by an application of the second law of thermodynamics." "All we need is an

PHYSICS: C. BARUS

by

an

application

of thesecondlaw of

thermodynamics."

"Allweneed isan

expression

for

dS,

theincrementin the

entropy

caused

by

motionof the

pis-

ton. If (pis the

change

in the

energy

of the

system

which

accompanies

the transferenceof each electron from the hot

body

tothe

surrounding enclosure,

then

dS =

[d(n

v

p)

+

pdv],"

etc.

In

writing

this

equation,

which is fundamental to his

argument,Richardson

treatsthecaseofametal

emitting

electrons

precisely

^{as one}treatsthecaseof a

body

ofwater

giving

offsteam to

push against

^a

piston.

^That

is,

he treats theemission of electrons as a

process strictly comparable

with

evaporation.

Butthere is an

important

differencebetween thetwo

processes.

In

evapora-

tion the

thing given

off is of the same substanceas thatleft behind. In the emission of electrons this is not true.

Evaporation

leaves the constitution of the

remaining liquid unchanged.

Emissionof electrons

continually changes

the constitution of the

emitting metal,

unless other electronsare

put

^{into the}

body

tomake

good

theloss. Whena

body

emitsacertainmassmof electrons under the conditions described

by Richardson,

^the

system

underdiscussion takes in

something

^{more than heat}

energy;

it takes in

substance,

themassm

of electrons. There isno

analogue

tothis in the

process

of

evaporation,

and itremains to beshown that the

equation

which I have

quoted

from Richard-

son,

an

equation

thatholds

beyond. question

for thecaseof

evaporation,

holds also for thecaseof emission of electrons.

One

cannot, according

^to

my view,

meetthis

difficulty by supposing

the

body

ofmetalmade

very large,

so

large

that thestatic

charge produced

onit

by

the emission of a massmof electrons without

compensation

would be

negligibly

small.

For,

if the loss of the electrons is not made

good,

the

mass-law,

re-

quiring

thatn, X nishall remain constant within the

metal,

will cause ioni- zation there

proportional

to

m,

without

regard

to the amount of the

metal;

and thisionization will introducea

consumption

of'heatforwhich Richardson has

made,

I

think,

no

provision.

ON THE

EQUATIONS

OF THERECTANGULAR INTERFEROMETER BY CARL BARUS

DEPARTMENT OF PHYSICS, BROWN UNIVERSITY

.Communicated December8, 1917

1.

Auxiliary

Mirror.-Itis desirable

to deduce the

fundamental

equations

more

rigorously

thanhasheretofore' seemed

necessary.

.

Figure

^{1 is}

supplied

for this

purpose,

and

represents

the moresensitive

case,

where in addition to themirrors

M,M', N,N' (all

butM

being necessarily half silvers),

there is an

13

auxiliary

mirror

mm, capable

of rotation

(angle a)

about a vertical axis A.

The mirrors M-N' in their

original position

are

conveniently

^at 45° to the

rays

of

light,,

whilemmis normaltothem.

Light arriving

atLis thus

separated by

thehalf silverNat

1,

into the two

components 1,2,1,9,3,T

and

1,6,7,6,3,T, interfering

in the

telescope

atT.

When mm is rotated over a small

angle a,

these

paths

are modified to

1,2,2',4,4',5,T2

and

1,6,7',8,T1.

T1and T2 enter the

telescope

in

parallel

and

produce

interferences visible in the

principal

focal

plane, provided

the

rays

T1 and T2 are not toofar

apart,

in

practice

not more than 1 or2 mm. In- terference

fringes

therefore will

always, disappear

if the

angle

^a is

excessive,

butthe limits are

adequately

wide for all

purposes.

Theessential constants ofthe

apparatus

are tobe

(9,1)

⁼

(6,3)

=

b; (1,2)

=

(6,7)

=

c; (9,3) =(1,6)

=

2R,

R

being

^theradius of rotation.

JiL^^-lA-X css'//^

^"

d-

When themirror mm is rotated tom'm' over the

angle a,

the new

upper path

will be

c + R tan a

+

d

+

e

+ g,

where

(2',4)

⁼

d, (4,4')

⁼

e, (4',5)

⁼ g,

the

plane (8,5)

=

q

normal to T1 and T2

being

the final wave front. The lower

path

is

similarly

2 R

+ (c

- R tan

a) +

d' to the same wave front

(8,5)

where

(7',8)

⁼ d'. Hence

(apart

from

glass paths

which have been

treated,

the

path

differencenX

(n being

the order of

interference)

shouldbe nX= 2R

(tana

-

1) +

d^- d'

+

e

+ g.

