The alterations in
the mean length in the two cases are given by
61/1 = F ngph/E,
where 7 is Poisson’s ratio, while h is the distance of the centre of
gravity from the horizontal plane through the points of suspension
in the first case and through the points of support in the second.
When a solid of any shape rotates with uniform angular
velocity » about a principal axis of inertia through its centre of
gravity the volume v is increased, the increment being given by
bv = wT /3k,
where J is the moment of inertia of the body about the axis of
rotation.
When a right cylinder or prism rotates about its axis it
shortens, and the mean shortening (— 6/) taken over the cross
section is given by
— 61/1 = nw’pk’/E,
where « is the radius of gyration of the cross section about the
AXIS.
When a rectangular parallelepiped 2a x 2b x 2c rotates about
the axis 2c, the mean increment 26a in the distance between the
faces perpendicular to 2a is given by
éa/a = wp (a’ — nb*)/3E.
Thus the tendency to increase in length in material lines
perpendicular to the axis of rotation becomes reversed when the
dimension perpendicular to this and to the axis of rotation is
sufficiently great.
A homogeneous sphere, whether isotropic or aecolotropic, owing
to the mutual gravitation of its parts suffers a diminution im
volume given by
— §v/v = gp R/5k,
where F is the radius and g “gravity” at the surface. This
suffices to prove that the application of the mathematical theory
of elasticity to the earth, treated as a homogeneous solid, violates
the fundamental condition that the strains must be small, unless
the material be assumed to offer a much greater resistance to
compression than any known material under normal conditions at
the earth’s surface.
The change in volume due to the mutual gravitation in its
parts in any very nearly spherical body, when isotropic, is shown
322 Mr Chree, Changes in the dimensions of Elastic Solids. [Mar:7,
to be the same as in a sphere of the same material of equal
volume, and it is thence concluded that the spherical is a form in
which the reduction of volume due to gravitation is in general
either a maximum or a minimum. The reduction of volume is
calculated for a gravitating ellipsoid, and it appears that the
sphere is the form in which, when the volume is given, the re-
duction is a maximum. In a nearly spherical ellipsoid whose
principal sections through the longest axis are of eccentricities ¢,
and e, the reduction in volume is given by
— dv/v = (gp R/5k) {1 —(e,* — €,7¢,? + ¢,)/45},
where Ff is the radius, and g the value of gravity at the surface,
in a sphere of equal volume and density.
In a given volume of an aeolotropic material a very slight
assumption of an ellipsoidal form, insufficient to produce an
appreciable effect if the material were isotropic, increases or
diminishes the diminution in volume due to mutual gravitation
according as it consists in a lengthening or a shortening of those
material lines which are parallel to directions in which the
linear contraction under uniform normal pressure is above the
average.
(6) On the law of distribution of velocities in a system of
moving molecules. By A. H. Leany, M.A., Pembroke College.
1. The following proof appears briefly to establish the fact
that Maxwell’s law of distribution of velocities gives the only steady
distribution. The proof is a little shorter than the ordinary proof
as given by Boltzmann, even if Mr Burbury’s variation of it as
published in the Philosophical Magazine tor October 1890 be
adopted.
Let a particle whose velocity is OP in magnitude and direction
strike a particle whose velocity is op. Suppose the particles to
belong to different systems, and let the number of particles of
the first kind which have velocity components lying between &
and €+d€, 7 and n+dn, Cand 6¢+d£, where &, n, & are the com-
ponents of OP, be F(OP)dEdndé Let f(op) dé’dn' dg’ have a
similar meaning when applied to the particles of the second
system. Then the number of impacts which particles with ve- —
locity OP have with particles of the second system which have
velocity op will, in the interval dt, be per unit volume
F (OP) d&dndé. ms’udt . f (op) dé’ dy’ df’ ........: (1),
since each particle in the unit volume strikes, on the average,
as udt f (op) dE’dy’dg’ in the interval dt; the particles being re-
garded as hard spheres, the sum of the radii of two spheres, one
1892.]_ Mr Leahy, On the law of distribution of velocities. 323
of each system, being s, and w the velocity of a particle of the
first system relative to the velocity of a particle of the second
system.
Suppose now that after encounter the velocities of the particles
become OP,, op, respectively, so that the components of OP, lie
between & and &+dé,, n, and n,+dy,, § and €,+d€,; and &/,
m,, €, have similar meanings as the components of op, Since
the velocity of the centre of gravity is unchanged by the impact,
the condition that the required change shall take place is that the
direction of w shall lie within a cone of solid angle dS making an
angle @ with the line of centres at impact. A coilision such that
velocities OP, op before collision may become velocities OP,, op,
after collision may be called a “collision of a given kind,” and
since, as Mr Burbury has pointed out, all directions of the relative
velocity after encounter are equally probable if the molecules
behave as hard elastic spheres, the whole number of encounters of
the given kind in the interval dt will be
F (OP) f (op) d&dndg.d&’dn’'d&’ . ws*udt. — ities (2)
per unit of volume.
Conceive now that the velocity of every molecule of the system
is suddenly reversed in sign, the molecules of the solid boundary
of the system, if such exist, being similarly reversed as to the
direction of their velocities, the position of every molecule being
unaltered. The physical properties of such a medium will of
course differ in many respects from those of the original medium,
but it will at any rate have this property, that all particles whose
velocities in the original medium changed from OP to OP, in the
time dt will in the second (or as we may call it the “reversed ”’)
medium change in a second interval dt from —OP, to —OP.
Now, since the distribution of particles is perfectly regular in
space, the distribution of velocities in the original medium
“behind” any class of molecules is the same as the distribution
“in front” of the same class. Hence the number which in the
reversed medium change their velocity — OP, for — OP by striking
particles with velocity op, is in the interval dt
F(—OP,) f(- op,) dé dn,df,. dé dn ag) swat. &...(3),
where w’ is the velocity of OP, measured relatively to op, and dS’
is the angle of the cone, whose axis makes an angle @ with the
line of centres, within which the direction of the relative velocity
w must lie in order that the collision may be one of the given
kind. Since the number which change from velocities OP to
velocities OP, in the original medium by collisions of the given
324 Mr Leahy, On the law of distribution of velocities [Mar. 7,
kind is equal to the number which change from OP, to —OP in
the reversed medium by collisions of the same kind, expressions
(2) and (3) are equal. Also w’ in (3) is the same as w im (2) since
the velocity of the centre of mass is unchanged and the spheres
are elastic, and by ordinary geometry dS’ is equal to dS and
dédndé. dé'dy'd&’ is equal to d&,dn,dg&.d£,'dn,dé’. Hence, since
F(—OP,) in the reversed medium is equal to F/(OP,) in the
original one,
F (OP,) f (op,) =F (OP) f (op),
and therefore since the kinetic energy is unchanged by the impact
we have, by the ordinary methods, Maxwell’s law of distribution
namely
OP?
FOP) = Ae
In order to examine the validity of the above proof the
assumptions underlying equations (1) and (2) should be further
considered. In equation (1) the assumption is that, if a number
of particles are distributed uniformly throughout a medium,
and if the velocities are so distributed that the number per
unit volume which have velocity op is f(op) d&'dy’dé’, then the
number in the volume F (OP) d&dnd{.7s*udt, which may be
written do,, is f (op) dE'dy'd&’. do,. This assumption is equivalent
to two others, first, that the particles are distributed throughout
the medium with perfect uniformity so that we can safely take
the number of molecules of a given kind in an element of volume
to be proportional to the volume of the element if the element is
large enough to contain a great many molecules; secondly, that
the particular volume do, is large enough for the first assumption
to be applied to it. In order to prove result (2) we must suppose
that the number of molecules within the volume
F (OP) dédndé. 7s’ udt . cos 6,
which may be written do,, is proportional to do,. Thus, since the
volume do, is greater than do,, the whole assumption that we
make is that do,, and consequently do,, is large enough to contain
a very large number of the molecules which are distributed so that
the number which have a velocity op is given by the function
S (op).
