HI

PHILOSOPHICAL

TRANSACTIONS

OF THE

ROYAL SOCIETY OF LONDON

SERIES A.

CONTAINING 1'APEES OF A MATHEMATICAL OE PHYSICAL CHAEACTEE.

VOL. 205.

LONDON :

PRINTED BY HARRISON AND SONS, ST. MARTIN'S LANE, W.C.,

iit ^rbinarg to |jis gjajtstg.

MAY, 1906.

Q «f I

CONTENTS,

(A) VOL. 205.

List of Illustrations page v

Advertisement . vii

I. On the Normal Series satisfying Linear Differential Equations. By E. CUNNING-

HAM, B.A., Fellow of St. John's College, Cambridge. Communicated by Dr. D. H. F. BAKER, F.R.S page 1

II. Memoir on the Theory of the Partitions of Numbers. — Part III. By P. A.

MACMAHON, Major P. A., Se.D., F.R.S 37

III. Atmospheric Electricity in High Latitudes. By GEORGE C. SIMPSON, B.Sc.

(1851 Exhibition Scholar of the University of Manchester). Communicated by ARTHUR SCHUSTER, F.R. S. (51

IV. The Halogen Hydrides as Conducting Solvents. Part I. — The. Vapour

Pressures, Densities, Surface Energies and Viscosities of the. Pure Solvents. Part II. — The Conductivity and Molecular Weights of Dissolved Substances. Part II f. — The Transport Numbers of. Certain Dissolved Substances. Part IV. — The Abnormal Variation of Molecular Conductivity, etc. By B. D. STEELE, D.Sc., D. MC!NTOSH, M.A., D.Sc., and E. H. ARCHIBALD, M.A., Ph.D. (late 1851 Exhibition Scholars). Communicated by Sir WILLIAM KAMSAY, K.C.B., F.R.S. 99

V. TJie Atomic Weight of Chlorine: An Attempt to determine the Equivalent of

Chlorine by Direct Burning with Hydrogen. By HAROLD B. DIXON, M.A., F.R.S. (late Fellow of Balliol College, Oxford), Professor of Chemistry, and E. C. EDGAR, B.Sc., Dalton Scholar of the University of Manchester . . 169

a 2

VI. Researches on Explosives. — Part III. By Sir ANDREW NOBLE, Bart., K.C.B.,

F.R.S., F.R.A.S. page 201

VII. Colours in Metal Glasses, in Metallic Films, and in Metallic Solutions.— II.

By J. C. MAXWELL GARNETT. Communicated by Professor Larmoi; Sec.RS. 237

VIII. On the Intensity and Direction of the Force of Gravity in India. By Lieut.-Colonel S. G. BURRARD, H.E., F.R.S. 289

IX. On the Refractive Index of Gaseous Fluorine. By C. CUTHBERTSON and

E. B. II. PRIDEAUX, M.A., B.Sc. Communicated by Sir WILLIAM RAMSAY, K.C.B., F.R.S 319

X. Modified Apparatus for the Measurement of Colour and its Application to the

Determination of the Colour Sensations. By Sir WILLIAJI DE W. ABNEY, K.C.B., F.R.S. 333

XL The Pressure of Explosions. — Experiments on Solid and Gaseous Explosives. Parts I. and II. By J. E. PETAVEL. Communicated by Professor ARTHUR SCHUSTER, F.R.S 357

XII. Fifth and Sixth Catalogues of the Comparative Brightness of the Stars — in

Continuation of tfiose Printed in the 'Philosophical Transactions of the Royal Society' for 1796-99. By Dr. HERSCHEL, LL.D., F.R.S. Prepared for Press from the Original MS. Records by Col. J. HERSCHEL, R.E., F.R.S. 399

XIII. On the Accurate Measurement of Ionic Velocities, with Applications to Various Ions. By K. B. DENISON, M.Sc., Ph.D., and B. D. STEELE, D.Sc. Com- municated by Sir WILLIAM RAMSAY, K. C.B., F.R. S. 449

XIV. On Mathematical Concepts of the Material World. By A. N. WHITEHEAD,

Sc.D., F.R.S., Fellow of Trinity College, Cambridge 465

Index to Volume 527

LIST OF ILLUSTRATIONS.

Plates 1 to 13. — Sir ANDREW NOBLK : Researches on Explosives. — Part III.

Plates 14 to 20. — Lieut. -Colonel S. G. BURRARD on the Intensity and Direction of the Force of Gravity in India.

Plate 21. — Mr. J. E. PETAVEL on the Pressure of Explosions.— -Experiments on Solid and Gaseous Explosives. — Parts 1. and 11.

ADVERTISEMENT.

THE Committee appointed by the Royal Society to direct the publication of the Philosophical Transactions take this opportunity to acquaint the public that it fully appears, as well from the Council-books and Journals of the Society as from repeated declarations which have been made in several former Transactions, that the printing of them was always, from time to time, the single act of the respective Secretaries till the Forty- seventh Volume; the Society, as a Body, never interesting themselves any further in their publication than by occasionally recommending the revival of them to some of their Secretaries, when, from the particular circumstances of their affairs, the Transactions luid happened for any length of time to be intermitted. And this seems principally to have been done with a view to satisfy the public that their usual meetings were then continued, for the improvement of knowledge and benefit of mankind : the great ends of their first institution by the Royal Charters, and which they have ever since steadily pursued.

But the Society being of late years greatly enlarged, and their communications more numerous, it was thought advisable that a Committee of their members should be appointed to reconsider the papers read before them, and select out of them such as they should judge most proper for publication in the future Transactions ; which was accordingly done upon the 26th of March, 1752. And the grounds of their choice are, and will continue to be, the importance and singularity of the subjects, or the advantageous manner of treating them ; without pretending to answer for the certainty of the facts, or propriety of the reasonings contained in the several papers so published, which must still rest on the credit or judgment of their respective authors.

It is likewise necessary on this occasion to remark, that it is an established rule oi the Society, to which they will always adhere, never to give their opinion, as a Body,

upon any subject, either of Nature or Art, that comes before them. And therefore the thanks, which are frequently proposed from the Chair, to be given to the authors of such papers as are read at their accustomed meetings, or to the persons through whose hands they received them, are to be considered in no other light than as a matter of civility, in return for the respect shown to the Society by those communications. The like also is to be said with regard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public newspapers, that they have met with the highest applause and approbation. And therefore it is hoped that no regard will hereafter be paid to such reports and public notices ; which in some instances have been too lightly credited, to the dishonour of the Society.

PHILOSOPHICAL TRANSACTIONS.

I. On the Normal Series Satisfying Linear Differential Equation*.

By E. CUNNINGHAM, 13. A., Fellow of St. John's College, Cambridge.

Communicated by Dr. H. F. BAKER, F.If.S.

Received December 14, 1904, — Read December 15, 1904.

CONTENTS.

Section Page

1. Introductory ........ . .................. 1

2. The equations to be considered .and the canonical form of a linear system of equations . . 3

3. The solution in view ........................ 4

4. The unique determination of the determining matrix when the roots of the characteristic

equation are unequal ....................... 5

5. The completion of the solution in the same case ...............

6. The general case; restriction on systems considered .............. 10

7. The matrix x ; preliminary assumptions as to its form ............. 11

8. The difference equations for the coefficients ; equations of condition ......... 11

9. On certain operators Ar and their application ......... ....... 14

10. The particular case when the roots of a certain equation are unequal ........ 16

11. The non-diagonal elements of x in this case ................. 19

12. The complete solution for p — 1 in this case ................. 21

13. The solution forp = 1 in the general case ................

14. Resumption of most general form .................... 25

15. Application of the method to a particular equation .............. 29

16. On the method to be adopted when certain equations of condition are not satisfied; sub-

normal forms ......................... 30

17. Complete solution of a certain equation of the third order ............ 34

1. THE present paper is suggested by that of Dr. H. F. BAKER in the 'Proceedings of the London Mathematical Society/ vol. xxxv., p. 333, "On the Integration of Linear Differential Equations." In that paper a linear ordinary differential equation of order n is considered as derived from a system of n linear simultaneous differential equations

or, in abbreviated notation,

dx/dt — ux,

VOL. CCV.— A 387. B 21.6.05

2 MR. R CUNNINGHAM ON THE NORMAL SERIES

where u is a square matrix of n rows and columns whose elements are functions of t, and x denotes a column of n independent variables.

A symbolic solution of this system is there given and denoted by the symbol fl(u). This is a matrix of n rows and columns formed from u as follows :— Q(<£) is the matrix of which each element is the (-integral from t0 to t of the corresponding element of <£, (j) being any matrix of n rows and columns ; then

wQw...ad inf.,

where the operator Q affects the whole of the part following it in any term. Each column of this matrix n(n) gives a set of solutions of the equations

dx/dt = ux,

and since fl(n) = 1 for t = t0, these n sets are linearly independent, so that fl(u) may be considered as a complete solution of the system.

Part II. of the same paper discusses the form of the matrix Sl(u) in the neighbour- hood of a point at which the elements of the matrix u have poles of the first order, or in the neighbourhood of which the integrals of the original equation are all " regular."

It is there shown that if t = 0 be such a point, a matrix

0* c,.2(t/t0)e-

0.

can be found, in which all elements to the left of the diagonal are zero, in which c^ = 0 unless 0t— Q} is zero or a positive integer, such that fl(u) is of the form

where G is a matrix whose elements are converging power series in t, and G0 is the value of G at t = ta.

The form of <f> is such as to put in evidence what are known as HAMBURGER'S sub- groups of integrals associated with the fundamental equation of the singularity ; the method is, in fact, a means of analysing the matrix fl (?t) into a product of matrices, of which one is expressible in finite terms and shows the nature of the point as a singularity of the solution.

The object of the following investigation is to see how far, under what conditions, and in what form, such an analysis can be effected for equations having poles of a higher order than unity in the elements of the matrix u.

It is known that if in the neighbourhood of infinity the equation is of the form

X

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 3

pr being a polynomial of degree pr, and Pr(l/.x) a series of positive integral powers of l/x, the equation has a set of formal solutions of the form

r =

where ilr is a polynomial of degree p+\, provided a certain determinantal equation has its roots all different.

The case in which these roots are not all different is discussed by FABRY (' These, Facultd des Sciences, Paris,' 1885), where he introduces the so-called Subnormal Integrals, viz., integrals of the above form in a variable xr'k, k being a positive integer.

The investigation carried out in the following bears the same relation to the discussion of these normal and subnormal integrals that Part II. of the paper quoted at the outset bears to the ordinary analysis of the integrals of an equation in the neighbourhood of a point near which all the integrals are regular.

2. Throughout the discussion the neighbourhood of the point t = 0 will be under consideration, the coefficient pr being supposed to have a pole of order tsr at this point.

Let p+l be the least integer not less than the greatest of the quantities rav/r. The equation may then be considered as belonging to the more general type

_ A

where Pr(0 is holomorphic near t = 0.

This equation may be reduced to a linear system of simultaneous equations as follows (vide ' Proc. Lond. Math. Soc.,' vol. xxxv., p. 344) :—

Put x, = z, o-2 = ^+1z(1>, ... srr+1 = tr(f+1Wr\ r=l,..,,n-l.

The n equations then satisfy the system of n equations

0 0 0

(n-5

dx	' 0 l 0
dt	tpv
	o p+l l o
	£±1 1
i	o,
	Qi
	tp+l'

where Qi...Qn are series of positive integral powers of t. This system belongs to the more general form

B 2

ME. E. CUNNINGHAM ON THE NORMAL SERIES

where ap+1 ...ft... are square matrices of constants.

The most general equation of this form will be considered.

If p. be any matrix of constants and y = px, the n quantities y satisfy the system

a£ or

Let /i be now chosen so that (p^fT1) is of canonical form as follows :-— (i.) It has zero everywhere save in the diagonal and the n-1 places immediately to the right of it ; (ii.) The diagonal consists of the roots of the equation ap+l-p\= 0, equal roots occupying consecutive places; (iii.) The elements to the right of the diagonal consist of (e,-l) unities, then a zero, (e3-l) unities, a zero, and so on (' Proc. Lond. Math. Soc.,' vol. xxxv., p. 352).

Form now the matrices (/lo^'1) ...(/tftfT1) ; the equation is then replaced by an equation of exactly similar form, the matrices a,,... being still any matrices whatever, but Oj,+1 being of the canonical form.