The

figure

inview of the laws ofreflectionthen

gives

^usinsuccession d =

(b +

^c

+

Rtan

a)/(cos

^2a

+

sin

2a),

d' =

(b +

c- Rtan

a)/(cos

2a

+

sin

2a),

e = 2

R/(cos

2a

+

sin

2a),

g

= 2Rsin2a

(1 +

tan

a) (cos

2a ^- sin

2a)/(cos

2a

+

sin

2a),

q = 2 Rsin2a

(1 + tana).

PHYSICS: C. BARUS

Toobtain

g

it issufficient to treat thesimilar

triangles (3,8,9')

and

(9,8',9')

where h =

(9,4), h'

=

(3,8),

k =

(9,9'),

1 =

(9,8') may

be foundin

succession

asthenormaldistancebetweenthe

mirrors

Mand M'is

RV/2,

^{so that}

finally g

=

(h- I)

sin

(45

^-

2a), q

=

(h

-

)

cos

(45

-

2a).

Ifthese

quantities

^are introduced into the above

equation

fornX we

may

obtain aftersomereduction

nX = 4 Rsina

(cos

a - sin

a).

Since nX = 2AN cos

i,

AN

being

the normal

displacement

ofthe mirror

M',

and i =

45°,

the

corresponding equation

tothe second order ofsmall

quanti-

ties^ais

AN/Aa = 2R

(cos

^{a -} sin

a)/cos

i =

2A/2

R

(1

^{- a -}

a2/2).

Ifais

sufficiently small,

thecoefficient is

simply

2

R/cos

iasusedheretofore.

Thereremain the

glass paths

which for the

rays

dandd'are

compensated.

Additionally

^the

upper ray

hasa

glass path

3

displaced

^to4'. Thelower

ray

has the fixed

path

at

1,

and this is

equal

to the other at

1,

since the

angles

are45°. Thus the variable

part

ofthe

glass paths

at 3 to4'is

uncompensated

and the

angle

^of incidence

changes

from 45° to 45° ^- 2a. The

reflecting

sidesof the

plates

are silvered. Hence e

(sin

i ^- cosi tan

r)

2Aa mustbe addedtothe

equation.

2.

Rotating

Doublet.-The second

case, figure 2,

inwhich the

auxiliary

mir- ror ofthe

preceding apparatus

isomitted

is, curiously enough, inherently

sim-

pler. M,M', N,N',

are

mirrors,

halfsilveredat

(1)

and

(3)

and the twolatter on averticalaxis A and

rigidly joined by

therail

(2,3).

The mirrors

being preferably

at

45°,

the

component rays

are

1,2,3,T

and

1,5,3,T,

themirrorM'

being

onamicrometer with the screw normal to the face. The

ray parallelo- gram

^{is made}

up

asbefore of

(1,2)

= b =

(3,5)

and

(1,5)

= 2 R =

(2,3).

When the rail

(2,3)

^{is rotated} over an

angle a,

themirrors take the

position

N1and

N1'

atan

angle

a to their

prior position

and the

angle

of incidence is now 45° - a. The new

paths,

if

(4,6)

is the final wave

front,

are thus

(1,2,2', 6,T2)

and

(1,5,4,T1).

The

rays T1

and

T2

are

parallel

andinterfere in the

telescope.

Hence the

path

difference introduced

by

rotation is

(n being

the order of

interference)

nX= b+Rtana

+ (2 R/cos a)

cosa-

(2

R

+

b- Rtan

a)

= 2 R tan

a,

for the

triangle (a,7,2')

isisosceles and its acute

angles

each a.

The

rays

T1 and T2 have now

separated

and the amount

(4,6)

is also 2 Rtan a. When this exceedsafewmillimeters theinterferences vanish.

A correction musthoweverbe

applied,

sincein the

practical apparatus

the mirrorsrotate at a fixeddistance

apart.

Hence themirror

N1

mustbe dis-

placed

toward the

right (shortening

the

path) by

thenormal distance e

(R/cos

^{a -}

R)

cos45°

andthemirror

N1'

towardtheleft

by

thesame amount. The

path

difference

15

introducedisthusadecrementand is twice the 2ecos

(45°

^-

a)

ofeachmir- ror. Thus the total correction to besubtracted from the

equation

is after reduction 2 R

(1

^-

cosa) (1

+

tana).

Hence the

equation

^becomes

ultimately

nX = 2 R

(sin

^a

+

cosa -

1).

Tothesecond order ofsmall

quantities,

ifi = 45° is the

angle

of

incidence,

andAn thenormal

displacement

of

M',

AN/Ac

=

R(1 + Aa/2)/cos

^i.

As all the mirrors receive the

light

on their silvered

sides,

M

originally compensates

N if the mirrors are identical in thickness and

glass.