To estimate the number of these molecules, suppose #’ (OP)
Vee eer mero
to be equal to 458 @) which is Maxwell’s law of distribution.
Ta
Then, taking hydrogen as an example, since* Nzs° is 7:0 x 10°
* These numbers are calculated in accordance with Professor Tait’s results,
Edin. Trans, xxx. p. 91.
“a
Vet
1892. ] im a system of moving molecules. 32:
approximately, the volume is
7-0 x 10° 0” d&dnd
Gomer a? , ET dielhy ee 7
pal a
cubic millimetres. Now a cubic millimetre of hydrogen at atmo-
spheric pressure contains about 9°76 x 10" molecules. Hence the
number of molecules in do, is
op q
G8 x 10". 61). as udt. oF cos 0.
Suppose wu to be equal to & times a which is in hydrogen equal to
762 x 10° millimetres per second; we get the whole number of
molecules in do, to be
— d&Ednd& 10% kat 2S cos 6
. oe ; Arr :
52-6
dt being measured in seconds; and this number must be large
in order that equation (2) may accurately give the number of col-
lisions of the given kind.
~The smallest value which we can ascribe to dé will depend
upon the magnitude of the limits dé, dn, dg, dS which define the
encounter. Suppose that d& =dy =df=a/1000; suppose also that
dS/47 =10°%. Let us also suppose OP not to be greater than
2a, a supposition which excludes less than 0°5 per cent. of the
whole number of molecules. Suppose also that & 1s greater than
10°, so that the relative velocity wu is not less than a/100. These
suppositions give the whole number of molecules in do, to be
greater than 9°6 d¢.10", so that this number is more than a
million if dé is not less than 10° of a second. This estimate of
the limiting value of dt is perhaps too small as we have taken
the limits of d&dyndf& exceedingly small, but it will appear that
dt must not be taken indefinitely small and should at any rate
be greater than the mean time between collisions, which is of the
order 10° of a second. With the above proviso as to the value
of dt, it appears that result (2) can be taken to be accurate, and
since result (3) is merely an application of result (2) to the re-
versed medium it appears that the assumptions made can be
relied on.
2. It has throughout been assumed that the distribution of
velocities is “steady”, and the proof shows that Maxwell’s law
gives the only possible steady distribution. It is however desirable
to show if possible that the system must ultimately acquire a
steady distribution. Now the steadiness of the distribution has
been assumed twice, first when the assertion is made that the
326 Mr Leahy, On the law of distribution of velocities [Mar. 7,
number of molecules with velocity op struck by molecules moving
among them with relative velocity w is proportional to wdt. 7s’,
secondly, when the distribution in the “reversed” medium was
taken to be the same as that in the original one. If the distri-
bution is not steady, expression (2) must be amended by inserting
a factor (1+ vdt), and expression (2) must contain a factor
(1+v'dt); where v, v’ depend upon the variations of F and f.
The result obtained as before will be that
F(OP,) f(op,)— F (OP) Ff (op)
will not be zero but equal to wdt where w depends upon », v’, F,
and f. Integrating equation (2) and using the proposition that
all directions of encounter are equally probable, we get the usual
result
e F(OP) = F(OP) dédndt | |[ag'an’ag aris
a | | |For) sen) — F(OP) (op) ie
where d&’, dn’, df’ are elementary increments of the components
of op, and the double integral is taken for all possible impacts
between particles whose velocities before collision were OP, op;
OP,, op, being the velocities which the particles acquire if their
relative velocity falls within the cone of solid angle dS.
Since the subject of integration in the double integral is equal
to wdt, ay (OP) must contain dt as a factor and will be very
small when dt is very small. But, since there is a limit to the
minimum value of dt, this does not prove < FOP) to be zero;
that is we cannot in this way prove the ultimate distribution to
be steady, although its variation from the steady state must be
small when the distribution of the particles is regular throughout
the space considered.
Boltzmann’s proof would show that the function H which he
has introduced will continually diminish until the steady state is
obtained, but I think that it assumes equations (1) and (2) to be
absolutely true, which they appear to be if the motion is from
the first assumed to be steady. The proof that the motion of the
particles finally must attain a steady state is apparently still
wanting, although the above argument shows that the divergence
from the steady state must ultimately be small. It is not im-
possible that (OP) may ultimately be periodic with a period of
magnitude of the same order of magnitude as the time of free
path. But the assumption that the motion of the particles is
1892.] in a system of moving molecules. 327
ultimately absolutely steady is after all not greater than the
assumption that it is ultimately perfectly regular, and if the
regularity of the distribution both in space and time is assumed
Maxwell’s law of distribution appears, from the above, readily to
follow.
Monpay, May 2, 1892.
Pror. G. H. DARWIN, PRESIDENT, IN THE CHAIR.
The following were elected Fellows of the Society :
Thomas Clifford Allbutt, M.D., F.R.S., Fellow of Caius College,
Regius Professor of Physic.
David Sharp, M.A. (M.B. Edin.), F.R.S., Curator in Zoology.
J. C. Willis, B.A., Gonville and Caius College.
The following was elected an Associate :
A. Antunis Kanthack (M.B. Lond.), St John’s College, John
Lucas Walker Student in Pathology.
The following communications were made to the Society:
(1) The application of the Spherometer to Surfaces which are
not Spherical. By J. Larmor, M.A., St John’s College.
The ordinary form of spherometer, which is used for measuring
the curvatures of lenses, rests on the surface to be measured by
three legs which are at the corners of an equilateral triangle ; and
the mode of using it consists in finding the length of the ordinate
drawn up to the surface from the centre of the “triangle formed by
the points of support, by means of a micrometer screw moving
along the axis of the instrument.
In the actual use of the instrument the surface to be measured
is assumed to be spherical ; and the question has apparently not
occurred to examine the character of the results which may be
derived from its application to a surface of double curvature.
On actual trial with such a surface, for example the cylindrical
surface of an iron pipe, it appears at once that when the centre is
set at a given point the instrument may be rotated anyhow on its
axis without affecting its reading. It therefore measures some
definite quality of the double curvature of the surface at the
point. There is a temptation to hastily assume that the plane of
support is parallel to the tangent plane at the centre of the instru-
328 Mr Larmor, The application of the Spherometer [May 2,
ment, that it is in fact the indicatrix plane of that point, and to
deduce that the reading gives the mean of the principal curva-
tures of the surface; this result is correct, but the assumption just
mentioned is erroneous.
To obtain a rigorous investigation, let us assume that the
points of support form an isosceles triangle, let the base subtend
an angle 2¢ at the centre of the circumscribing circle, and let ¢ be
the radius of this circle and hf the ordinate drawn from its centre
up to the surface. If this ordinate is taken as axis of z, the equa-
tion of the surface will be
2 2
.
L
Beat wth
ii sl LBS FRET
where (p, g, — 1) is the direction of the tangent plane at the origin,
and #,, R, are the radii of principal curvature. As the three legs
rest on the surface, we have
2 oe
h=cp cos (0+ 4) +cqsin(6+4)+4$e° {= Sate Loe:
h = ep cos (@— a) +oqsin (@—a) + Jo O94 SEO OI
: . 0} ster Gg
h=—cpcos@ -—cqsin@ i eas
y, q A at 20 \ R, ai Rk, ?
where 7 + @ is the azimuth of the vertex of the triangle of sup-
port. We are to eliminate p, g, and so connect h with R,, R, and
6. By addition of the first pair of relations
2h = 2cp cos @ cos a+ 2cq sin @ cos a+ de? i‘ es =
1 1 )
“7 E- a cos 24 cos ie ;
therefore by use of the third
2h (1 + cos a) = 3¢? {( 1
1
R, ae) (1 + cos a)
at (x a a) cos 20 (cos 24 + cos ah,
1 2
or on rejecting the factor 1 + cos a,
Me cig he sci ee!
a (zr +z) + aie Be 26 (2 cosa—1).