3. The equation being denoted by

dy/dt = uy,

if 77 be any solution of the equation

(A) drjjdt = U7)-r)x,

X being an arbitrary matrix, we have

so that i/n (x) is a matrix satisfying the equation in question.

In what follows we are concerned with the form of a solution more than the actual convergence and existence of the same. It is therefore important to notice that if 77 be a diverging power series formally satisfying equation (A), 770 (\) may be still considered as a formal solution of the original equation, the only condition necessary to secure its actual existence being the convergence of 77.

If 77 be convergent, the solution may be particularized by adding the factor ij0~lt i.e., 170 (x)^)"1 is the solution reducing to unity at t = ta.

The main investigation to be carried out is that of a simple form for the matrix x, such that the subsidiary equation (A) may have a formal solution in the form of a

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 5

matrix whose elements are series of positive integral powers of t, reducing for t = 0 to the matrix unity.

4. Owing to the much greater simplicity of the case in which the equation \ap+l—p\ = 0 has all its roots different, it will be treated first separately. The result obtained is as follows :—

A matrix x can be determined uniquely of the form

Xi+i i XP i i Xi

tn+ 1 ' j« i • * * i , )

where XT>+I ---Xi are Matrices of constants in whicli all elements save those in (he diagonal are zero, such that there is a formal solution

where the matrix rj is made up of series of positive integral powers of f — generally diverging — and reducing for / = 0 to the matrix unity. Consider the equation

(B) $- «•-

where

r = I.,..., p+\.

The roots of otp+l—p\ = 0 being unequal, the matrix «y)+1 will have zero elements except in the diagonal; the diagonal elements will be /31; p.,, .../>,„ the roots of the equation.

If the equation (B) is satisfied by the matrix

i) = (x, y, z...}, where x, y, z ... denote columns of elements of the form

X = X

y = y«+yit+...,

the coefficients xr, yr... being columns of constants xra, xr\ xr2, &c., these constants satisfy the following equations :—

X. (ap+l — 61p^l)x0 = 0,

a+1-0nz, + -<V)*0 = 0,

MR. E. CUNNINGHAM ON THE NORMAL SERIES

+i*+- +(«1-011)*« = 0,

A precisely similar set of equations gives the relations connecting the constants

The equations X just written determine uniquely a set of values for 0lp+l...0ll and the coefficients .r0 ....

The first of these equations gives

m

Since r,, is to be equal to : \ve must have 6\,+1 = /31( and these equations are

\0/ then satisfied. , ,

Similarly the first of the ?/ equations with ?/„ = gives 02p+} = p2 ; and so for

W the other columns.

The second of the equations X written more fully gives

<'-0; = 0, (pr-pi)*,r + Vr = 0, * = 2. •••> w.

These then determine xl save for its first element, in place of which a unique value is given for 0pl.

The third equation X in full gives

x1" + a1V1 = 0, r = (2, ..., n).

Of these, the first gives 0lf-i, while the following determine sc2 save for its first element, but only in terms of the yet undetermined .r,1. Of the next group, the first equation is

This equation apparently involves the unknown .r,1 explicitly, and also, in x22...x2", implicitly.

But the whole coefficient of xrl is

„ U

pS-p,/ 2

'i n

2 2 '

EO that ^-2 is given independently of o^1.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 7

The remaining equations of this group give x3 except for x3\ in terms of x^ and x,1.

Proceeding in this way as far as the (p+l)th equation of X, x^...xp are all found except for their first elements, while the first elements of these equations give 6lpJrl...0i, 0lp+r being not a priori independent of x^...xrl hitherto unknown.

It has been shown above, however, that the determination of 6lp^1 does not require a knowledge of x±.

In general, in fact, ()lp-r is given independently of x^.-.x,1.

To prove this, the way in which x^ enters into .x-*r+1 will be first considered. This may be stated .as follows : —

The coefficient of x± in x\+l is equal to that part of xrk which is independent of

For r = 1 this is at once seen by writing down the equations

In general the equation for .rAr+1 is

Assuming the statement above to be true for 1, 2...r, and that 9V, ..., 0p-r+i are independent of x1l...xlr-l, the above equation shows that the coefficient of x^ in — (pt— pi)xkr+l is the part independent of a?!1,..., ,rrl in

r— I

, V /-,!* .''j- - r,

li i \a p-r-n+l^j T...-rt* p_r+< + iXs j-,

i.e., u

so that under the above assumptions the statement holds for 1, 2..., r+1. Also, under the same assumptions, from the equation giving 0l;,-r, viz.,

we deduce that the coefficient of x/ in ^p-,. is the part independent of x^ in

This expression differs only from the left-hand member of the equation for 6lp-r+i by multiples of x^^.x1^^ and therefore, on the assumption that this equation gives ^P-! independently of a?/..., the part independent of these quantities in the above expression must, when 6lp-r+i is determined, vanish, so that 6lp-r is independent of x,1.

8 MR. E. CUNNINGHAM ON THE NORMAL SERIES

Now the way in which the successive equations follow one another shows that the coefficient of aV in 6* is equal to that of a:,1 in 0ft+r_,.

Thus 0',,-r+i being independent of a^1, 0lp-r is independent of x2\ and in general, 0lp-,+k(k = l...r-l) being all independent of a^1, Qlp-r does riot contain x^..^,^.

Thus, if the assumptions made on p. 7 are satisfied for any particular value of r less than p, they are satisfied for a value of ?• one greater than that value.

For ?• = 1 the statements have been justified, and it follows therefore that fl1^...^1 are all determined uniquely without the knowledge of a;,1, x2\... from the first (p+l) of the equations X, and by the same equations x^-.x^ are found, except for their first elements, the expressions obtained containing those first elements.

5. Consider now the (p + 2)Ul equation X in regard to its first element.

As before, this will be independent of xpl...x3l; but on account of the extra term arising from dy/dt, which now enters for the first time, the coefficient of aV is not zero. 'It is, in fact, — 1.

Thus, the quantities 6ll...6lp+l being now known, this equation gives a^1.

Similarly, the next group's first member will contain the term — ZxJ but will not contain x-}.,,xlp+lt and will therefore give x2l after o^1 is found.

Thus all the elements x^ are determined successively, and returning to the expressions for a?/ (?'>l) in terms of these and substituting the values so found, all these are given also.

The equations for the columns y, z, &c., being treated in the same way, give the corresponding O's uniquely, and also the coefficients in the series of which these columns are composed.

Thus it is shown that when the "characteristic equation" «p+i— p\=0 has its roots all different, the equation

dy/dt = ny, where

ap+i being in its canonical form, possesses a unique formal solution in the form

where the elements of Xp-- Xi llot "i the diagonal are zero, and the elements of r? are power series in t, reducing for t = 0 to the matrix unity.

The matrix n^£« + ... + £j can at once be written in the form a>/a>0, where o> is a matrix whose non-diagonal elements are zero, and whose kih diagonal element is

e p-tr and w0 is the value of CD at t — tlt.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 9

If the series t) happen to be convergent, the solution which reduces to unity at t = t0 can at once be written in the form i)<ato0~1r}0~1.

Applied to the system formed from a particular linear ordinary equation we have at once the result referred to on p. 3 (v. SCHLESINGER, ' Lin. Diff.-Gleichungen,' vol. I, pp. 341 ff.*).

As a simple example of the application of the method we may take the well-known equation Putting Wi = w and W-T, = s2 dw/dt

dz*

. w.

which in matrix notation is

The equation p

a. -p

to canonical form is

du/dt =

= VM, say. The subsidiary equations are

t = - - + + y wl + ± w,,

\Z Z j Z

w.

= 0 gives p = ± \/a, and the matrix //, needed to transform the first matrix

, so that the equation is transformed to

i

+ JL I__£_N

which give as the general relations connecting the coefficients of the first column of ?;, putting i> = 1 + -J A,

(i.)-

- Tia^,1 + ^2 + j-Zj- (a;1,,.! - x2,,,!) = 0

- 2

-> - m

^-^ (^,.-1 - »2«-i) = 0 I

- 2

Hence

and therefore

-

which with the first equation gives

2 V« . n (x

Thus a recurrence formula is established for the quantities xnl-xn2 in terms of which a;1,^! and a;2,,+i can be at once expressed.

* With reference to SCHLESINGER'S demonstration of this result, see a note by the author in the 'Messenger of Mathematics," January, 1905.

VOL. CCV. — A. C

10 MB. E. CUNNINGHAM ON THE NORMAL SERIES

The series for x will terminate if for any value of n

y + (q-n)(p + n-l) = 0, ? f if

for some value of n.

The series for y will similarly terminate if

(-A + 2/»-l)2-(4y + l) vanish for some value of n.

Both these are certainly satisfied if

X = q and 4y + 1 = p2, where p, q are any integers of which one is odd and the other even.

6. We pass now to the case where the characteristic equation \ap+1-p\ = 0 has its roots not all unequal, and the analysis becomes a good deal more intricate with the less simple canonical form of the matrix ap+1 as stated on p. 4. It will be remembered that the numbers e1; e2... there used are the powers of (p^-p) in the elementary divisors of | ap+1-p \ with respect to the root pl of this equation of multiplicity I In the case of the system obtained on p. 4 from a single equation of order n, we may prove that ^ = I, e2 = e3 = . . . = 0.

For the matrix (ap+1— p) is of the form

/ 0	i 0	0 . 1 .	. . 0
\:	b	• •	. 0 1 . . k

The minor of the quantity " k" in the determinant of this matrix is simply (— p}"~\ Thus the elementary divisors are certainly merely unity with respect to any non-zero roots.

If there be a multiple zero root, however, since the minor of "a" is unity, the elementary divisors with respect to this root are all simply pu.

Thus for such a system we have for each multiple root e3 = e3 . . . = 0 ; so that in the canonical form of ap+1, if

Pi = pi+i> ap+i' ' = 1» and if

Such systems being by far the most important in practice and also considerably simpler to work out, the full discussion will be restricted to systems of this type. It may be pointed out that the most general system can be solved by means of the solution of systems of this more restricted type, for from the general system

SATISFYING LINEAE DIFFERENTIAL EQUATIONS. 11

ft *T' tt

-T- = jj+i %, where u is a power series in t, a linear equation of order n and rank p

near t = 0 can be obtained for each row of the matrix x, and this equation can be solved by the solution of a linear system of the restricted type in question.

Of the matrix ^ to be used here, the following properties will be presupposed :—

(i.) It is to be of the form %£±± + %£ + ...+& } where each of the matrices

V V V

XI---XP+I has all elements to the left of the diagonal zero. (ii.) The diagonal elements of these matrices are to be numerical constants

denoted as before by 0r* (r = 1, ... , p+ 1 ; s = 1, ... , n). (iii.) All the other non-zero elements of x^Xs-'-Xp-t-i are t° be constants, while

the other elements of ^i may contain t, but only to positive integral

powers (cf. the matrix x in Dr. BAKEE'S paper, loc. cit.).

8. As before, the matrix 77, which is a formal solution of the subsidiary equation

drjjdt = UTf)—r)x, will be supposed to be formed of the columns

y = 2/0

and the equations for the coefficients xryr... are the same as the equations X (p. 5).

But the detailed form of these equations is quite different. The first of them ( «,,+! — 0lp+i)x0 = 0 is still satisfied by

3. — | 0 I #' — n

° ~ y i ' ^ p+i ~ PI-

Supposing now p, to be a root of multiplicity el} the second equation X is in full

= °= = 0,

where

pf,+i = ••• = pt^,,^

C 2

12 MR. E. CUNNINGHAM ON THE NORMAL SERIES

These equations manifestly determine x^ except for its first and second elements, the second being known as soon as 0pl is.

We are also at once faced with a condition necessary for the possibility of the solution under the assumptions made as to the form of x, viz. :—

«/' = 0.

This condition arises from the e,"1 equation of the set, and as, in the ensuing discussion, the e^'1 equation of each set is most important we shall here introduce a notation for it, viz., Xr will stand for the e/11 equation of the (7-+l)th set; i.e., of the

set

(af+l-0lp+l)xr+... = 0.

This equation will not contain any element of xr.

A similar notation will be adopted for the equations Y, Z... for the coefficients in the other columns of TJ.

If the second element of the first row of XP+\ t»e C2\, the equations Y are

(^-H-tf'W l)2/0-^0 = 0,

(a.j,+i-8a1,+i)yi+(a.f-0p)y0-clslx1 = 0,

Of these the first is satisfied by

provided we take c21 = 1 = corresponding element in ap+l.