But the transmission

at

3 varies as the

angle

of incidence

changes

from i = 45° to 45° ^- a. The

glass path

^here decreases

by

^e

(sin

i - cos i tan

r) Aa,

whereeis the

plate thickness,

^rthe

angle

ofrefraction. The

path

difference asabovereckonedhas thus beenincreased

by

thisamountand this

quantity

is to beadded to the

right

hand member. Theeffectwillnot

usually

exceed

afew

per

centof the air

path difference,

and the ratio is the same as above.

3. Ocular Micrometer.-It has been stated that the motion of the

fringes

acrossthe field of the

telescope, T,

is

astonishly

swift. Henceit is often de- sirable to insertamicrometer

here,

asthe

displacement

^of

fringes

can thus be much more

accurately

and

easily

measured than at the micrometer

along

the normal of the

opaque mirror, M',

^{of the}interferometer. If thelatter is of the

type using

an

auxiliary mirror, mm, figure 1,

^the

fringes may

even be estab- lished of a size to

correspond

with the ocular

micrometer, by rotating

the

auxiliary ^mirror;

^but this is not

usually necessary.

^A

good

ocular

plate

micrometer was at hand

dividing

^the width of field

(about

1

cm.)

into 100

parts,

thedivisions

being

^0.1mm. One-tenth of this is

easily

estimated

by

the

eye

inviewof the

strong eye

lens. The

light

from the collimatorat L should

completely

fill the

field,

^aconditionwhich

may

be fulfilled

by suitably placing

the

former, modifying

its

objective.

After

completing

such

preliminary

ad-

justments

with the

fringes,

made

very sharp

and the ocular scale

equally so,

this is to be

placed

at

right angles

to the

fringes.

Let Ae denote their dis-

placement

measured in centimeters on the ocular scale and AN

(cm.)

^{the dis-}

placement

of the

opaque

mirror M' of theinterferometer. The

question

is whetherAe andAN are

nearly enough proportional quantities

^for

practical purposes.

A number of such standardizations were carried out

throughout

1 cm.of

Ae,

twoof whichareshown indetail in

figure

^{3. The}fluctuationof data is due toair currents across the interferometer. It was not

easy

toob- viate

these,

and it was not

thought necessary

for the

present purposes.

Other- wise the datawould have been smooth. There isnodoubt thatalinear rela- tion

may

beassumed. In curvea the

readings

of the interferometer micro- meter

increase,

incurveb

they

decrease. If themeansbetaken from doublets far

apart

^theratiosare

(a) AN/Ae

⁼

0.00310; (b) AN/Ae

⁼

0.00310,

16 C.

PHYSICS: C. BARUS

and

they happen

to

coincide.

ThusAe

is

323timesas

large

asANandcorre-

spondingly easy

to measure. The

impossibility

of

setting

themicrometer for AN

accurately enough,

since it is

graduated

to

only

5X10-5cm.is

completely

obviated inAe.

Moreover,

as2ANcosi X

(i being

the

angle

of incidence

450,

andXthemean wave

length),

we nowhave

0.0061/Ae

cosi =

X;

sothat the

fringe displacement

^Ae = 0.014 cm. measured on theocular micrometer

corresponds

tothewave

length

of

light

in the interferometer measurements.

This ismore than one scale

part.

There ishoweverno

difficulty

in

making

the

fringes larger

and

obtaining

^amuch more sensitive

apparatus

in

propor-

tion. The

achromatic fringes, moreover,

when

properly produced,

containa

distinctive central black

line, compatible

with the measurementof 0.1 scale

part,

ashere

given; i.e.

measurementto afew millionths ofa centimeterare thus

easily feasible under proper surroundings. The apparatus

will

be used

elsewhere.

9

7

3

'd

-I i,%

A~~~~~

^A..,

~ ^/*a

1 -/I s 4-

If

Ago,

the

angular fringe breadth,

is

given, Ae/AN may

be

computed

from the

equation

inthe

earlier paper, to be

Ae/AN = 2LAp cos i/X

or

LAp

=

X/(2

cos

i.AN/e)

as theradius is the

length L

=

19.5 cm. of the telescope. Hence the fringe

breadth in

centimeters is, if

X= 6X

10-Lcm.,

i=

45°,

and

10IAN/Ae

=

3.1,

LA^o

= 0.014

cm.,

the value

actually

observed.

Thus if Ap is given or measured,

Ae/AN

may

bededuced.

The

question finally

to bedetermined is thus the value and the

meaning

of the

fringe

breadthAp.

Since

2 AN cos is nX

17

i

Adrxlow ^10,

if AN =

ANo

is constant and also

X,

we

may

thenwrite A(e = di/dn =

-X/2 ANo

sini.

FurthermoreifA0is the

angular displacement

of

fringes

AO =

nA<p

= -

(AN/ANo)

cot

i,

if nX is

replaced by

its value and AN is small

compared

with

ANo.