The value of therefore depends on the azimuth 6 except in
one case, when « is 47 so that the triangle of support is equi-
lateral, which is the case referred to above. The quantity involved
1892.] to Surfaces which are not Spherical. 329
in the formula is then at ; and by referring back to the
1 2
original case of a spherical surface we see that the instrument
measures the arithmetic mean of the principal curvatures.
Thus for example the equilateral form of the instrument may
be conveniently used to measure the curvature of a cylindrical
lens or a cylindrical pipe, but for that purpose its indication must
be doubled.
The equilateral form will be of no use for testing deviation
from sphericity at a given point of a surface. The isosceles form
may however be so used, the difference of the extreme curvature-
indications given by it for any point being by the above formula
that is directly proportional to the difference of the principal
curvatures. The curvature may thus be completely explored *.
In all these formulae the usual assumption is made that the
span of the instrument is small compared with the radii of curva-
ture of the surface.
If the instrument had four legs at the corners of a rectangle,
there would be only two positions in azimuth, corresponding to the
sections of greatest and least curvature, in which it would rest
firmly at a given point on a surface, with all its legs in contact ;
and the plane of contact would in this case be parallel to the
tangent plane at the summit of the surface. The readings for
these positions would give
cos’a sin®a4
Re Rh,
ae sin? a . cos” a
R, Pe
where @ is an angle made by a diagonal of the rectangle with a
side; so that the values of both principal curvatures might thus
be determined.
* Mr H. F. Newall informs me that an isosceles spherometer is used by
Dr Common for exploring the curvatures of his large specula.
330 Prof. Thomson, On the electric strength of a gas. [May 16,
May 16, 1892.
Proressor G. H. DARWIN, PRESIDENT, IN THE CHAIR.
The following were elected Fellows of the Society :
W. Robertson Smith, M.A., Fellow of Christ’s College, Professor
of Arabic.
J. K. Murphy, B.A., Caius College.
The following communications were made to the Society :
(1) Recent advances in Astronomy with Photographic Illustra-
tions. By H. F. NEwAtt, M.A., Trinity College.
A series of photographs was exhibited by the lantern and
described, to illustrate recent progress in astronomical photography.
The series included some interesting specimens taken with the
Newall telescope, in which the object glass is not specially cor-
rected for photographic purposes.
(2) On the pressure at which the electric strength of a gas is a
minimum. By J.J. THomson, M.A., Cavendish Professor.
The author showed that when no electrodes are present, the
discharge passes through air at a pressure somewhat less than
that due to 1/250 mm. of mercury; the discharge passes with
greater ease than it does at either a higher or a lower pressure.
Mr Peace has lately shown that when electrodes are used, the
critical pressure may be as high as that due to 250 mm. of mercury :
so that as the spark length is altered the critical pressure may
range from 250 mm. to 1/250 of a mm. It was pointed out that
this involved the possession by a gas conveying the discharge of a
structure much coarser than any recognized by the Kinetic Theory
of Gases. The author suggested a theory of such a structure and
showed that the theory would account for the influence of spark
length and pressure on the potential difference required to produce
discharge.
(8) On a compound magnetometer for testing the magnetic pro-
perties of tron and steel. By G. ¥F. C. SEARLE, M.A., Peterhouse.
When a bar of iron or steel is subjected to the action of a
longitudinal magnetic force, H, it is found that the intensity of
magnetisation of the iron thereby produced depends not only
upon the value of H at the instant, but also upon the series of
1892.] Mr Searle, On a compound magnetometer. 331
values which H has previously assumed. Thus if the magnetising
force is made to undergo a series of changes in a cyclical manner,
the curve representing the relation of J to H will be of the form
of a loop, which however degenerates into a straight line when
the maximum value of H does not exceed 04¢.G.8. units*. This
dependence of the intensity of magnetisation, due to a given
magnetic force, upon the previous magnetic history of the iron
has been called by Ewing hysteresis. The curve ABCDEFA (fig. 1)
Hysteresis Curve for Annealed Steel Wire.
taken from Prof. Ewing’s book on “Magnetic Induction in Iron
and Other Metals” will serve to give a general idea of the relation
between Z and H for a piece of annealed pianoforte steel wire
when the magnetic force is made to pass repeatedly through a
complete cycle of changes. The maximum values of H and J in
this curve are about 100 and 1100¢c.G.s. units respectively. The
curve OB gives the relation between J and H for a piece of steel
which has never previously been magnetised or which has been
completely demagnetised by continued reversals of a magnetic
force whose amplitude has been slowly diminished to zero.
These hysteresis curves are of great interest from the practical
as well as from the philosophical point of view, since, as has been
shown by Warburg and by Ewing+, the area of the curve, when
estimated on the proper scale, represents the energy expended
per cubic centimetre of the iron in carrying H through its cycle
of changes.
In determining the form of the hysteresis curve, a specimen
of the material in the form of a wire is placed inside a long
uniformly wound solenoid through which a current can be sent.
The current gives rise, inside the solenoid, to a uniform longi-
tudinal magnetic force whose value can be calculated from the
equation H = 4rni,
* Lord Rayleigh, Phil. Wag. March, 1887.
+ J. A, Ewing, ‘“‘ Magnetic Induction in Iron and Other Metals,” § 79.
Vor VIL. PT. VI. 26
332 Mr Searle, On a compound magnetometer for [May 16,
where # is the strength of the current, and n the number of turns
of wire upon the solenoid per centimetre of its length. The
solenoid is placed near a suitable mirror magnetometer; and a
small coil, which is joined up in series with the magnetising sole-
noid, is so adjusted that it exactly neutralises the action of the
solenoid itself upon the magnetometer. Thus when the specimen
of iron is placed inside the solenoid the deflection produced is
due entirely to the magnetisation of the specimen. From the
observed deflection the value of the intensity of magnetisation, J,
can be determined. The magnetising current also passes round a
suitable galvanometer by means of which its strength, 2, can be
measured. The strength of the current is gradually varied by
means of a resistance box in the circuit, and the simultaneous
readings of the galvanometer and magnetometer are noted. The
values of H and J deduced from these readings are used as abscissa
and ordinate in the construction of the hysteresis curve. This
process naturally involves a good deal of labour.
I have endeavoured to construct an instrument which should
perform simultaneously the functions of both galvanometer and
magnetometer and should cause a spot of light to trace out a
hysteresis curve upon a screen. One method of attaining this
end is to provide a mirror with two independent motions about
two axes mutually at right angles, the motions about these two.
axes being governed by two small magnets. One of these magnets
must be acted on by a magnetic force proportional to the
magnetising current, and the other by a magnetic force propor-
tional to the intensity of magnetisation of the specimen.
fe
Fig. 2.
B
D re
This idea was put into practice in the following manner. AB
(fig. 2) is a thin aluminium wire about SO centimetres long. This
is suspended by one end A bya silk fibre from the support K.