Considering each of the columns in succession we have thus, with i)0 = 1. XP+I = ap+\-

The second of the equations Y gives

-X11 = 0,

-V-xf = o,

atp8"'- fl^* = 0,

(j°^i-/'i)2/1ei+1 + <'+1-*iei+1 = 0,

which, when xl is known, determine yl save for its first element, and its third until OP is known.

The exceptional equation Y,, af'-xS = 0, gives us again a necessary condition for the possibility of the solution in view, a/^' + a'-1 = 0,

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 13

Similarly the equation Zl gives

0 = a/' "-y,'' = «/•" + <• "-'-a^'-1 = a/^' + a/'''-'*^'.'.-3,

and so on for the equations for each of the first et columns of r).

For the (ej + l)"1 column, however, the non-diagonal term of x*+i = 0 and the equations for this column do not contain the elements of the preceding column.

In fact, the e2 columns beginning with the (cj + l)"1 form a group related to one another in just the same manner as the first ex are. We obtain from them, as from the latter, the conditions

€, + !,<! + t, A

a.p \J,

and so for the columns associated with each group of equal roots. These and other conditions which arise in the course of the investigation will be called " equations of condition." Supposing those already found to be satisfied, we may return to the solution of the equations X, and of these the following statements are to be proved : —

I. The first e[ — 1 equations of the (r+l)th set determine xr2...xrf< in terms of ^...aj1,-! and 0lp+l...0lp-r+1 ; the equations from the (ej+l)"1 onwards give .r/1+I....rr" in terms of the same quantities.

II. When the values thus found are substituted in the equation Xr+1, the resulting left-hand member is independent of the undetermined quantities .r,1...^1, Bpl...0\-r for all values of r up to (e2— 2), but for r = Ci — 1 it is independent of all save Bpl ; in fact, the equation Xei is an algebraic equation of degree et for 0pl and contains no other undetermined quantity.

III. Supposing one root of this to be chosen for the value of Bp, and the equations Y to be treated in the same way, Yei_j will be an equation of degree (el — 1) in Bp2, whose roots are exactly the remaining roots of Xv

IV. Similarly ZE|_2 furnishes an equation of degree EI— 2 for Bp\ whose roots are the remaining roots of Y,^, and therefore of X£i, and so on.

Thus Bpl...Bp' are the roots of the equation Xei.

V. The values of #,,_!... subsequently obtained in association with each of these roots will be the same in whatever order they are taken.

Of these I. does not require proof.

With regard to II., the proof that the equations do not contain the undetermined a;'s follows exactly the same lines as the corresponding proof when the characteristic equation has its roots all diiferent (vide pp. 7-8).

The proof that they do not contain 6P . . . until r = ej — 1 requires considerations of a different kind involving the equations X, Y, Z... simultaneously.

14 MR. E. CUNNINGHAM ON THE NOEMAL SERIES

9. Consider the system of equations X1 derived from X by changing 6J into 6f> and x into x1, viz. :—

X1. K+i-^VOxo1 = 0,

and let these be treated in exactly the same way as the equations X, the undetermined elements of x1 being supposed the same as those of x.

From the two sets of equations X, X1 let a new set be formed by subtracting corresponding members of X and X1 and dividing each remainder by 0P-0P2, and let

this new system be denoted by

X-X1 _ Q

The expressions for x1 obtained from X1 in terms of 01p+l... will be identical with those obtained from X in terms of the same with 0pl changed into 0P".

x—xl Let AP.T denote the expression ' ' •

Vp Up

Then A;)X is the system of equations

0 = 0, («„+, - 0Vi) A/d ~X0 = 0,

a1 /9* ^A/y-l-//v— $ \ \ T:-. — T,-, = 0

\ w+l " p + 1 / *-*/>'^'2 ' \ p up I *-*p'*Jl *JU\ v?

Further Ap(;r1) = [ M = y0> and 0lp+1 =

Thus these equations are identical in form with the equations Y, except that 02p-k is replaced by 6lp-k, &>0. Thus if from Y the y's be calculated as in I., p. 13, and from APX the quantities \x be similarly determined, the only difference between Aa?r+1 and yr will be in the substitution of x±... for t/i1... and 6lp-k for 02p-k, A;>0, and

Thus if we substitute the values of Apa; thus found in ApX^! the result will differ from Yr only by the same substitutions.

In a similar way, denoting by APY the difference equations

Y-Y1

0P*-0P3

APY will differ from Zr_j only by the substitution of y?... for z?, tfv-v for 0^..., and 0*+l for 0J3, and so for the remainder of the first el columns.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 15

In exactly the same way if X2 denote the system derived from X by the change of p_! into &p-i, and the equations

Y X-X2

p-1^ ~" ai on

be formed, the quantities A,,_1a.v+2 differ from yr only by the substitution of 6p for 9P\ 0lp-2 for 02p-2... and 6^ + 2 for 6*. The same operator Ap_j applied to the equations Y connects the columns y and z, and so on.

Similarly, operators Ap_3...Aa may be defined.

X —

Lastly, an operator Aj may be defined so that the equation AxX is -^-r • where

"i — PI +p

denotes the equations X with O-f—p substituted for 8*. Then the equation Yr will, when y^,... are replaced by aV,... and Bp2,..., 022 by 0P1,..., 02l, become the equation A?Xp+r.

Consider now the equation Zl ; it is independent of #/, Zi1,..., and therefore reduces to a simple numerical constant which must be zero (p. 13).

But Y is a polynomial in 6*. It can therefore be only of the first degree, since APY2 is independent of 0P3 ; is, in fact, the same as Z,, viz., zero, so that Y2 does not contain 0P2. It must then, like Z,, be only a constant, and must therefore vanish identically, so that Y2 = 0 gives a further " equation of condition."

Hence again X3 cannot contain 0pl, and the operator Ap_! connecting it with Y, shows that it cannot contain 0lp-i. Thus X3 again must be a vanishing constant, giving another "equation of condition."

Similarly, starting with the corresponding equation of the fourth column, we find APX4 = 0, so that X4 must be independent of 6P. Also A^X, will vanish identically, so that X4 is independent of 6lp-1, and similarly it is independent of ffip.a....

Thus if ej > 4, X4 reduces to a mere constant which, as before, must vanish.

The process may clearly be carried on as far as the e^'1 column, so that the equations Xi.-.X,^! all give equations of condition, as do also Y^.-Y^-js, Zj.-.Z^-a, etc. Starting now from the second equation of the e/'1 column,

where ^ denotes the e^ column of TJ, it follows that the third equation from (e^ — l)th column must be a quadratic in 9p-^~l, independent of &1, {= <f>2(0p'~l) say}, and such that

Thus if 0/1"1 is one root of </>2 = 0, 0/' is the other.

16 MR E. CUNNINGHAM ON THE NORMAL SERIES

Similarly, the fourth equation from the preceding column must give a cubic for ep'~\ (fo = 0), such that

Thus 0/'~2, 0/1"1, 0ptl are the roots of <£3 = 0.

Eventually the first column gives an equation of degree et for 0P (viz., Xei), of which the roots in any order are a possible set of values for 0pl...0pf'. Calling this equation (f> (6} = 0, and denoting its roots in some assigned order by <rlt cr2, ...cr€], let us consider the values determined for 0lp-i... by taking 0pl = ov.. and 0P'' = <rtl.

10. Again, as prior in order of simplicity, let the case in which the roots of <j>(0) are all different be taken first.

It has been shown above that the equation Yr, when x^... are substituted for y?... and 0P\ 6lp-2... for 0P2, 02p-2..., becomes identical with Ap_iXr+2.

Now, Y£I_! is merely a polynomial in 6P, independent of?/!1... and 02P-1..., and vanishing for 0P2 = cr2, o-3...crei.

Let Y.,., = *„_,(*/).

Thus Xei+1 is linear in 6lp^ the coefficient of the same being ^,^(0^); the part independent of 6lp^ contains only 0P, which is now a determinate quantity.

If the roots of <j)ei(0p) are all unequal, ^.^^^O and 6lp-^ is given uniquely ; and similarly Yt] = 0 is a linear equation for 02p_i in terms of 0P2 and 6P, Z€]_! a linear equation in 03p-i, and so on.

It is important now to consider whether the order in which the roots o-1...o"ei are taken is of significance in the solution ; that is, whether the value of ff'p-i associated with a particular root a-k is the same whichever column of the dependent variables this root is associated with, and whether a change in the order necessarily implies a distinct solution, because, if so, the solution would appear to be by no means unique.

The equation X,i+1 giving 0lp^ is, we have seen, of the form <^1(o-1)^1p_i + »/>(o"i) = 0,

where #(8) = Thus

Now fifa) is save for a constant factor

(0-1-0-2) (o-j-o-a)...^-^), so that ApX€i+1 is

= 0.

But the equation Y€I, which is independent of 0*,_,,... becomes, when 0pl(= a-,) is

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 17,

substituted for 0P', the equation Ap_1X,i+2, so that when 6* is substituted for 6a'n\ Yt| it becomes symmetrical in 6lp^ and 0ap-i. Y€I must therefore be of the form

A(<rlfir,)tflr.1+B(aI,<r1)«V.i+0(<rtoiri) - 0, where

A(o-i, o-j) = B(o-i, o-j) and

A(<TI, o-2) + B(o-,, o-2) = (o-j-o-g)... +(0-3-0-.,).... Hence

A{o-!, o-j) = (o-i — 0-3) (o-i — 0-4)... = B(cri, o-,).

Now before the values of #/, #/ are substituted in Y€I the coefficient of 62P^ must be a function of 0pa only.

Therefore B must be a function of o-2 independent of cr,.

Hence

B = (cr,-o-:i)...(o-,-o-ti) and

A = (cTi — 0-3)... (o-j — 0-t|).

The equation for $2p_i is therefore

By virtue of the equation giving #^-1, therefore, we have

(o-2-o-3). ..(o2-o%i)6V1 + ^-- = 0

tra — o"!

as the equation for ^_!.

But this is identically the equation that would have been obtained for 0lp-i if 0-,, o-2 had been interchanged. Thus the value of 6P-^ associated with the root cr., is unaltered by this interchange.

We have further to see that the same permutation does not alter the 6p^l for the subsequent columns.

In just the same way as above the equation for 63p-i is shown to be

( — W — } O2 + ' — } 6S + °'2~g'1 (r3~Crl _ Q

which gives

independently of the order of o-j and o-2.

The same holds for 6kp-i, k < c^ and a similar proof for the interchange of any other pair of the roots o-.

Thus supposing the roots of <f> (6P) = 0 to be all different, there is associated with each a unique determinate value of 0f-i.

VOL. CCV, — A, D

lg MR. E. CUNNINGHAM ON THE NORMAL SERIES

These quantities then being determined, consider now the equation X€,+3, Y,1+1... and first it must be pointed out that the relations established between the equations X Y... through the operators A (pp. 14-15), where the quantities 6P, 0^... were considered as independent, are still valid when 0P^, &c., are determined as functions

of 6

The operator A, in the first place becomes replaced in the equation X,i+2 after that

giving 0',-! bJ ^ + Vi--VF?F- But ViX.,« vanishes owing to the choice of

"p Up

0Vi. so that the value of 6\^ being substituted in Xe,+2 the operator Ap may be still applied to establish a relation with Y,I+I. We have further

VA.« = &-i(V).

while Ap_3X,,+2... vanish identically because of the vanishing of Y,,-2, Yei_,.... Thus the equation X,i+2 is of the form

in which 0pl and 01p-l are to have their determined values, so that the equation may be written

The operation Ap having been shown to be applicable to the equation in this form, reasoning exactly as above shows that the equation for 02P_2 reduces to

so that the values of (Jp-2 associated with the roots o-1; cr2 are independent of the order in which these roots are taken, and likewise the values of 6*p-.2... will be unaltered by a permutation of the same. The same may be shown in the same way of a permutation of any other two consecutive roots, viz., that such permutation gives rise to a corresponding permutation of the 0^_2.... Identical reasoning leads to an identical conclusion with regard to Qp-.a...02.

Eventually we come to equations giving #,. When dpl...02l have all been found, the equation 'Kfi+p.l is of the form ^-i^ij^+x^i) = 0, where, as before, the coefficient of 6* is not zero, so that 6* is determined like the rest ; while 0l2...0i1 are found respectively from Yei+p_2....