If it is

not,

since

ANo

involves

AN,

we must statethecasethus:

ANo+AN

-A =cot i

dN/dNo,

JANo

or on

integrating

and

expanding

thenatural

logarithm

- AO = coti

[AN/ANo -(AN/ANo)22 + ...]

and Ae = LAO.

In the above measurements

A(p =

LAp/L

= 7.2 X 10-4 whence

apart

from

signs

ANo

=

X/2A(p

^sini = 0.06

cm., nearly,

whereas the maximum

displacement

^AN

throughout

the whole series

(equiva-

lenttothe

telescopic

field

width)

doesnotexceedAN= 5 X 10-3cm. Hence

(AN/ANo)2/2 may

here be

neglected

to about

1/300

and

(again apart

from

signs)

since i =

45°,

Ae =

L(AN/ANo)

⁼ 325

AN,

asitshould

be; i.e.,

therelation ofAeandAN is

practically linear,

ifthe dis-

placement

ANis notexcessiveor

goes beyond

the

equivalent

of field width.

As the determination of

ANo

is inconvenient we thus come back to the

practical equation already used,

^or

AN/AO =

X/2Aqp

cos

i,

orif

Ld(p

= beandAeand8earethe

fringe displacement

andthe

fringe

breadth measuredonthe sameocular

micrometer,

AN/Ae

X/2

^{de cos}

i,

With this deduction the

equations

of

long

distance

interferometry, etc.,

form in termsof 6e the

fringe

breadth andthe

fringe displacement

Aewhich

may

be recorded

here,

d

being

the

distance,

Aa =

(X/2R6e)Ae;

^{d =}

(bRbe/X)/Ae.

4. Collimator Micrometer.-For

many purposes

even better conditions are obtainable

by replacing

theslit ofthe collimator

by

a

plate glass

^microm-

eter. The

magnification

in such a case is

usually greater

^{and since the} tele-

scope

now containsnofiducial

lines,

it need not be

fixed,

but

may

be shifted

PHYSIOLOGY:S. HATAI

at

pleasure

so

long

asthe

collimator is

fixed. This is often a

great conveni-

ence in

working

with the achromatic

fringes;

but I will

pass

these

details

overhere.2

1These PROCEEDINGS, 3,

1917,

(563). In this note the glass

paths,

whichareofless

importance than theairpaths, ^{are toomuch}accentuated.

2Abridged

from a

Report

tothe Carnegie Institution ofWashington,D. C.

THEBRAIN

WEIGHT

IN RELATION TO THEBODY LENGTH AND ALSO THEPARTITIONOFNON-PROTEIN

NITROGEN,

IN THE

BRAIN OF THE GRAY SNAPPER

(NEOMAENIS GRISEUS)

BY SHINKSHI HATAI

TORTUGAS LABORATORY, CARNEGIE INSTITUTIONOF WASHINGTON, AND THE WISTAR INSTITUTE OF ANATOMY AND BIOLOGY

CommunicatedbyA. G.Mayer,December24, 1917

The

predatory

fishcalled the

gray snapper,

N.

griseus

was

mainly

^usedfor

the

present investigation,

which was conducted at

Tortuga's,

Florida in the summerof 1917. The

following

arethemore

important

facts

brought

out.

1. The Brain

Weight

in Relationto

Body Length.-Altogether

observations have been made

upon

74 brainsof the

gray snapper.

Itwas found that the relation of brain

weight

tothe

increasing body length,

from 150mm.

upward,

is

practically

linear and

may

be

satisfactorily expressed

asy ⁼ a

+ bx,

where

y represents

brain

weight

in

grams

and xthe

body length

in

millimeters,

and

aand barethe constants with the

vales,

ⁱⁿthis

instance,

-0.333and0.00433

respectively. (Body length

is measured fromthe

tip

of thesnout tothe crotch of thetail.)

It iswell knownthatintheadult

stage

therelation between brain

weight

and

body length

or

body weight

is

practically linear,

evenin the case ofsome mammals1 butit isremarkable tofind the linear relation in

fish,

when

they

aresosmall. This linear relation

during

the

period

of

early growth probably

*means that in the

snapper

the brain reaches its structural

maturity early,

andthat the

subsequent

increase in

weight

indicates

merely

a

uniform

swell-

ing

ofthe nervous tissue asawhole.

On account ofscantiness of thedata for

specimens

less than 200 mm. in

body length,

I amunable to

present

a

complete

record of the

growth

of the brain.

However,

it

appears

from the

general

trendofthe

growth-curve

that withthe

possible exception

ofthe

very early period,

the

relation

betweenthe

brain weight

and

body length

islinear.

Kellicottt

studied the

growth

ofthebraininthe smooth

dogfish

in

respect

to

the body weight,

andfound the

graph

toresemble that of the

mammalian brain (logarithmic curve)

andthus to

differ strikingly

fromthat for the

gray

19