»
1892.] testing the magnetic properties of iron and steel. 333
Near its top the wire carries a small magnet C whose axis is at
right angles to the wire. The lower end B of the wire carries a
small fork DBE, also of aluminium wire, across which the silk
fibre DE is stretched. Attached to this fibre by means of wax is
a small plane mirror F, such as is used in reflecting galvanometers,
carrying a small magnet whose axis is at right angles to the fibre
DE. Attached to the bottom edge of the mirror is a disk of
thin mica about 1 inch in diameter. When the plane of the
mirror is vertical, the plane of the disk is horizontal. Close
beneath the mica disk is placed a piece of cardboard in a horizontal
position. The mica disk, owing to the close proximity of the
cardboard, very rapidly reduces the mirror to rest. The mirror is
fixed to the fibre so that the centre of gravity of the mirror and
mica disk is slightly below the line DE. Thus the controlling
force acting on the lower system consists partly of gravity and
partly of the magnetic force due to the earth and to any control
magnets which may be required to bring the mirror into any
desired position. The mirror now possesses two independent mo-
tions, the one about the axis AK and the other about the axis DZ.
The apparatus is set up as in fig. 3*, in which the suspended
part has been turned through a right angle so that the mirror is
now seen edgeways. The plane of the paper is supposed to be a
plane through the wire AB at right angles to the magnetic
meridian. The magnet C is therefore at right angles to the plane
of the paper. The mirror Fis shown slightly tilted. The coil ZL
‘ * This figure is purely diagramatical and does not represent the relative pro-
portions of the separate parts of the apparatus,
26—2
334 Mr Searle, On a compound magnetometer for [May 16,
is placed near the magnet C' and deflects it through an angle
proportional to the strength of the current in L, the deflections
being kept very small. The solenoid M is placed in a vertical
position east or west of the mirror, its upper end being about in
a horizontal line with the mirror. The small coil V can be
adjusted so that the effect on the magnet F of the solenoid itself
is completely neutralised. is a resistance box for varying the
current, P a battery, and Q a commutator. A lens S of about 40
inches focal length forms the window of the case in which the
suspended part is hung. A lamp and screen are placed at about
40 inches from the lens. Cross wires are placed in front of the
lamp and a sharp image of these is thrown upon the screen by
the action of the lens and mirror. The spot of light may be
made to take up any desired position on the screen by properly
adjusting small permanent magnets in the neighbourhood of the
two magnets C and F. I had expected that a good deal of trouble
would have been caused by change of zero in the vertical direction
owing to changes in the silk fibre on which the mirror is strung,
but I was agreeably surprised to find that the spot of light would,
if the mirror were disturbed, return to the same horizontal. posi-
tion to within ;, inch. The zero position seemed to be quite
permanent.
In order that the two motions of the spot of light should take
place in horizontal and vertical lines, the axis DEH must be ad-
justed so as to be accurately perpendicular to the axis of suspension,
AK. The necessary fine adjustment is easily made by slightly
bending the suspending wire near the point B. I found that a
small block of cork formed the best means of connecting the wire
AB with the fork DBE. To get rid of any secondary effect of
the coil Z upon the lower magnet a second small “ compensating”
coil may be included in the circuit. In order to bring the spot of
light quickly to rest a suitable mica vane was attached to the
vertical wire AB. This rapidly stops the motions in azimuth.
When I exhibited the instrument to the Society, the magnet
C was slightly affected by the induced magnetization of the speci-
men of iron in the solenoid. This effect can not be compensated
by another coil, since a coil through which the magnetising current
flows will not imitate the magnetic behaviour of the iron. To
remedy this defect I have fitted a second magnet of moment
nearly equal to that of C to the vertical wire a short distance
below C, its axis pointing in the opposite direction to that of C.
The effect of the magnetised specimen on the astatic system is
very small and I hope that all trouble from this source has now
been got rid of.
The indications of the instrument can easily be reduced to
absolute measure (at least approximately) in the following way.
1892.] testing the magnetic properties of iron and steel. 335
Suppose, for instance, that it is desired that a movement of the
spot of light through 10 centimetres horizontally should corre-
spond to a magnetic force inside the solenoid equal to 100 C.G.S.
units. A known current is sent round the coil J, and this coil is then
so adjusted that the spot of light is deflected through the proper
distance corresponding to the calculated value of H. To standardise
the vertical motion of the spot, a magnetised steel wire may be
placed inside the solenoid M, through which no current is passing,
and the solenoid is then adjusted until the spot of light shows a
deflection in the vertical direction of 10 centimetres for each 1000
C.G.S. units of intensity of magnetisation of the steel wire. The
intensity of magnetisation of the steel wire can be determined by
the use of an ordinary magnetometer. If the cross section of the
wire to be tested is different from that of the steel wire an
appropriate factor must be introduced.
Although for very accurate observations in the subject of
hysteresis the use of two separate instruments, galvanometer and
magnetometer, will probably still be necessary, yet I think that
the instrument I have described may be found useful as a means
of rapidly gaining an approximate knowledge of the form of the
hysteresis curves for various samples of iron and steel without
any calculation. For this purpose it may be a useful instrument
in the Lecture Room.
May 30, 1892.
Proressor G. H. DARWIN, PRESIDENT, IN THE CHAIR.
The following communications were made:
(1) The hypothesis of a liquid condition of the Earth’s interior
considered in connexion with Professor Darwin’s theory of the
genesis of the Moon. By Rev. O. FisHer, M.A., F.G.S., Hon.
Fellow of Jesus College.
IN a series of papers in the Philosophical Transactions, Parts I.
and II. 1879, Professor Darwin has developed the theory of tidal
action in the solar system.
At p. 23 of his paper on Bodily Tides of Spheroids he
gives “a dynamical investigation of the effects of a tidal yielding
of the earth on a tide of short period according to the canal
theory.” A numerical estimate will afford a clearer idea of the
effects produced.
336 Rev. O. Fisher, On the hypothesis of a [May 30,
The symbols used are
h =the depth of the canal.
a = the earth’s radius.
w = the rotational speed referred to the moon (23 h. 56 m.
= lunar day at present).
¢@ — wt = the longitude west of the moon.
e = half the lag of the bodily tide.
2H =the greatest range of the bodily tide at the equator *.
2 :
T => x the moon’s mass x a” + (her distance)’.
—
The result obtained for the height of the wave relatively to
the bottom of the canal is
dé
h — h rr ’
where
d&) «+s Lose ((sadhaaas el a
ia we Soh c cos € — 5 gl) cos 2 (¢ — at)
Zi 5 sin ¢ sin 2(6 — wt} +.
Taking up the investigation from this point since the land
partakes in the rise and fall of the bottom of the canal, the
measurable height of the tide will be the difference between the
depth of the canal and the height of the water above the bottem
of it, which will be — h ds
dz’.
Hae ah lg Sere cos 2(¢—@ 9 Sin esin 2 b—at)- ,
—-h + 4gk
= Fara gh 2 08 2 — ot) = 4 —5 cos 2(9 — at).
For - me
Farhi write H, for 2(¢— wt) write 0.
Then the height of the tide will be expressed by
—H jeos (0 — ay see
0
“— cos a!
Ta
49h ; :
=—H fang 0 (cos €— 5) + sin @ sin al ;
| OT }
_ * Elsewhere Prof. Darwin uses E as the ratio of the bodily tide in the case of
viscosity to the like in the case of fluidity.
+ Loc. cit. p. 26.
1892.] liquid condition of the Earth's interior. 337
, ; 4qE
We must now estimate the numerical value of F aie
“b
From the definition of t already given we have
4gf _42 (distance)’ 1 earth’s mass
5 t 53 moon’s mass@ a’ =
_ 8 earth () E
15 moon \a/ a’
8 E
eer 0: Mere (ourze5e/ 20902404 «
= 045474 EF;
a foot being the unit.