All the 6's being now determined, if we pass to the equation X,1+p and follow the same argument that was required to prove the preceding equations independent of Xj1..., the coefficient of x± is found to be the left-hand member of X.1+p_i with (^ + 1) substituted for 0^, i.e., it is simply $,^(0-1), which is not zero. Thus X.1+p determines the first of the undetermined elements x± ____

Similarly in X,1+p+1 the coefficient of x2l is 2<£Ci_1(o-i), so that by this xal is given, and so on for the succeeding equations in turn.

SATISFYING LINEAR DIFFERENTIA!, EQUATIONS. 19

In order to proceed to the determination of the second column it may be noticed at once that the coefficient of yrl in any equation is exactly equal to that of xrl in the corresponding equation from the first column with o-j and cr2 interchanged, which includes the interchange of 0lp-k and 62p-k. In the equation 'Y,i+p_l, therefore, the coefficient of y^ is identically zero, while the unknown 6-Js are now all determined. The closer consideration of this equation is deferred for a moment.

The coefficient of y± in Y,i+p = coefficient of x* in X,i+p with o^ and <r2 interchanged

= (<ra— <7i)(era— o-8)...(crg— o-.,) i= 0.

Similarly the coefficient of y% in Yn+p+1, and in general of ykl in Y +p+k_-i, is not zero, while Yei+p+i_! does not contain any element ym* for which m>k.

For the third column the equation Zei+p_3 determines 0^, and the two equations following, Z,i+p_2, Z.i+p_!, are still independent of z^, z21..., while the equation Z.j+p+A-! contains z^...z^, the coefficient of zk being &(cr3— o-i)(o-3— cr2)(cr3— o-.,)

In the same way, if the elements of the etih column be denoted by £ and the associated equations by ft, the equation £lp determines 0^', ft;j+i...ft;)+6i_1 do not contain fi1..., and ft,+,, contains &1 only.

11. So far the matrices XP---XI nave been taken to be simply diagonals. It will now be shown that the insertion of constants to the right of the diagonal in the first ev columns of XP can be carried out in such a manner as to affect none of the conclusions hitherto made, while they may be chosen so that the equations Y^^, Z,i+p_2, Z.^p-j, &c., are all satisfied.

Denoting by a,-,- a constant in the iih column and _/th row, i>y, i^f\, ,/<fi, the Y equations become

(0) (a?)+1-^41)7/0-.r,,=0.

(1) (ap+1~0p^)yl + (ap-0p)ya-xl-a!>lx0=0,

(r+l) (ap+l-0p+l)^r+1 + (ap-0p)>/r...-.rr+l-a.-\rr = 0.

These equations are to be treated just as they were before the constants a were introduced — the same elements remain undetermined as before, but at each stage the quantities found presumably contain a21.

We see at once from equation (1) that the coefficient of «21 in y^ (the first element being excepted) is simply y0. In fact, it can be shown step by step that the coefficient of a21 in yr+l (the first element always excepted) is exactly that part of yr which is independent of azl with 0^ increased by 1 ; and therefore the coefficient of a21 in Yr+1, when the values ofyr2... as far as they are known are substituted, is equal to that part of the left-hand member of Yr which is independent of a21, #/ being increased by unity.

Now Y! is independent of a21 and of 0^ and vanishes, thus Y2 is independent of a21,

D 2

20 MR. E. CUNNINGHAM ON THE NORMAL SERIES

and is therefore the same as if a21 were zero. It is also independent of ft,2, so that Y3 again is independent of a21.

Thus until k is so large that Y* does not vanish independently of 0?, Yi+1 is independent of a21, and therefore the same as was obtained in the foregoing, where a21

was neglected.

Thus the insertion of a21 in XP does not affect any of the equations Y^-.Y.^-* and therefore the values of 0P2...6* are independent of a21.

But in the equation Yei+7,_! the coefficient of <x21 is the left-hand member of Y,1+P_2

with 0*+l for 6i*

= (o-2-o-3)(cr2— or4)... ^ 0.

Thus a21 can certainly be chosen so that the equation Yei+p_! is satisfied.

Having determined a21, it is at once seen from p. 19 that the following equations now determine yS, yj... without ambiguity ; for since 0*...$? are independent of a21, the coefficients of y,\ &c., are those found there whether a21 be zero or not.

In the same way for the third column, with a32, a31, taken into account, the equations become

= 0,

and just as before the first equation in which «:<2 occurs with a non- vanishing coefficient is the one following the equation from which -0* first does not vanish out identically, viz., Zti+;)_2; while «31 will occur first in the equation homologous to the Y equation in which «21 first occurred, viz., Z,i+p_i ; in fact, in Ze|+p_2, a32 will occur multiplied by the left-hand member of Zei+p_3 with 0^+1 put for 0f, and in Zei+p_2, «31 will be multiplied by the left-hand member of Yei+p_2 with 0^+1 put for 0* : both these factors are other than zero.

Thus a:!2 can be chosen to satisfy Zej4p_3, and subsequently a31 to satisfy Z,i+p_2! while the preceding equations are quite independent of them both ; just as for y, then, Zf1... are given in succession without ambiguity.

Treating the remainder of the first Ci columns in just the same way, all the elements of these columns are found in succession, and the solution is complete as far as these columns are concerned.

The t-2 columns associated with the next group of equal roots may be treated in the same way, the singular equations being in this case the (e! + e2)th of each set; constants aij will again be chosen in the matrix x to the right of the diagonal, Ci + e3>: i> £i + l, el + e2>j>fl+l, to satisfy certain equations as above, and so for each root in succession.

Thus if the various equations for Op associated with the different groups of equal roots of the characteristic equation have their roots all different, and the " equations

SATISFYING LINEAll DIFFERENTIAL EQUATIONS. 21

of condition " (p. 15) for each root are satisfied, a formal solution of the linear system has been found in the form

nti(^+Xz+ +Xi\ ^ \tp+l V tr

where the elements of rj are series of positive integral ascending powers of t, and XI---XP-I have all elements zero save those in the diagonals, which are made up of determinate constants ; and XP consists merely of square matrices about its diagonal of CD e2... rows and columns respectively, each of which has zero everywhere to the left of the diagonal and determinate constants everywhere else. The elements of 77 are in general divergent.

The matrix II above will be known as the " determining matrizant." As occasion will be found later to discuss a more general matrizant, nothing further will be said of it here except for the case in which p = I , which will be worked out fully in order to make clear the march of ideas in the more general case.

12. For p = 1 the equation Xti is an equation of degree el for 0^, Yfi_! is of degree el — I for 0^, and so on.

-.6(8*)- 0 theiY = , — <P \Vi ) - u, t i ,,-i —

_ , i

so that the remaining roots are those of Y^ diminished by unity.

Similarly the roots of Y€I_J are those of Zei_2 diminished by unity, and so on, so that the roots of iffi) = 0 are <V, 0,2-l, ..., ^"-e.-l.

The equation

,i+1 s .r1

X,i+a is z^011 + 2) + .r1V(011) + X1(0i1) = 0,

and so on. The equations for y^... are of the same form, with 0^ for 0,1. We suppose therefore in the first place that #/, 9-?... do not differ by positive integers or by zero, so that the coefficients of the first terms in these equations are all other than zero, and all the x's and y's are determined uniquely. The quantities a being then determined, as above, the solution is altogether determinate.

If p = l and 0ll...0l'1 do not differ among themselves by integers, then the solution is of the form

21

in which a.i] = 0, unless #2* =

+ + sav

+3+ y<

22 MR. E. CUNNINGHAM ON THE NORMAL SERIES

Now fl (w+cr) = Sl(w)a{Sl-1(w)<rQ(w)} (Dr. BAKER, loc. cit., p. 339), and

~l, where o = / e ' t6'1 0

and wu is the value of w at t = C, so that

n(|-4+£+*

\^ « t' t

I ,.\-»<+el* \ / /t\ -«,'+«,' //X-Si'+Si1

f /°'(f) ' ° '"A /°'a21(f 'a (f) '

/ \tn/ \ / \Co/ \co/

- W O J M //\-9|2 + «,3 4-1 //\-«,2 + «,s

•?«.o ,i ....r«U - .*"© •

I

there being no exponentials in the last matrix since

«'>' = 0, unless 6>/ = <?/.

(/I /3 \ /p\

-| + — )n( -, ), r is a matrix having zero in and to the left of the t* 1 1 \taj

diagonal, so that Q ( — ) has zeros in the same places. \t /

r /r\

-gQljj) therefore has zeros in and to the left of the diagonal, and also in the (n— 1)

£ \ L '

places to the right of the diagonal, and also wherever F has a zero, and so on.

r r

Thus Q-^Q-^... vanishes after a finite number of steps. Further, none of these

l> v

expressions contain log t, since F contains no integral powers of t. Thus

721, 731-

all the places which were occupied by zeros in F being also occupied by zeros in this, and yy contains only a finite number of powers of t, positive or negative, and no logarithms.

We may specify a little more exactly the form of the term y{j. A typical term of T/t2 is

and Bj—0i is not an integer and c,-, is unity or zero.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 23

The corresponding term in Q (T/t2) is

C, -l + A, -l

L Vw L \to '

r r

It follows that in -g Q -g the r"1 column is a sum of terms belonging to the indices

If V

Q\—6i, where ,s- = 1, 2...T, and so for each operation Q. Thus finally we have the result—

/r\

The term in the i*1'1 row and/1' column of H(-a) is a sum of terms belonging to the

(ft 0 ^ -j + — ) the /h column is a sum t t /

of terms belonging to the indices 0ll...0lj.

13. Supposing still that p = 1, let the indices 0ll...0le* cease to differ by other than integers, and likewise the other groups. Let them be arranged in groups differing by integers, so that their real parts are in descending order of magnitude in each group.

Then no root of ^(^i1) = 0 will exceed Q* by an integer, and therefore ^(0^ + k)^ 0 — k a positive integer, so that the equations Xf]+i, ... do not fail to determine x^ —

If, however, 6* = 0il—k, $ (0^+k) — 0 ; so that the coefficient of y^ in Yti+A vanishes, leaving y^ unknown. We take ?/// = 0 as the simplest assumption, and the following equations then give ylk+\, &c., all without ambiguity. We are, however, left with Ytl and Yei+Jl in general unsatisfied. Of these one can in general be satisfied without affecting the rest of the argument by an adjustment of the element x/1-

It has been seen that a constant a21 in the matrix xi occurs first with non- vanishing coefficient in Y£I.

Clearly, then, if we introduce a21tk, it will leave all the equations to Ye]+Jl_i unaffected, and add to Y,i+A the left-hand member of Y6i_j with 0^ + k+l for 6*.

But Y ! is an equation of degree €l— 1 of which d* is the greatest root, so that the multiple of a21 added to Yei+A is not zero. Thus a proper choice of a21 satisfies Y,1+4.

Again suppose 0* = 0*—^ = 0ll — kl — k2, ^>0, £2>0.

Then the equations Z.i+t, Zti+,i+,a fail to give z\, z\+ki; but a31, «21 can again be determined so that, if a31«*1+*% a3V' occupy the places above 0^ in XL the equations Z.I+AI, Z€|+AiH.tj are satisfied, 0^ being unaffected and zlki, z\i+tl being taken zero.

Suppose then 0l\..01k form the first group of 6^... 6^ differing by integers. Then treating the first h columns all in the same way, the ^h(h — l) equations Y€I, Z.,, Z,,-!... must be satisfied identically when the 0^ have been all determined, and must be added to the equations of condition already found.

Suppose now 0lh*l...0lh+k form the next group of roots differing by integers and consider the (h + r)lh column r<k. The equation giving 6>,A+I is that indicated by the suffix ^-(h+r-l), and those following this up to that with the suffix el are independent of the undetermined elements of r).

24 MR. E. CUNNINGHAM ON THE NORMAL SERIES

Further r— 1 of the equations subsequent to these fail to determine the appropriate element as above, on account of the quantities 01h+s—0]h+r being positive integers for s = 1, 2...(r— 1). These (1 — 1) equations are satisfied by putting terms <«,*+*-«l*+raA+r,A+*(£ _ j , — i) in ^i( wnile of the other h + r—l equations constants a*+r'*(s = 1...A) can be found to satisfy h. Thus (r—l) equations of condition are found from this column, and therefore ^k(k— 1) from this group of roots ; and so for each group of roots.

Assuming all the equations of condition to be satisfied, we have now the following formal solution

where ft is as follows : — •

The square matrices about the diagonal of h, k... rows and columns respectively, corresponding to the groups of 6*... which differ by integers, are of the form

and all other elements, to the right of the diagonal and within the matrices of e^. rows and columns about the diagonal corresponding to the groups of equal roots for # are numerical constants, and all others to the left and right of the diagonal are zero. Applying now the formula

the solution is put in the form

'j ( tjt^j ' ' . .	\		/°	Cai	cM..-\l
	\	1			\ |
'» o,	(t/tu)~e''* *'",...	_J_ _	0	0	e»...
	1	i	i
	/ /		\		I
	/		\ ,		• • / J

where in the last matrix all elements are zero that were zero in Xi, and ci} is a constant if ^-^ is a positive integer, but otherwise is a numerical multiple of t*-*.

The expansion of the matrizant can be effected as on p. -22, with the result that in the. expanded matrix the first h elements of the first row contain Iog(t/t0) to the powers 0, 1...A-1 respectively, while the rest of the row is free from logarithms ; the second row begins with zero, then unity, and the next (h- 2) elements contain log («/*„) to powers l...(h-2) respectively, and so on, the Ath row being entirely free from logarithms. In the (A+l)th row in the k places beginning with the diagonal term occurs to powers 0, !...(&-!), and so on.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 25

14. Returning now to the general case left out of consideration on p. 16, in which the roots of <f>(0p) = 0 are not all unequal, we suppose the roots of this equation arranged in groups, of which the members are all equal.

If o-j = or2 = ... = ov, <£ei_i (<TI) = ^-^o-a) = 0, since the roots of <j>ei-i(0) = 0 are o-2, ...<rr ; and again, <f>fi_2(a2), <jin_8(<78)...^(n_r+1)(<rr_1) are all zero, where $,,-2 = 0....<^>ei_r+1 = 0 are the equations for dp, 6P...6P.

The equation for 6lp--i (p. 16) reduces therefore simply to t|>(cri) = 0 and fails to determine 6lp-i ; but, o-j being already known, this must be merely an " equation of condition " among the coefficients.

Similarly in the equation X,I+2 the coefficient of 6lp-2 vanishes, and this equation therefore is of the form

-i, 01 = 0 or _1, = 0.

Now the operator Ap_! acting on this equation, since cri = cr2, gives the equation Yf|> which, as has been seen (p. 17), is linear in 02p-l, the coefficient being ^>ei_2(cr2).

If r = 2, this does not vanish, and therefore Xe|+2 is a quadratic for 0\,-i, of which, owing to the relation through &p-i, 6lp-i, 02p-\ are the two roots.

If, however, r>2, the equation YEi must become an equation of condition, since <£ti_2(o-2) = 0, and therefore also Xei+2 becomes independent of 61p_l and gives another equation of condition.

Carrying on this reasoning step by step, we find that X^+i...X,1+r_j are all independent of 0lp-i, 6lp-2..., while Xfi+r is of degree r in O1^ and independent of Olp-2.... If any root of this equation be taken for ffip^ the equations Yt]...Yti+,._a arc independent of 02p-i, 62P-^..., while Yn+r_2 is an equation of degree r— 1, which, since it is derived from X,i+r by the operator A;)_1; has for its roots the remaining roots ofXe,+r.

Choosing one of these for 62p-i, Ze]+r_4 gives an equation of degree ? — 2 whose roots are the remaining, and so on.

Similarly, if trr+1 = ... = ov+s, 6p-lr+l...0p-lr+:i are given as the roots of an equation of degree s, and so for each group of equal roots cr.

Consider now one such group with the values of ^p_,r+I, ...0p_1r+* obtained.

Let the equations of which these are the roots be

Then, if the roots of $s be all unequal, say = TJ...T,,

^-i(ri)M=0, ^-2(T2)^=0, .... but

^_1(r,) = 0,/^l.

The subsequent equations are then seen by the application of A3 to be VOL, CCV. — A, E

2C, ME E. CUNNINGHAM ON THE NOKMAL SERIES

which, since the coefficients of 8p-2 do not vanish, at once give the values of 0P_/+1... ; these, as in the case of 0P^ when the o-'s were unequal, can be shown, if the Toots T^... undergo a permutation, to undergo the same permutation, so that the same 0P,2 is associated with any particular T in whatever place this T is taken.

If, however, the roots T fall into groups of which the members are equal to one another, these equations again resolve themselves into equations of condition owing to the vanishing of ^^(TJ), &c. ; and, as before, the quantities 0p-2 fall into corre- sponding groups given as the roots of equations of degrees equal to the numbers in the respective groups.

The process may clearly be carried on as far as the determination of 02 by the use of the operators A,,_2...A2.

A further remark should be made as to the finding of 0lt in connection with the operator A1( which has been defined to be such that

Suppose that 02...62 are given by a set of equations

a*^1) = 0, ^(di) = 0, ..., Wl(0/) = 0,

where the affixes of the w's denote the degrees of the equations, and the roots of each equation are the remaining roots of the preceding after any one of them has been chosen.

Suppose that of these 02 ...#/ are equal, so that

Then 6,^(0;) = 0, o,*_a(0a8) = 0, ..., w,_A+](0/-i) = 0, but <o,_A(0/) =/= 0. Then if the X equations following u>k be denoted successively by

i«* = 0, 2w,t.= 0, ...,

i(ak is independent of x^, &c., by virtue of &>* = 0 and the preceding equations, and therefore

Aid*,) = «*_»(#)« 0, since 0J = 0P>, 0^ = 0^, ...,

so that ,o)A is also independent of 6? and must therefore vanish identically when 0al is determined. Hence also 2o>k is independent of*!1, &c., and therefore

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 27

But in the same way

so that !<«*_! (022) must also vanish when 0./ is found and thus iwA_1(^21) == 0, so that 2a>k is also independent of 0^. Ultimately we have

and o>k_h(02h) =£ 0 and is independent of 0f.

Thus hG>k(Oi] is an equation of degree h for (V, since 0% = 02h, 03l = 03h....

Suppose its roots are vlf v2, ... vh-

Take 6± to be v^ Then A-IW*-! is an equation of degree h — l for 0*, and its roots are (p+va)(p+va)..., as shown by effecting the operation Aj.

Choosing one of these again to be 0*, say p + v2, A-aw*-a is an equation for 0,3, whose roots are (2p + vs), (2p + v4).... Thus the quantities 0ll...0l'1 are r1( (_p + ra), (2^+r,)..., (h-l)p+vh.

In order to particularise the order of the roots v1} v2, ..., they are arranged as soon as found as follows : —

Let all those roots which differ from one another by integers be grouped con- secutively and let the arrangement in each group be such that 6r~l — 9r = 0 or a positive integer. Suppose, now, the equation k(ak(0il) = 0 is the equation X?.

Then the coefficient of x± in X?+1 is Aw/l(#11+l) which, since no root of Aw*(0) = 0 exceeds 0,1 by a positive integer, does not vanish. X7+1 is moreover independent of xal, ..., owing to the equations k-i<ak = 0, ..., being satisfied independently of 0,\

Thus X?+1 determines x^.

Similarly, X?+2 gives xal, the coefficient being AwA(011 + 2), and so forth.

Of the Y equations, that determining 0^ is obtained from X7 by the operator At. It is, in fact, Y?_p.

The coefficient of y^ in Y^.^+j is equal to the coefficient of o^1 in X?_/)+1 with 0^, 0./, ..., substituted for 0^, 02l, ____ But X?_p+1 vanishes identically as far as 0^ is concerned and 02, ..., are the same as 0^, .... Thus Yg_p+1 is independent of?/!1. Similarly, Yg-p+2...Yq-l are all independent of the undetermined elements of y.

Suppose now 0\—0i — X (a positive integer).

The equation Y?+ft contains y^, ..., yk, the coefficient of the last of these being AeoA(0!2+&) which vanishes for k — X.

If, now, in the matrix *%r(r =P> P~ 1--.2) the second element of the first row be cr21, and in ^ be c21^, the constants c will, as before (p = 1), affect first the equations Yj-p+j..^.! and Y?+A respectively, entering into these with non- vanishing coefficients. Let cr21 be determined then to satisfy the first p—l of these equations; Y?+1...Y?+x_j

E 2

28 ME. E. CUNNINGHAM ON THE NORMAL SERIES*

then give ^...yV-i. Y,+A then fails to give y,\ but c21 can be chosen to satisfy the equation and y? may be taken zero.

The following equations then give the remaining elements in succession.

This leaves the equation Y, in general unsatisfied, and a further equation of condition is therefore necessary.

Similarly, if 0,"-0,8 = fi (an integer) of the equations Z, we can, by proper choice Of C32(r _ £,...2), satisfy the p-l following that which determines 6^, viz., Z,-^ ; and just as a proper choice of the constants or21 enabled us to satisfy Yg-p+1...Yq-i, the constants cr31 can be chosen to satisfy Zf_f+i...Z2-i.

Thus two equations, Z,,p, Zg, are left unsatisfied in general. The two remaining constants, c,32 and c?\ are utilised to satisfy the equations Z,+(1, Z?+A+^, in which the coefficients of z* and z\+li vanish respectively. To do this the terms c31^" and c**"*' are inserted in the third column of Xi-

If then the indices O^.-.B,' be equal, or differ from one another by integers, exactly similar treatment applies for each of the first I columns of 77, the iih column furnishing (i— 1) equations of condition.

For the (Z+l)tlh column, however, 01r-01'+1, (r^h) is not equal to zero or a positive integer. Thus hwk (&i+l + m) does not vanish for any value of m, and the Ip constants crl+l''(r =p...l, s = l...l) can be determined to satisfy the Ip equations between Uq-,,, and Uj+1, U denoting an equation of the l+l"1 column, and, in particular, ~Ug-ip being the equation determining 0/+1 and U?+1 determining u^.

For the next column, however, 0ll+l — 01l+i may be a positive integer, X1 say, so that h<ak (#1't+2 + V) = 0, and the (</ + X1)"' equation, instead of determining the appropriate element of •>?, can only be satisfied at the expense of the 5th, by making the element above 9^ in ^ — c'+2'm^'. The </th equation then becomes a further equation of condition. Thus we shall obtain r— 1 equations of condition from the (/ + r)th column, associated with an index belonging to the second group of indices ^ differing among themselves by positive integers ; and so on through all the indices as far as #/'.

A similar treatment is now applied to the columns (h+l)*,.(h+k), for which 0/'+1 = erh+2. . . = 6rh*k (r = p. . .2) ; 6Vl+1 is given as the root of an equation of degree k ; and the minors of the determinants %„ whose diagonals are 0rh+l...6rh+k, have the elements to the right of the diagonal suitably adjusted as above, while one equation of condition is furnished in connection with every difference 01*+r-r#1A+s, which is an integer.

Supposing these equations all satisfied, consider the expansion of the matrix

--- + }> which is effected in just the same way as for p = 1 (p. 22).

P + l Q

If to = 2 p where Qr is a matrix made up simply of the diagonal terms 0r1...0r", the application of the equation

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 29

o (tv+<r) = n (w) n [rr1 (w) o-n (w)]

resolves the required matrix into a product of two, of which the first, 11 (w), consists simply of diagonal terms of which the rth is

The second matrix has zero everywhere in and to the left of the diagonal ; and, since in the matrices ^ the element in the rth row and sth column was zero unless ®m = Qm' '' (m = p+l, •••%), it contains no exponential expressions. It can therefore be completely integrated in finite terms, just as was done in the case p = 1 (p. 23).

15. A simple example may be appended of the application of the method to a particular system. Consider the equation

Putting iji - y, y, = y?y\ we have the linear system

y'=t o,

= 17 ° l }L+(Q °'U+/ ° °\~L

The characteristic equation is

-0 or

-P, -S-p

giving equal roots - 1 for p.

With p = ( j the transformed system becomes

OM./O .

= uy say. Considering now the subsidiary equations

Then the equations to be satisfied are

II. (1) xj - (1 + #1) *il = 0 '2) (2 - 61) x£ -1=0,

I. and II. (2) give (6»!- 1)2 = 0.

We take 9\ — 1 therefore, so that a^1 = 1 and x-2 = 2xil.

Again

III. (1) xo? - &i zj2- 0, (2) (1 - 6>,) xj - a,1 = 0,

of which (2) gives Xil = 0, so that x£ = 0.

30 ME. E. CUNNINGHAM ON THE NORMAL SERIES

Similarly xl* = xt* = Q, and so on.