If the interior is considered liquid, the bodily tide may be
taken equal to the equilibrium tide, which would be about 13 feet
from highest to lowest}, and # would be half that, so that
The Jag of such a tide would be small. Darwin seems to con-
sider 14’ as an admissible value for 2e in that case, and cose would
be 09999668. Hence cose would be greater than 5 — and we
may assume
7
and by substitution and reduction we obtain for the height of the
measurable tide
-H{1- Beane 5 a(S} cos {2(¢ — wt) — D},
or, cos e being very nearly unity,
=a at co s {2(¢ — wt) — D},
= — H x 060211 cos {2(¢ — wt) — D}.
Hence the tide would be 3ths of what it would be on a rigid
earth.
It is evident that tan D is small. Hence low water will occur
a little west of the moon.
* “The Moon” by Proctor, Tab. rv. p. 313.
+ Thomson and Tait, § 804, 2nd Ed.
338 Rev. O. Fisher, On the hypothesis of a [May 30,
In forming the potential of the protuberance of the bodily tide
the earth has been taken as homogeneous. But the superficial
parts having only half the mean density, it seems that the value
of z gh ought to be taken at one half that assumed, and then
5 ot
4 gE
T
: = 0:19894,
5
and we find for the tide
— H x 0°80106 cos {2 (¢@ — wt) — D} ;
which shows that it will be diminished by only {th of what its
height would be if the earth was rigid.
We learn from this expression that high ocean tide will occur
when 2(¢—ot)—D=7, that is when $— ot = = a to the
west of the moon; and high earth tide will occur when ¢ — at = — > ;
or : to the east of the moon. Hence the crests of the ocean and
earth tides are separated by the obtuse angle ul aa ae so that
the tidal protuberances of both of them, which are nearest to the
moon, are to the east of it; and the effects of the couples caused
by the moon’s attraction upon both of them will be to retard the
earth’s rotation.
Prof. Darwin remarks, that the expression for the height of the
ocean tide as affected by the bodily tide is subject to a modifica-
tion of the same form on the equilibrium theory as on the canal
theory, with the exception of a change of sign. Hence on that
theory also, which neglects the inertia of the water, and therefore
less nearly represents the case of nature, the ocean tide would be
diminished by the same factor, and therefore only to the small
extent of about one-fifth, as has been now shown would be the
case on the canal theory.
The lag of the bodily tide has here been put at 14’, because
Darwin has shown* that, on the hypothesis of approximate
liquidity, the reaction of the moon on the bodily protuberance
with that amount of lag would account for the unexplained
acceleration of the moon’s mean motion at the present time of
4 seconds in a century. It is evident that if the lag of the bodily
tide is larger, cos 2e will be smaller, and the reduction of the tide
in the canal will be still less,
* “Precession of a viscous spheroid,” § 14.
1892.] liquid condition of the Earth’s intervor. 339
The above appears to be a sufficient answer to the objection
brought against the theory of internal liquidity that in such a
case there could be no measurable ocean tides.
Prof. Darwin appears when he wrote to have held the view
that the earth must be very rigid probably in consequence of his
investigation by which he had proved that on a solid globe
nothing short of a high degree of rigidity could sustain the
weight of continents and mountains. This necessity is of course
entirely removed by Airy’s hypothesis that the crust is supported
in a state of approximate hydrostatic equilibrium on a yielding
nucleus *.
Assuming therefore the necessity of a high degree of rigidity,
Darwin finds a certain coefficient of viscosity, which according to
his calculations would cause the obliquity of the ecliptic to
increase most rapidly at the present time (p. 526), and uses this
particular value in his numerical calculations. Thus, when he
estimates the length of time, since the moon may have been
detached from the earth, at about 57 million years+, the estimate
depends upon that particular value of the viscosity. So also do
his estimates of the amounts of heat generated by tidal friction
within the earth during certain intervals of time dating from the
same epoch. And in short all the numerical results in Table Iv.
at p. 494, depend upon the particular assumed high degree of
viscosity. It cannot therefore be too carefully borne in mind by
Geologists that none of those numerical estimates, which relate
to time, are applicable to the case of a liquid interior.
With respect to the obliquity of the ecliptic, it seems probable
that it may have originated when the moon broke away from the
earth, however much the amount of it may have since changed ;
for the rupture must have occurred at what was then the equator ;
but the alteration in the principal axes of the earth owing to its
removal must have caused the axis of rotation to shift its place
within the mass, so that the plane of the moon’s orbit would
represent that of the original equator, while the plane of the new
equator would have become oblique to it.
Although, as just mentioned, the amounts of heat generated
in the earth during certain intervals of time depend upon a
particular assumed value for the viscosity, not so the whole
amount since the rupture. Darwin says “According to the
present hypothesis [of the generation of the moon] looking for-
ward in time [from that epoch], the moon-earth system is from
a dynamical point of view continually losing energy from the
* Phil. Trans. Roy. Soc., vol. 145, p. 101. See also a lecture by Sir G. B. Airy
“On the interior of the Earth.”’ Nature, vol. 18, p. 41, 1878.
+ See ‘“‘Precession of a viscous spheroid and remote history of the earth.”
Phil. Trans. Pt. u. p. 531, 1879,
340 Rev. O. Fisher, On the hypothesis of a [May 30,
internal friction. One part of this energy turns into potential
energy of the moon’s position relatively to the earth, and the
rest develops heat in the interior of the earth.” It is evident
therefore that, knowing the initial and present circumstances,
it is possible to estimate the total amount of energy converted
into heat without knowing the lapse of time in which it has
occurred. Darwin finds the common period of rotation, when
the moon separated from the earth, to have been 5h. 36 m.,
taking the viscosity at that time as small, the earth being sup-
posed to have been “a cooling body gradually freezing as it cools.”
The present rate of rotation relative to the moon (the lunar day)
is 23h. 56m. The total heat generated in the earth in the
course of this lengthening of the day if applied all at once would
he says* be sufficient to heat the whole mass of the earth about
3000° Fah. supposing it to have the specific heat of iron. In
Table Iv. of the former paper+ he had given 1760° Fah. as the
temperature corresponding to a period of rotation of 6h, 45 m.,
so that it appears that the additional 1240° must be due to
the loss on the difference between 6h. 45m. and 5h. 36m.,
or 1h. 9m., and he remarks that, “The whole heat generated
from first to last gives a supply of heat at the present rate of
loss for 3560 million years. This amount of heat is certainly
prodigious, and” he adds, “I found it hard to believe that it
should not largely affect the underground temperature”; but a
further calculation led him to believe that it need not do so,
for he found that 0°32 of the whole heat would be generated
within the central eighth of the volume of the earth, and only
one-tenth within 500 miles of the surface. The heat generated
at the centre is 3,4 times the average, that at the pole 1/24
of the average, and at the equator 1/122 of the average; and it
turned out that the heat, being so centrically produced, would, on
account of the slowness of conduction, not have had time to reach
the surface in the 57 million years postulated. This conclusion
depending on conduction would of course be true only in the ease
of a solid earth, the interior of which had the particular viscosity
which has been assumed, on which the 57 million years depend.
In connection with this point a serious difficulty seems to
arise. Lord Kelvin, in his well-known paper “On the secular cooling
of the Earth§,” held that, when according to his view it solidified
in a comparatively short period of time, the interior was at the
temperature of solidification suited to the pressure at every depth,
and, because the cooling would not even yet have penetrated to
*
‘‘Problems connected with the tides of a viscous spheroid,” p. 592.
+ ‘‘On the precession of a viscous spheroid,” p. 494.
t+ p. 561.
§ Trans. Roy, Soc, Edin. vol. xxu1, pt. 1., p. 157, and Nat. Phil., App. D.
1892.] liquid condition of the Earth’s interior. 341
any great depth, it ought to be so still if it is solid) How
then, it may be asked, could this enormous amount of heat be
perpetually being communicated to the central parts, and they
still remain solid? It seems that they must have become heated
far above the temperature of fusion appropriate to the pressure,
and must now be liquid; as nearly all geologists believe.
I think I have proved in the Physics of the Earth’s Crust*
that, if the crust is as thin as geologists suppose, and if the age
of the world is anything approaching to what geological pheno-
mena appear to indicate, then there must exist convection currents
in the interior, which prevent the crust from growing thicker
by melting off the bottom of it nearly as fast as it solidifies.
But I made no suggestion to account for such currents being
maintained. Here however we appear to find the explanation.
This centrically generated heat would be amply sufficient to
support fusion, and to keep the currents in action. Indeed the
difficulty is rather to see what would become of it all. Darwin’s
result, regarding the localization of the heat generated, does not
depend upon the viscosity, for the coefficient (v) which is intro-
duced into the calculation does not appear in the final result +;
but it applies only to the heat generated within the earth by
the action of the tidal couple upon the substance of the interior.
The distribution of heat within the earth caused by the tidal
couple will still follow the same law if only a portion of it is
generated within the earth, and the rest within the water of the
- ocean. Suppose for instance that the earth was either perfectly
rigid or perfectly fluid. In either such case no heat would be
generated within the earth. But without doubt the friction of
the oceanic tidal flow would, in a sufficiently long time, reduce the
speed of the rotation}. The heat in that case would be generated
only in the water, and be radiated into space. But besides friction
there seems reason to believe that some amount of heat may be
generated in the ocean owing to the fact that the speed of the
forced tide wave differs from that of the free wave with which a
disturbance would travel round the earth under the influence of
gravity alone. The question is an interesting one, and the following
attempt is made to solve it.
We have ¢ the west longitude of P the place of observation,
wt the moon’s angular distance west of the prime meridian. Then
the moon is ¢— a¢ east of P.
* 2nd Ed. pp. 77 and 349.
+ ‘* Problems connected with tides of a viscous spheroid,”’ p. 558, equation (28), viz.
aE: n\2 BTN J218 ue r\?
——. {3-5 (£) | -3(z) sin? 9 {32 (26+sin 6) (¢) |.
where H is the average loss of heat throughout the earth.
+ Sir W. Thomson on. “Geological Time.” Trans, Geol. Soc. of Glasgow,
vol. m1. pt. 1., 1868, p. 6. .
342 Rev. O. Fisher, On the hypothesis of a — [May 30,
Suppose, as is usual in the canal theory, that AP is developed
into a straight line, and that the earth is at rest, and the moon
moving westward above AP. Then, if AP=a, the attraction
of the moon on the water at P will be in the direction to diminish
x, and will be negative. The moon’s differential horizontal attrac-
tion at P will therefore be
= ne sin 2(¢— at),
which for shortness write — w sin 2p.
Let, as before, the depth of the canal be h, and its width
unity, Le. one foot, and let y be the height of the tide above the
undisturbed water.
Then we have for the accelerations on a unit particle of water
at P
X =—psin 2,
Z=-—4.
And s=ah, .. de=ady.
Now the work on a unit particle
i | (Xdx + Zdz)
=p [c pa sin 2dr — gdz)
=p (‘st cos 2y — gz) +0.
1892.] liquid condition of the Earth’s interior. 345
At an angular distance of 45° from the moon the water is
at rest, and its depth is the mean depth of the canal, viz. h. This
makes z the depth at P;
O=—pgh+C.
Hence the work on a unit mass of water in the column at
eae
p (5 cos 2yr — g (z -h))
ome ay
pa
=p "Gg cos Iyt+g
= pa a ee gh cos 2v.
Therefore the work on the whole column of unit width is
P 2 a= gh™ oe ty)
=p er cos 2p (1 - —=— Ee aah cos ni
=p is h ao cos 2 — e er oar 5 (1 + cos th.
To obtain the work done on a length AP of the canal we must
multiply this by ady and integrate, whence, putting for y its
value ¢— ot, and taking the integral from ¢=0 to t=t, we get
a =
2
work on AP =p Fae h ss =i {sin 2 (f — wt) — sin 24}
Ea “ge \,
de aw — gh ie
ljpa aw_=s\* ha... ,
ars For =) 5 {sin 4(¢—wt)—sin 44} | .
The work on the whole canal will be given by putting ¢ = 27,
and will be,
work on the whole canal = p \- n h era al sin 2wt
{he __ae ) ha ot
( 2 a’w*—gh/ 2 *
: — sin soot}
+ — |
wa aw ) si
"4AX2 @o* — gh
B44 Rev. O. Fisher, On the hypothesis of a [May 18,
This work is done by the moon upon the whole mass of water,
while she traverses the interval AQ.
Hence the work done while she makes a complete revolution
will be given by putting wt = 27, and it will be
Ha ao i ha 9
(5 a’w’ — gh} 2 fe
This work will accumulate once every lunar day.
To obtain the corresponding rise of temperature, we know that
a weight m raised through s feet is equivalent to heat sufficient to
warm m pounds of water through s/772 degrees Fah.; so that to
find the rise of temperature produced by the work W upon a mass
m of water we have
W=mgqs,
and the equivalent rise of temperature in the water will be
In the present instance
m = p27ha,
and therefore the rise of temperature in the water in a lunar day
will be
pa aw i 1
( 2 @w’ — gh] 29772
We know that
Ma? _ 1 24
LDP 182 aie
; _3 g
Hence p= z 182 x 10° -
And w = (000072924 radian,
a = 3959 miles,
h= 4 miles,
g = 32 feet per second.
Reducing to feet, the rise of temperature in the water of the
equatorial canal in degrees Fah. comes out about
0:000006° Fah.
in one year, or 6° in a million years.
We see then that under the present circumstances a very
small portion of the heat generated about the earth in this manner _
would be taken up by the ocean, and radiated into space, irre-
spective of the friction of the water. But Darwin informs us
* Thomson and Tait, 2nd Ed. p. 383.
1892.] liquid condition of the Earth's interior. 345
that, looking backwards, the moon’s orbital velocity increases very
rapidly. Now @ is the earth’s angular velocity minus the moon's
orbital velocity. If then retrospectively the moon’s orbital velocity
increases more rapidly than the earth’s angular velocity, » will
diminish.
If we put “= = = ) ,
2 a*w* — gh
idiot Bo. (42) 2a* w (a*o* + gh)
dw 2 (a’@* — gh)’
Hence, so long as a’w’ is greater than gh, wu will increase as @
diminishes. Moreover the moon’s distance diminishes. Hence,
(uwa/2)? varying inversely as the sixth power of the distance will
increase very rapidly; so that on both these accounts the heat
generated in the water per lunar day will rapidly increase. It
must not however be forgotten that the length of the lunar day
increases, so that fewer of them go to a year.
The above expression would become infinite if
o = WV ghia;
that is if w = 0'000039,
whereas at the present = 0:000073.
But such a result cannot be relied upon, because the same value
of @ would make the expression for the height of the tide infinite,
whereas in the formation of the differential equation from which it
is found it has been assumed to be small. But it is evident that
the generation of heat in the water must increase as that value of
@ is approached, and that something of the nature of a catastrophe
will have happened at that juncture, because, going back in time,
when that epoch has been passed the expression for the height of
the tide is found to have changed signs, and consequently high
and low water will have interchanged places then.
If is less than Vgh/a, then du/dw becomes positive, and the
heat generated in the water rapidly diminishes as » diminishes.
We know that Vgh is the velocity of the free wave, with which
a disturbance in the water would be propagated under the influ-
ence of gravity alone.