The equations for the second column are

I. (1) y21-«i1-X = 0) (2) 3-02-Zi2 = 0, (2) gives 02 = 2 = 1 + 0i and (1) gives J/21 = A.

II. (1) yj - (\ + Oa) yil - z,2 - A- xS = 0, (2) (2 - 0,) yj - xf - W = 0, (2) gives A. = 0 and (1) gives y? = 2yJ and also yj = 0.

III. (1) y? - %!2 - zi3 - W = 0, (2) (1 - 6S) </22 - 2/11 = 0,

so that yi1 = 0 and ?/22 = 0.

It is easily seen that all the remaining terms vanish. Thus the solution reducing at x = x0 to the matrix unity is

1/1 o\\/i o\-i

2J/U V

where

Thus

fl 0

f nr /o i\

T-lo *-tt o ^ bo oj

16. The number of conditions found in the course of the analysis shows that the solution in this form — which may be called the " normal" form, by analogy with the name "normal integrals" of linear equations — is by no means always possible. As many writers have pointed out, there is a much more general type of solution than the normal series for the ordinary linear equation, in the form of a normal series in a new independent variable x1'*, k being a positive integer (CAYLEY, HAMBUBGER, FABKY, &c.).

The method developed in the foregoing is peculiarly adaptable to the investigation of these integrals, inasmuch as the transformation to a new independent variable is very simply efiected.

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 31

If in the linear system

-| = u(t)y we put t = (j>(z),

we have without any calculation the new system for y as a function of 2

Suppose now (f>(z) = zk. Then the transformed system is

If, then, the original system is of rank p, so that

the new system is

dy Vka^_ Jca ,_

«+-«+ -

and is of rank kp.

If, now, we were to put z = F'1*'' . z1, the form of the equation would be unchanged, while the coefficient of zi;!cp+l in the right-hand member would become the original canonical matrix a.p+l. This is, however, not necessary, as the whole investigation could be carried out equally well if any constants whatever replaced the unities to the right of the diagonal in ap+1.

It may now well happen that though all the equations of condition found for the general system are not satisfied, those associated with the new system are all satisfied, so that the new system possesses a solution in normal form. If this is so, the original system will be said to admit of a solution in subnormal form. In fact, an integer k can always be found such that this is so, owing to the vanishing of the coefficients of z-*f+r{r= 0) !...(£_ 2)}.

In the first place, all the conditions arrived at from the equations X,, Y,, ... will be satisfied (p. 13), for the coefficient of z~kp is identically zero; in general, the left-hand members of X,, Yr . . . are rational integral functions of the elements of the matrices A^, A^-j, ..., Atp_r+1, if A,n stands for the matrix multiplying z~'", and contain no term independent of these elements.

Now the conditions found on p. 13 arise from the equations Xj.-.X,,-!, Y^-.Y^-a, ..., and therefore involve the matrices A^, ..., AA?_ei+2. These conditions will therefore all be satisfied if k > ej. Similarly, the analogous conditions for the second group of equal terms in the diagonal of ap+1 will be satisfied if k ^ e2, and so on.

Consider first, as being simplest of explanation and as containing the essential features, the case in which all the roots of the characteristic equation are equal, so

32 ME. E. CUNNINGHAM ON THE NOEMAL SERIES

that E! = n. It will be shown that a subnormal form satisfying the equation certainly exists if a/" i= 0 with k = n.

We know from the foregoing investigation that 6\p is given as the root of an equation of degree n, and that, if the roots of this equation are all different, no more conditions than those just mentioned as satisfied are necessary to ensure the existence of the subnormal solution. But in this case the equation for 0\p is particularly easy to construct. We have, in fact,

nxf-01,, = 0, x? = xf = ... = a," = 0,

nxf-ffi^' = 0, ira« = ... = x» = 0,

nxS-ffinStf-ffiv-M* = 0, xf = ... = xs» = 0,

rn Q\ ™ n-l_f)2 ™ »-l _ A

X B_i — U ,,pJin-2 " np&n-X ... - U,

— u ,,,ixnn-l — .. . + A n(p_])+i = 0,

from which at once we have

-+I - u'

Tlius, unless AlnB(p_1)+1 = 0, the values of 6ln), are all different, and a solution in subnormal form is therefore possible, as stated above, with the independent variable changed to xi;n.

If, however, A1",,^-]^] = 0, we have

(n _ _ tin _ f\

\J np • ... -- I/ np - U,

and it will be found that the same conditions are necessary between the constants A.n(p— 1)+1 as were found previously (p. 13) between the constants otp, An(p_,)+1 being the same as n . a.p, e.g., from the equations

-1.^,,*, = 0, A.2'\(p-l)+l-nxn» = 0, we have

1Mp_1)+1 = 0, i.e, ap2'n + ap1'"-1 = 0, and so on.

Consider now what happens when these conditions are not all satisfied. Suppose, for instance, a/^ + a/'"-1 ^= 0. Let the original system be transformed by the change of variables

Then from the equations

= zk, k = n—l.

,(?,-1)+12,2 = 0, we obtain the equation for 6lkp

SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 33

(&\p)n -^fo (A'."-^ _1+1 + A2'". _ ) = 0 K" Ic

and since the last written coefficient is not zero, the roots of this equation are all different, and therefore the transformed system possesses normal integrals free from logarithms.

Again, suppose a.p<n = 0 and that a.pl-*~1 + a.p*1 also vanishes; then the roots of this equation all become zero, and we find that the condition a.p'n~* + a.p'n~l + a*'n = 0 is also necessary. If this is not satisfied and the original system be transformed by x = 2*, k = n — 2, the equation for 6lKp becomes

ai \« //}i \2

V kp\ I u kp \ ( \l,n-2 , A2'"-1 J. A :!. " x

— i— I ~~ —r~ I \-a- Dtt-sm-r-tt- »(/t-2>+i + -rt- :

p(k-2) + \ ~ V>

which has two zero roots and the rest all different. If the one condition necessitated by the equality of the two zero roots is satisfied, the solution is again found. If this condition is not satisfied, the transformation z = £ effects what is required.

Suppose now all the conditions of p. 1 3 are satisfied. Then whatever value of k be taken, we have 6\p = 0, so that, as fur as we have seen, the transformation does not render the solution any nearer.

We must, in fact, proceed to consider the further conditions for the case when the roots Op are equal (p. 13).

Suppose, for instance, the first of these conditions is not satisfied, then putting k = n, we shall have 9lkp = ...— d"Kp, the conditions then necessary before the deter- mination of Oty-i will be satisfied, and we shall eventually obtain a binomial equation for 8kp-i of degree n in which the constant term does not vanish ; the roots of this equation being all different, the subnormal integral exists.

Thus we may go through all the equations of condition in turn.

In the more general case, where the roots of the equation for 6P fall into more than one group of equal roots, the procedure is exactly similar.

For example, suppose that a.p'*\ a/1"1"1''1'1''4, ... are all different from zero. Then the preceding work suggests that a solution may be ftnind in which the first el rows- proceed according to powers of £' % the next e2 according to powers of £i;'a, and so on.

The whole would thus be of normal form with the variable tlik, k being the least common multiple of e^ e2 —

In fact, if we change the independent variable to tllk, k having this value, the matrix Aip-,1+i is identically zero, and the indices G^p.-.O'^ are the roots of — (^)'' + 0 = 0.

These roots are all equal, and the corresponding equations of condition are all satisfied owing to the vanishing of the matrices other than AA.r+I.

If, now, we form the equation for #%,_! we have

T.v 2 ffL _ 0 Z™ 3_f)l 2 _ A

KX2 — V kp-\ — U, /CX4 — (7 Ap-l.t-2 - - U,

~" /cp-lX"2<l-2 + A ''kp-2^ + 1 = 0,

VOL. CCV.^ — A. P

34 ME, E. CUNNINGHAM ON THE NORMAL SERIES

If - = 2, A.l\.p-2ti+l = k.ctp> and therefore if aple' ={= 0, the ej roots of this equation are all different. If, however, - > 2, A*p_2ei+1 again is identically zero, and the

necessary equations of condition are again satisfied.

Proceeding thus we find, in fact, that if apfl ^ 0, 6*kp) ..., 6'kp-k,^i', s= l.-.ei all vanish and that &\-p-kei+i is the root of a binomial equation of degree e1( whose roots are all different, and so for the other divisors of k. If, however, a.p'1 = 0 we have the same equations of condition again necessary, viz., a/'1 and a/''1"1 = 0, &c.

Assuming, then, that aplt> ^ 0, we find, without difficulty, that all the quantities 9* vanish for s = I...e1; save those for which r is of the form —k(p—m/e1) + l, so that the exponential arising in the first ej rows involves only ?*'> and not tllk.

The discussion of whether the solution of the subsidiary equation proceeds according to powers of tl!'1 only in the first EJ columns will not be carried out in full here. It is enough to know that, provided a,,'1'1, a//1+1'<r'")"<- do not vanish, a subnormal form certainly exists satisfying the equation.

If, however, one or more of these quantities does vanish, and one of the consequent equations of condition is not satisfied, we may, as on pp. 31-33, find a new integer /-, such that the necessary conditions for the existence of the subnormal form are satisfied.

17. As a concluding example consider the system derived from the equation of third order and rank one

Clsot3 + Cl3iZ2 •

~ — ~ .'/ T which with

gives

if-'

I •£" \ / Z

.\-oss -a-2-2 O/ \~0ss -BO, 4/ \-fflgi - «20 O/ \-a30 0 0;

The characteristic equation is - p3 - pa?, + a33 = 0.

We shall confine ourselves to the case in which this equation has three equal roots. These must all then be zero, and a™ = 0, a33 = 0.

For the equation then to possess a normal solution we must have 001 = 0, a32 = 0. Supposing these conditions not satisfied, put z = t3 ; then the equation becomes

0 1 ON / 0 0 Ov

! 1 0 0 I

\ - a3> - 0-21 4/ The subsidiary equations then become

= 0, 3x.2s - OJxi* = 0 ;

i - (9,1 = 0, 3«33 - OJx-p - O^XT? = 0, - ^31// - O'Sxi3 - 3a33 = 0.

			.'/ T- -	-4	«•		—y = v,
				yi =	y, »-«y, ys=.Y'
/	0	1	°\, /	0	0 0\ / 0	0	°\ /	0	0	0
	0	0	«i*	0	2 oU+l 0 1 •>	0	oU«/ 0 +«	0	0	0

SATISFYING LINEAE DIFFERENTIAL EQUATIONS. 35

The last gives - ,', (0s1)3- 3«32 = 0, so that 6*3!= - 3(rc32)S, where any cube root may be taken, the other roots giving #32, 033. Further

3z42 - 03W - Wx£ - (1 + <V) .-Ci1 = 0,

3/43 - flsW - Wtf + (1 - 011) Zj2 = 0,

- tfsW - ^2S + (3 - 6V) »;,8 - SasaKi1 - SoziX;8 = 0 ;

of which the last gives

-fcV.fW-aaieV-o,

so that ftj1 = «2i/(«8o)J, and #.r, 0o3 are given by taking the other roots for («32)*.

Lastly the equation

- #3W - foxa3 + (2 - 6V) z23 - 3a82a;3i - 3a.nxJ = 0 gives

,3

which gives

so that 6'11 = 5. Similarly, 0,!i = 018 = 5, and a subnormal form exists satisfying the equation, of which the

•>'»/ _ _f^]_

first column has the determining factor c2-' ";E'' tS, and the 'other columns have the same factor with the other cube roots of «so.

We may remark that this agrees with the results obtained for this equation by the ordinary methods (FoRSYTH, "Linear Differential Equations," § 99) under the assumption that «32=^0. We have shown that this is a necessary condition for the existence of the subnormal form in the variable /-— ,~S satisfying the equation formally, unless we have also <i21 = 0.

If, however, «32 = 0 and n^i^O, then, as we have seen above, the transformation / = il will give us a system admitting of 3 normal solutions; the equation for 0\ is, in fact, (#i)3- 4«2]0i = 0, giving 0} =0 or ± 2rt21*.

We see, in fact, that, when (130 = 0 and ff-Ji^O, the characteristic index of the original equation is 2, so that there will be one regular form satisfying the equation, i.e., an expression of the form ./-PP (./ X

If a3v, a2i arc both zero, the equation is of Fuchsian type. Thus the normal or subnormal forms are found in all cases.

F 2

C 37 ]

II. Memoir on the Theory of the Partitions of Numbers. — Part III.

By P. A. MACMAHON, Major R.A., Sc.D., F.R.S. Received November 21, — Read December 8, 1904.

SINCE Part II. of the Memoir appeared in November, 1898, the following papers by the author, bearing upon the Partition of Numbers, have been published : —

" Partitions of Numbers whose Graphs possess Symmetry," ' Cambridge Phil.

Trans.,' vol. XVIL, Part II. ; " Application of the Partition Analysis to the Study of the Properties of any

System of Consecutive Integers," 'Cambridge Phil. Trans.,' vol. XVIII. ; "The Diophantine Inequality KX^/JHJ," 'Cambridge Phil. Trans.,' vol. XIX. : " Combinatorial Analysis. The Foundations of a New Theory," ' Phil. Trans.

Roy. Soc. London,' A, vol. 194, 1900.

In the present Part III. I consider problems of " Arithmetic of Position." In particular. I define a " general magic square " composed of integers and show that for a given order of square it is possible to construct a syzygetic theory. Such a theory is worked out in detail for the order 3 as an illustration. I further discuss the problem of the enumeration of the squares of given order associated with a given sum. I show that there is no difficulty in constructing a generating function for such squares even when the construction is specified in detail, and I obtain an analytical expression for the number when the sum, associated with rows, columns and diagonals, is unity or two.

§ 9.

Art. 124. A "general magic square" I take to consist of n2 integers arranged in a square in such wise that the rows, columns and diagonals contain partitions of the same number, zero and repetitions of the same integer being permissible among the integers.

An ordinary magic square I define to be a general magic square in which the n2 integers are restricted to be the first n2 integers of the natural succession.

We may regard general magic squares as numerical magnitudes. To add two such magnitudes we add together the numbers in corresponding positions to form a

VOL. CCV. — A 388. 30.6.05

38 MA JOE P. A. MAcMAHON: MEMOIE ON THE

magnitude which is obviously also a general magic square. We can, therefore, form a linear function of magnitudes of the same order, n, the coefficients being positive integers, and such linear functions will denote a general magic square.

The magnitudes, of the same given order, can be taken as the elements of a linear algebra, and since arithmetical addition can be made to depend upon algebraical multiplication, the properties of the magnitudes can be investigated by means of a non-linear algebra.

Art. 125. The properties of a general magic square can be exhibited by means of homogeneous linear Diophantine equations, and it thence immediately follows that there must be a syzygetic theory of such formations. There exists a finite number of ground forms, corresponding to the ground solutions of the equations, and the method of investigation determines these and the syzygies which connect them.

Generally speaking, there is a syzygetic theory associated with every system of linear homogeneous Diophantine equalities or inequalities, and it is because invariant theories depend upon such systems that they are connected with syzygetic theories.

Art. 126. The method of investigation about to be given applies not only to magic squares of different kinds but to all arrangements of integers, which are defined by homogeneous linear Diophantine equalities or inequalities, whose properties persist after addition of corresponding numbers.

For example, the partitions of all numbers into n, or fewer parts, are defined by the linear homogeneous Diophantine inequalities

«1>:a2>a:,...5:a,,, and if another solution be

A=±&>&. ..==&,

we have

and since the property persists after addition, a syzygetic theory results.

This is one of the simplest cases that could be adduced and is at the same time the true basis of the Theory of Partitions.

Many instances of configurations of integers in piano or in solido will occur to the mind as having been subjects of contemplation by mathematicians and others from the earliest times. These when defined by properties which persist after addition of corresponding parts fall under the present theory.

Art. 127. There is no general magic square of the order 2 except the trivial case a a

, but we may consider squares of order 2 in which the row and column

Cf Ct

properties, but not the diagonal properties, are in evidence. Let such a square be

«2

a*

THEORY OF THE PARTITIONS OF NUMBERS, which must clearly have the form

a, a,

! »2 «1

and we may associate with it the Diophantine equation

and regard a,, «2 and a5 as the unknowns.

The syzygetic theory is obtained by forming the sum

for

all solutions of the equation, and the result is

i

l_

a'

auxiliary quantity and the meaning of the prefixed symbol fi is that n of the algebraic fraction in ascending powers of X,, Xa, X5 we are to rms only which are free from a.

Where (6 IS an ctLi^vmc^i y vjucmuiuy CHAH LUC meet

after expansion of the algebraic fraction in as retain those terms only which are free from a. The expression clearly has the value

1

i-x^.i-x-X

denominator factors denote the ground solutions

The

«1	aa	«5
1	0	1
0	1 1

and the absence of numerator terms shows that there are no syzygies. Thus the fundamental squares are

1 0 0 1

0 1

1 0

and this is otherwise evident. The case is trivial and is introduced only for the orderly presentation of the subject.

Art. 128. Passing on to the general magic squares of order 3 we have the square

«

«

40 MAJOE P. A. MACMAHON: MEMOIE ON THE

defined by the eight Diophantine equatious

= a!0.

= aio-

We require all values of the quantities a which satisfy these equations. To form the sum

for all solutions, introduce the auxiliary quantities

ft, b, c, d, e, f, f/, h

in association with the successive Diophantine equations. The sum in question may

be written

1

_ f (1 -adgX,) (I -fU'X3) (1 -rt/'AXs) (1 -MX,) (1-beghXJ

where after expansion we retain that portion only which is free from the auxiliaries. Remarking that

1 1

we eliminate the auxiliar ft and obtain

1

l_p222)/l_^£*L)/i-

Put now bd = A, be = B, If- C, cd = D, and we obtain

1

f/ AX.X.pW X^W. X3X

"BOD&A era/1

(1 -Br^X,) (1 -CX,) (1 -DAX7) (l -^2) f 1 -CDfM

\ A / \ A /

an artifice which reduces the number of auxiliars to be eliminated by unity.

THEORY OF THE PARTITIONS OF NUMBERS.

Remarking that

(l-P1P8)(l-P1P4)(l-P3Pa)(l-P2P4) 1

(i-p1p4)(i-psp8)(i-psp4)

We eliminate A and find

ft

^1-4 X.X.X.o ) (1 -BDX4XS) ( 1 -CD?X4X,) ( 1 - ^ 7 X2XU

\jLs(f/t

-Br///Xs)(l-

1

i \ / • /

1 "VVY lit ,/VVV \ / l TJTiV V \ / i l V V

1 — TTT AiAsAjo I""TOT AjAoAn, (1— .DlJA.,A8) l ~ 7TST~r -^-a-^-io C/i B« nrk^j,

- - X3X1U (1 -B^X6) (1 -CXH) (1 -DAX7)

Eliminating B from the first fraction and C from the second, we have

n l

( (} ODoX^^i n;,Y_\/i Y Y \ /i ^' YYY V
1 i~\ PY \ l (1-UA6)	l~\ ^<2Y YYY \ M YYYY 1 i *• ~9 AiAjAoAjoJ 1 1 - — A3A4 AgAjo 1 gDxxxxx /;

-CD<7X4X9) (1 -

~~rm~7 XjXj

(1-CXe) (1-/X1XBX9X10) ( l - -

tS

1-

k

VOL. CCV. — A.

/ 1 T^Ti Y \ i 1 "V V AT Y i

1 1 ~~ J-'/i-A.j ) II — — A.J AgJi.gA.jQ I

G

41

42 MAJOR P. A. MxcMAHON: MEMOIR ON THE

From the first fraction eliminate C, from the second C, and from the third B,

obtaining

1

I

(l-D/iX7)(l-

l -

l - - X3X4X8X10) ( 1 - 1 X2X4X«X10 9 / \ "

+ fl

TA1

— '—j— X^jA^XgXnj

-<7%X6X9X1U) 1 - X3X4

— X2A(jX]u

1 Y ^

/I "

(1 — (r/XiX3X4X5X8X9Ajo )

-D/».X7) 1 - -2 X1X4X8X9X1

X2X6X,

i

1 - 1 X3X4X8X10) ( 1 - 1

L(l-^X3X6X10j(l-

Eliminating D from each of the three fractions, we obtain

1

(1 -/

1 -

1^ 9*

I'

— -XSX4X8X1

1 \ / 1

1 — -A^-X^Ay-X^o 1(1 — — -A.3X4-X8X10 tj ' » i/

1 ^ / 1 *

1 — j X2X4X9Xi0 ) ( 1 — ^-2X]X2X4X6X8X9X10 )

fl I \ II /

T-Xj

(1 — X1X2X3X4X5XfiX7X8X9Xlu )

(i -f/^x.

(1-A2X3X8X7X10) fl-ixi

i - Ix2x6x7x10) (i - Ix3x4x8x

ts * \ y

111

THEORY OF THE PARTITIONS OF NUMBERS.

43

Art. 129. Before proceeding to eliminate g and h, observe that if we now put g = h = 1, we obtain the generating function for the solutions of the first six of the Diophantine equations corresponding to the squares which possess row and column but not diagonal properties.

Putting g = h = 1 , the generating function reduces to

— X1X5X9X10) ( 1 — X1X8X8X10) ( 1 —

^io) (1 — X3X4X8X10) (1 —

(1 — indicating ground forms

connected by the ground syzygy

. X3X4X8X10 =

X1X6X8X10,

X3X4X8X10,

corresponding to the fundamental squares

	100		010		0 0 1
	010		0 0 1		1 0 0
	001		1 0 0		0 1 0
	1 0 0		0 1 0		001
	001		1 0 0		010
	010		001		1 0 0
connected by the fundamental syzygy
	100		0 1 0		0 0 1
	010	+	0 0 1	~r	100
	0 0 1		100		010
	100		010		001
=	0 0 1	+	001	+	0 1 0
	0 1 0		100		100
each side being equal to
	1 1 1
	1 1 1	.
	111

Ill

•

This is the complete syzygetic theory of these particular squares of order 3.

G 2

44 MAJOR P. A. MACMAHON: MEMOIR ON THE

Art. 130. Resuming the discussion, we proceed to eliminate </ and h and remark that the second fraction may be omitted as contributing no term free from h. Eliminating g from the first and g from the third, we have

,,

i Y Y 2V 2v v 2V v 3\ /i Y Y 2Y Y 2Y 2Y Y 3\

L — AjAg A4 A5A8 AgAjQ J ^ 1 — AjA2 A5A<j A7 A9Ajy J

(l—h X3X6X7X10) ( 1 — j- X2X4X9.A

aV 2V 2V 2V 2V aV 2V 2V 2V 2V — Aj A2 A3 A4 A8 AB A7 A8 A9 At

r '' V V 2V 2V V 2V V 3\ /l Y Y 2Y Y 2Y 2

— Aj A3 A4 A5A8 A9A10 ) (i— A! A2 A5A6 A7

(1 —h X3X5X7X1(I) ( 1 — Y XiXtfXaXio j ( 1 — j-2 X1X2X4X))X8X9Xi0

Art. 131. If the diagonal property associated with (j is alone to be satisfied in addition to the row and column properties we may put h = I. Observe that the second of the three fractions cannot now be omitted. Simplifying we obtain

1 V 2V" 2V 2Y 2V 2V 2V 2V 2V 2V 6 — AI A2 A3 A4 A5 AK A 7 A8 A9 Am

( L— XiXgXgXio) ( I — X2X4X9Xn,) (l — X3X5X7X10)

II V V 2V V 2V 2V V .1\ /I V Y 2Y 2Y Y 2V Y . ^ i ^-1^-2 ^-5-^-6 A7 A9All( ; ( I — AtA3 A4 A5A8 A9A

Establishing the five ground products

X3X5X7XUI,

XY 2Y Y 2Y 2Y Y :! !A2 A5A8 A7 A9A,0 ,

XV 2V 2V V 2V V :t lA3 A4 ASA8 A9A10

connected by the ground syzygy

(X1X6X8X10):J (X2X4X9X10)2 (X3X5X7X10)2 = (X1X/X5X,2X7aX9X103) (X1X/X42X5X/X9X103) corresponding to the fundamental squares

100 010 001 001 100 010 010 001 100

1.20 102

^012 210

201 021

THEORY OF THE PARTITIONS OF NUMBERS. 45

connected by the fundamental syzygy

100 010 001

2001 +2100+2010

010 001 100

120 102

= 012 + 210, 201 021

involving the complete theory of the squares in which the property of one chosen diagonal is excluded.