The friction of the tides against the coast-lines will of course
have had some effect in retarding the rotation, but how much we
cannot estimate*.
We have seen that the fact that the speed of the forced tide
wave in the ocean differs from that of the free wave is a cause of
* Airy’s “ Tides and Waves,” § 544. Encycl. Met. quoted by Sir W. Thomson,
*“Geol. Time.” Trans. Geol. So¢c., Glasgow, 1868.
346 Rev, O. Fisher, On the hypothesis of a [May 30,
generation of heat (though small) in the water. A like cause
must be in operation within the earth, because the forced bodily
tide has a different period from the free gravitational oscillation.
The distribution of the additional heat from this cause would pro-
bably, if calculated, turn out to be different from that arising from
the internal friction produced by the tidal couple, which is the
source of internal heat contemplated in Prof. Darwin’s work.
It seems then that, unless the ocean tides have been in opera-
tion for a length of time exceeding any estimate hitherto suggested,
it does not appear probable that any considerable portion of the
heat, which according to Darwin’s hypothesis of the moon being
shed from the earth has been from first to last generated about
the earth, can be got rid of by that means. It follows that a
largely preponderating amount of it must have accumulated within
the earth. This as already remarked must have kept the deeper
parts constantly above the temperature of solidification for the
pressure, and is an argument in favour of present liquidity.
But if such is their condition we cannot appeal to the slowness
of conduction in a solid earth to account for this great amount of
heat not making itself evident at the surface; which it must have
done unless it has been prevented from accumulating faster than
it has been generated. There seem to be only three important
means of effecting this, viz. (1) conduction through the solidified
crust, (2) transference of heat to the surface by volcanic action,
(3) the conversion of heat into work against gravity, and against
the molecular forces, expended in modifying during geological
ages the condition of the crust.
The first and most obvious mode of escape of heat from the
interior, which we now regard as liquid, is by conduction through
the solidified crust. I have explained in my Physics of the Earth’s
Crust*, how it is the latent heat of the layer by which the crust
would have been thickened more than, owing to the action of the
hot liquid, it is actually thickened, which escapes by conduction
through the crust, and that this heat is abstracted from the interior
mass and lowers its temperature. I have also shown that the
mean fall of temperature of the interior from this cause, consider-
ing the store of heat to have been initial+ and the time elapsed
10 ti eer however that this assumption is not necessary, for the whole mass
remelted (using the symbols in Physics of the Earth’s Crust) is y x 4m (r -—k)?, and
it yields up A times that amount of heat.
This divided by the volume of the interior will give the mean fall of temperature
_ Ayden (r — k)? _ 1
7 ar0-e rae
or putting + for e
v
k
= 3hy Z nearly,
as on the hypothesis of the heat being initial,
1892. ] liquid condition of the Earth’s interior. 347
100 million years, may be put at about 209° Fah. This calculation
involves the assumption that the ratio of the rate of thickening to
the rate of retardation (or remelting) is constant, or, what is equi-
valent to this, that the thickness of the crust varies as the square
root of the time since it began to be formed. It was there shown*
that the assumption of constancy of the above ratio of the rates of
thickening and retardation of thickening is probable, because, if
their ratio varied as any power of the time, it would lead to
unnatural consequences. I now find that the assumption that the
store of heat was initial is not necessary, because the same formula
can be obtained without that assumption, so long as we adhere to
the other assumption that the thickness of the crust varies as the
square root of the time, or to its equivalent. Such a calculation
is sufficient to show that the amount of heat, carried off by con-
duction through the crust in an interval even so long as 100 million
years, must have been quite inconsiderable compared to the whole
amount generated in the interior.
As the most extreme case possible of volcanic action we can
estimate approximately the fall of temperature of the earth sup-
posing the whole of the water of the ocean to have been originally
in solution with the magma of the interior, and to have carried off
a corresponding amount of heat. Professors Riicker and Roberts-
Austen have determined the temperature of melting basalt to be
about 920°C.+ Now the total heat of steam at t degrees C. given
off in condensing into water at 0° C. is given by the formula
605°5 + 0°305¢7;
whence it appears that unit of vapour at 920° C. will have parted
with 886 units of heat in condensing to water at 0° C. Hence every
unit mass of the ocean on the hypothesis now made represents 886
units of heat removed from the interior of the globe; for remem-
bering that the main body of water in the great oceans is at very
low temperatures, and that a large volume of water at the poles is
frozen, 1t is not a violent supposition to assume 0° C. as the mean
temperature.
Now even supposing the ocean to cover the globe and to be four
miles deep, its volume will be about 0:003 of the whole globe. The
density of the globe is 55 that of water. Hence the mass of the
ocean is 0:°003/5°5 times the mass of the globe. Then, taking as
Darwin has done the specific heat of the globe to be that of iron, viz.
1/9, we get the mean temperature of the interior reduced by this
means by 4°°53 C. or 8° F., an inappreciable amount compared with
the 3000° F. attributed to tidal action, by which the earth is
estimated to have been heated.
* Appendix to Physics of the Earth’s Crust, p. 21.
+ “Nature,” vol. xutv:., p. 456. Also Phil. Mag., Oct. 1891.
t Tait’s Heat, § 166.
MOL. VIL PT. VI. ae
348 Mr Willis, On Gynodicecism in the Labiatae. [May 30,
On the above extreme hypothesis that the ocean consists of
condensed steam emitted from the interior, the solid ejectamenta
of volcanic action would have had a very subsidiary effect in re-
ducing the internal temperature.
There remains the consideration of the heat converted into the
work which has been expended in producing elevations of the
surface, in shearing and contorting the materials of the crust, and
in inducing molecular changes. The amount of this work has no
doubt been from first to last enormous; but it is easy to see that a
very inconsiderable fall of temperature throughout the interior
would represent a very great deal of such work effected. For
instance, the work of raismg through half its height a layer of
granite ten miles thick, weighing 178 pounds per cubic foot, would
represent the heat equivalent to a fall of temperature of only
one degree F. throughout the globe.
We have not then so far arrived at an answer to the enquiry—
What has become of all the heat generated by tidal friction ?
There appear to be only two replies to this question. One is, that
the solidification of the crust took place a very long while subse-
quent to the genesis of the moon, so that the still liquid surface
was able for ages to radiate directly into space the heat carried
up to it from below by convection during the time, when, owing to
the proximity of the moon, the generation of internal heat went on
most rapidly. The other answer can only be, that the moon was
not originally thrown off from the earth, but was left behind accord-
ing to the nebular hypothesis. In that case the whole amount of
tidal action would not have been so great, though nevertheless
sufficient heat may have been centrically generated by it to main-
tain those internal currents, which the theory of a thin crust and
liquid interior appear to necessitate.
(2) On Gynodiacism in the Labiatae. (First paper.) By
J. C. Wiis, B.A., “Frank Smart” Student in Botany, Caius
College.
In July, 1890, my attention was called, by Mr F. Darwin, to
the occurrence, on hermaphrodite plants of Origanwm vulgare in
his garden, of occasional flowers having one, two, three, or even
all, of the stamens aborted. JI found such flowers, on examination
of many plants, to be of fairly common occurrence. The corolla is
usually smaller than in the normal hermaphrodite flower, and
may even, especially in the case of the female flowers, be as small
as the corolla of a normal female flower (i.e. a flower on a female
plant). The aborted stamens are represented by small dark-
coloured bodies in the throat of the corolla, usually sessile, but in
some cases shortly stalked.
1892.] Mr Willis, On Gynodiewcism in the Labiatae. 349
Corresponding variations on the normally female plants were
of much rarer occurrence. Occasionally, however, I found a female
plant bearing a large hermaphrodite flower among the females,
and flowers with one or two stamens also occurred.
That these variations are not simply due to cultivation appears
from the fact that they are as common upon the wild form, which
I have examined at Abington (Cambs.) and Llangollen. Three
batches of plants gathered at the same time in August, 1890, gave
the following results :
Batch | Plants | Flowers I. Di | ae LVe Total %
A. 12 1146 8 8 14 37 67 5°85
B. 7 745 8 21 11 14 54 7°23
Cc 9 588 2 17 6 7 32 5:44
|
Total | 28 | 2479 | 18 | 46 | 31 | 58 | 153 | 6-17
. Plants from the ‘‘Labiatae’’ bed, Bot. Gardens, Camb.
iB. | GinOCOrE Bae ec EeOaCeee COMMIS Tce TIES aR Ser oteeeecatonce cease ener
BOER E cacieeciastet es Abington (Cambs.)
The numbers in columns I., IL., I11., IV., represent the numbers of flowers with
1, 2, 3, 4 aborted stamens, respectively.
It will be noticed that the number of flowers fully female is
greater than the number of any of the intermediate forms, and
this I found to be always the case, if a considerable number of
plants were examined. Some of the variations were very striking,
eg. on each of two plants in batch A there occurred a lateral
twig which bore female flowers only. I have observed the same
phenomenon on two or three other occasions.
Similar variations occur upon the hermaphrodite plants of
other Labiatae. Miiller*, Schulz+, and others have observed them
in some, and I have myself noticed them in Zhymus serpyllum,
Nepeta Glechoma, and N. Cataria (all in the wild state), besides
many garden plants of the order, e.g. Micromeria juliana, Nepeta
longiflora, Hyptis pectinata, Bystropogon punctatus, Mentha crispa,
Satureia hortensis and S. montana. The last-mentioned is ex-
tremely variable, at least in Cambridge, more than half the flowers
usually departing from the normal type.
During 1891 various observations were made upon these
abnormalities, with a view to discovering the conditions govern-
ing them, and also to throwing some light upon the origin of
* « Fertilisation of Flowers,” Eng. Ed. p. 476 (Calamintha Clinopodium).
+ ‘Die biologischen Eigenschaften von Thymus Chamaedrys Fr. u. T. angusti-
folius Pers.” Deutsche Botan. Monatsschr. m1, 1885, p. 152.
27-2
350 Mr Willis, On Gynodiacism in the Labiatae. [May 30,
gynodicecism. If Ludwig’s* view be correct, that the primary
cause is the protandry of the flower, rendering the stamens of the
earlier flowers useless, we might expect to find these variations
more frequent at the commencement of the flowering season.
This was tested as follows: Ten cuttings were taken from the
same parent stock, and grown under similar conditions: every
flower was carefully examined. The abnormalities were most
erratic in their occurrence, and I was unable to discover any con-
ditions governing this point. No two of the plants gave results
corresponding in any way, nor did the average follow any ap-
parent rule. Some bore about the same percentage of abnormal
flowers throughout the season, others bore them many at one
time, few at another: e.g. No. 1 bore 126 females altogether; 73 of
these appeared in three days, and of these 29 were upon one small
lateral branch of the inflorescence. It may however be noted that
the percentage of abnormalities was much lower than in 1890,
being only 2 per cent.; a result possibly due to the plants
being cultivated, and having no competition with one another for
space. ;
One plant was protected from insects by a muslin net through-
out the flowering season, and did not, though it bore hundreds of
flowers, set a single seed capable of germination. It should be
noted, that owing to the smallness of the meshes, the plant could
not be shaken by the wind.
Observations were also made (1891) upon Nepeta Glechoma
(wild). Two lots of plants were examined, one (A) growing on a
dry sunny bank, the other (B) in deep shade in a wood. The
latter commenced to flower 18 days later than the former and
were much taller and less hairy.
The numbers of plants in flower and the numbers of open
flowers were counted weekly during the flowering season, and it
was found, as Ludwig has observed in the case of thyme, that the
proportion of female to hermaphrodite plants in flower was greater
at the beginning than at the end of the flowering season. For
example in lot A, on the first day 6 female plants and one
hermaphrodite flowered; the percentage of females being 85°7. A
week later it was 33°6 per cent., and near the end of the season
was 23°6 per cent. In lot B, the percentages each week were 50°,
16°, 35°8, 28°5, 23°4, 19°2, 28°3.
It was noticed that the female plants generally bore more open
flowers at one time than the hermaphrodites. In lot A the
number of flowers on each female plant was (on an average for the
whole season) 2°40, and on each hermaphrodite 2:16. In lot B
the numbers were 315 and 216. The greater size of the her-
* “Ueber die Bliithenformen von Plantago lanceolata L. und die Erscheinung
der Gynodiécie.” Zeitschr, f. d. Ges. Naturw. x1. 1879. p. 441.
1892.} Mr Willis, On Gynodiecism in the Labiatae. 351
maphrodite flowers is thus to some extent compensated for by the
greater number of the females.
Abnormalities in the flowers of Nepeta, like those observed in
Origanum, We., are comparatively few and far between, but were
yet fairly often encountered.
During the course of these observations upon Nepeta an
interesting point was noticed. The protandry of the flowers
appears to vary according to the season: at the beginning of the
flowering season the stigmas begin to separate very soon after
the dehiscence of the anthers, while towards the end of the
season these processes are separated by a considerable time. I am
conducting further observations upon this point. If it should
prove general, it would, taken together with the negative results
of the above observations on Origanum, have a tendency to dis-
prove Ludwig’s view of the origin of gynodiccism. This point
however I hope to discuss in a future paper, when I shall have
concluded the further observations on Origanum, &c., which are
now being conducted.
(83) On the Steady Motion and Stability of Dynamical Systems.
By A. B. Basset, M.A., F.R.S., Trinity College.
1. The object of the present paper is to develop a method for
determining the steady motion and stability of dynamical systems,
by means of the Principle of Energy, and the Theory of the
Ignoration of Coordinates. The subject has already been discussed
by Routh*, but is treated in the present paper in a slightly
different manner.
Let the coordinates of a dynamical system consist of a group 8,
and a group of ignored coordinates y; and let « be the constant
generalized momentum corresponding to x. Then if the velocities
x be eliminated by means of the equations
a
dy”
it is well known that the kinetic energy of the system will be of
the form
T=2 +8,
where & is a homogeneous quadratic function of the velocities @,
and § is a similar function of the constant momenta x.
Also if © be that portion of the generalized component of
momentum corresponding to 8, which does not involve @, and
* Treatise on Stability of Motion.
352 Mr Basset, On the Steady Motion [May 30,
which is consequently a linear function of the x«’s, the modified
Lagrangian function is
L=T4 36048. (1),
where V is the potential energy, measured from a configuration of
stable equilibrium *.
The equations of motion of the system are accordingly
ee See 6) + ak dV _
dtd@ dt dé dé dO Ooh
From this equation it appears, that a steady motion may
usually be obtained by assigning constant values to the coordinates
@; whence the equations of steady motion are
d& dV
do dO ~ |
where the number of equations of the type (8) is equal to the
number of coordinates 6. .
O) i esseneene acess «5h (3),
2. Let there be m coordinates of the type 0, and n ignored
coordinates of the type x; then we have three cases to consider,
according as m is equal to, less than, or greater than n.
Case Il m=n.
In this case, the number of equations of the type (8) is equal
to the number of momenta «; hence these equations are sufficient
to determine these momenta. Accordingly the conditions of steady
motion are, that it should be possible, without violating the con-
nections of the system, to assign constant values to the 6’s, such
that the values of the m momenta x, furnished by the solution of
(3), should be real.
Case II. m