Art. 132. Resuming and finally eliminating h, we obtain

7l Y Y 2Y Y 2Y 2Y Y 3\ /i Y Y 2Y 2Y Y 2Y Y s\/i Y 2Y Y 2Y Y Y 2Y *\ \L — -A-i-Aa A5Afl A7 A9A10 ) {L — A,A3 A4 A5A8 A9Aui } (L — A2 A;!A, A5A7A9 A10 )

X2V V V 2V V 2V 3/1 V 2V 2V 2V 2V 2V 2V 2V 2V 2V *i\ l AsAgA^ A7Ag A10 ^ J — At A2 A:i A, A,s AH A7 A8 A., AH, )

"~ l~\ Y Y 2Y Y 2Y 2Y~Y 3\ /i Y Y 2Y 2Y Y 2Y Y *\ V1 — -A.j-A.ji A5A6 A7 A9An, ) ^1 — AjA3 A4 ASA8 A9An, J

aY 2Y Y Y 2Y Y 2Y 3\ /I Y Y Y Y Y Y Y Y Y Y :(1 — Aj AgA5Afl A7A8 A10 ) (i — A1A2A3A4Ar,AHA7A8A9All, ;

which may be written

aY 2Y 2Y 2Y 2Y 2Y 2Y 2Y 2

(1 — XjX2 XgXtf Xj XjX10 ) (1 — X^g X4'X6X8 XflXin' ) (1— Xj X3X5XH X7X8 Xln') i ( 1 — X22X3X42X5X7X92Xi03) ( 1 — X1X2X3X4X5XBX7X8X9X10 ) indicating the ground products

XY 2Y Y 2Y 2Y Y 3 lA2 AsAg A7 A9A10 ,

Xv 2V 2Y Y 2Y Y 3 lA3 A4 A5A8 A9Ai0 ,

X2Y Y Y 2Y Y 2Y 3 1 A-sAgAg A7A8 A1() ,

X2Y Y 2Y Y Y 2Y 3 2 A3A4 A5A7A9 A1() ,

Xj X2X3X4X5X6XjXgX9X1i|'

connected by the fundamental syzygies

(X1X22X5X62X7"X9X10'i) (X1X3 X4"XSX8 = (X12X3X5X62X7X82X103

corresponding to the fundamental general magic squares

46 MAJOR P. A. MACMAHON: MEMOIR ON THE

0 2 1 2 1 0

1 0 2

1	2	0	1	0	2				2	0	1
0	1	2	2	1	0				0	1	2
2	0	1	0	2	1				1	2	0
						1	1	1
						1	1	1
						1	1	1

connected by the fundamental syzygies

120 102 111 201 021 012 + 210=2111= 012 + 210. 201 021 111 120 102

Art. 133. If the sum of each row, column, and diagonal be 3«, the number of general magic squares of order 3 that can be constructed is, from the generating function, the coefficient of ar*" in the expansion of

(l-.x-H)-(l-;r<)-'\ and this is found to be

n2+(n+l)2.

Art. 134. The ordinary magic squares, the component integers being 0, 1, 2, 3, 4, 5, G, 7, 8, are eight in number and are easily found to be

723 =048, 561

and seven others obtained from

120 201 120 021 120 021

3012 + 012, 012+3210, 3012 + 210,

201 120 201 102 201 102

102 201 102 201 102 021 210+3012, 3210 + 012, 210+3210, 021 120 021 120 021 102

021 021 3210 + 210. 102 102

Art. 135. There is no theoretical difficulty in proceeding to investigate the squares

120 0 1 2 2 0 1

2 0 1 + 3012 1 2 0

THEORY OF THE PARTITIONS OF NUMBERS. 47

of higher orders, but even in the case of order 4 there is practical difficulty in handling the H generating function. There are 20 fundamental squares, viz. : —

1	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0
0	0	1	0	0	0	0	1	0	0	1	0	0	0	0	1
0	0	0	1	0	1	0	0	1	0	0	0	0	0	1	0
0	1	0	0	0	0	1	0	0	0	0	1	1	0	0	0
0	0	1	0	0	0	1	0	0	0	0	]	0	0	0	1
1	0	0	0	0	1	0	0	1	0	0	0	0	1	0	0
0	1	0	0	0	0	0	1	0	0	1	0	I	0	0	0
0	0	0	1	1	0	0	0	0	1	0	0	0	0	1	0
1	1	0	0	1	0	1	0	0	1	0	1	0	0	1	1
1	0	1	0	0	0	1	1	1	1	0	0	0	1	0	1
0	1	0	1	1	1	0	0	0	0	1	1	1	0	1	0
0	0	1	1	0	1	0	1	1	0	1	0	1	1	0	0
0	2	0	0	0	0	2	0	1	0	1	0	1	1	0	0
1	0	1	0	0	1	0	I	0	0	0	2	0	1	1	0
0	0	1	1	1	I	0	0	0	1	1	0	0	0	0	2
1	0	0	1	1	0	0	1	1	1	0	0	1	0	1	0
1	0	0	1	1	0	0	1	0	0	1	1	0	1	0	I
1	1	0	0	0	0	1	1	0	1	1	0	2	0	0	0
0	1	0	]	1	0	1	0	2	0	0	0	0	1	1	0
0	0	2	0	0	2	0	0	0	1	0	1	0	0	1	1

§10.

Art. 136. The direct enumeration of general magic squares of given order and sum of row.

Let hw denote the sum of all the homogeneous products w together of the

magnitudes

a,, «.,,...«„_!, «„.

If hw be raised to the power n and developed, the coefficient of

is the number of squares that can be formed of order n, so that the sum of each row and column is w, but in which there is no diagonal property in evidence.""

* "Combinatorial Analysis— The Foundations of a New Theory," 'Phil. Trans.,' A, vol. 194, 1900, p. 369 et seq.

48 MAJOR P. A. MAcMAHON: MEMOIE ON THE

In fact, if

(x-al)(x-a3)...(x-OLa) = x"-plXn-l+...+(-)a

and

w! DH, =

an operator of order w obtained by raising the linear operator to the power w symbolically as in TAYLOK'S theorem, then the number in question is concisely expressed by the formula

D»'V,

a particular case of a general formula given by the author (loc. cit.). Art. 137. To introduce the diagonal properties, proceed as follows:— Let /;.,,.(*) denote what !>,, becomes when Xas, /Aa,,_.,+1 are written for «„ an_,+i

respectively, and form the product hj-^ hw(2\ . . h,^"\ 1 say that the coefficient of

in the development of this product is the number of general magic squares of order n corresponding to the sum w.

To see how this is take 'n = 4, w = 1, and form a product

x and observe that, in picking out the terms

one factor must be taken every time from each row, column and diagonal ot the matrix.

Similarly, if n = 2, we form the product

x { Xaa/ + X^a2a3 + ^az2 + (Xa2 + /ua8) (a! + a4) + a^ + ajOt4 + a42} x Xaa2+XAaa + lt2a2+Xa + xa) (a1 + a4) + a1:J + «!«< + a/}

In forming the term involving

xy%V«»V

regard the successive products as corresponding to the successive rows of the square, the suffix of the a as denoting the column, and X, fi as corresponding to the diagonals.

Thus picking out the factors

XX2, /ua3a4, y*«2a4, a2a3)

THEORY OF THE PARTITIONS OF NUMBERS. 49

we obtain the corresponding square

2000

0011 0 1 0 1 ' 0110

These examples are sufficient to establish the validity of the theorem. . Art. 138. If we wish to make any restriction in regard to the numbers that appear in the sih row, we have merely to strike out certain terms from the function

7, (»)

nw .

E.g., if no number is to exceed t, we have merely to strike out all terms involving exponents which exceed t.

If the rows are to be drawn from certain specified partitions of w, we have merely to strike out from the functions

h (1) h <2) h (n)

"'HI ) n'w ••««««»

all terms whose exponents do not involve these partitions.

We have thus unlimited scope for particularising and specially defining the squares to be enumerated.

Let us now consider the enumeration of the fundamental squares of order n, such that the sum of each row, column and diagonal is unity. Observe that if the diagonal properties are not essential the number is obviously n \

Art. 139. It is convenient to consider a more general problem and then to deduce what we require at the moment as a particular case. I propose to determine the number of squares of given order which have one unit in each row and in each column, and specified numbers of units in the two diagonals.

Consider an even order 2n, and form the product

X (a, + Xa2 + a3+ . . . x (a, + a2+ Xa3+ . . .

-2 + Xa2)l_i + a2n) x (/*«! + a2 + «3 + . . . + a2;i_2+ «.,„_! + Xa2,,). We require the complete coefficient of

when the multiplication has been performed. Writing Sa =

VOL. CCV. — A. H

50 MAJOR P. A. MACMAHON: MEMOIR ON THE

the product is, taking the «th and 2n+l-tth factors together,

-!) a2+(X-l) a2B_i

Observing that we only require terms which involve the quantities a with unit exponents, the product of the first two complementary factors is effectively

and the complete product

has, on development, the form

s2" + A,*2"-1 + A2.s-2"-2+ . . . + A2,,,

where A,,, is a linear function of products of the quantities a, each term of which contains m different factors a, each with the exponent unity. Since, moreover, x"' gives rise to the term

ml ^0.^2... a,,,,

it follows that the coefficient of S«ia2...aa,, in the product is obtained by putting each quantity a equal to unity and sm = m\.

Hence, if S" = ml symbolically, the symbolic expression of the coefficient is

or

or writing

-s2— 4.S + 2 = o-2, .s— 1 =

This is the complete solution of the problem for an even order 2n. For an uneven order 2«+ 1, it is now evident that the symbolical expression of the coefficient of

is

{ o-2 + 2 (X + /i) a, + X2 + ^Y (o-i + V)»

the complete solution in respect of the uneven order 2n+l.

THEORY OF THE PARTITIONS OF NUMBERS.

51

Art. 140. To find the number of ground "general magic squares" corresponding to the sum unity, we have merely to pick out the coefficient of X/u, ; we thus find

even order 2n number is 8 ( „ ) a-an~3<ria,

w

uneven order 2n+l number is 8 (2) tr2n~2cr13+o-2",

wherein it must be remembered that the a- products are to be expanded in powers of s and then sm put equal to m I

In the general results the coefficient of

xy*

gives the number of squares in which the row and column sums .are unity and the dexter and sinister diagonals' sums are I, m respectively. I give the following table of values of simple a- products :—

0-2

9	4	4
44	24	16

«r.

_ 2_ 2 CTj <T2

(TjOj

265	168	116	80
1854	1280	920	672

The numbers cr/ = (s— l)p denote the number of permutations of p letters in which each letter is displaced and constitute a well-known series.

The remaining numbers are' readily calculated from these by the formula

O-/G-/ = o-/+2o-/-1-2cr/+1o-/-1-<o-/-1.

Art. 141. Another solution of the same problem yielding a more detailed result is now given.

For the even order 2n I directly determine the coefficient of

in the product above set forth.

H 2

52 MAJOE P. A. MACMAHON: MEMOIR ON THE

We have to pick out I X's and m /i's and to find the associated factors, 2n-l-m in number, which are linear functions of the quantities a.

In any such selection of I X's and m /u's there will be i pairs of X's symmetrical about the sinister diagonals and./ pairs of>'s symmetrical about the dexter diagonals, and the associated factors will depend upon the numerical values of i and j.

Consider then in the first place the number of ways of selecting I X's in such wise that i pairs are symmetrical about the sinister diagonals.

This number is readily found to be

n w—

i l-2i

With these I X's we cannot associate any /u which is either in the same row or in the same column as one of the selected X's.

Each of the i symmetrical pairs of X's in this way accounts for 2 /A'S, and each of the l—2i remaining X's accounts for 2 p's.

Thus we must select m p's out of 2n-2i-2 (l—2i) p-'s, i.e., m /A'S out of 2n-2l + 2i fi's.

We may select these so as to involve / pairs symmetrical about the dexter

diagonals in

(n-l+i\/n-l+i-j\ ,j

\ j !\ m—2j I

This number is obtained by writing in the first formula n—l + i,j and m for

n, i and I respectively,

and observe that we may do this because the selection of a symmetrical pair of X's or of one of the remaining X's results in the rejection of a pair of /x's which is symmetrical about the dexter diagonals.

Consequently the 2n—2l + 2i possible places for the m /A'S are also symmetrically arranged about the dexter diagonal. Hence the formula is valid.

We have established at this point that we may pick out I X's involving i symmetrical pairs and m /LI'S involving j symmetrical pairs in

. \ij\l-2ij \ J J\ m-2j j *

We must now determine the nature of the 2n— I— m associated factors, linear functions, of the quantities a.

In the matrix of the product delete the rows and columns which contain selected X's and /A'S. We thus delete l + m rows and l + m columns.

Consider the 2n—l—m remaining rows. There remain in these rows at most

2n— I— m elements a, because l + m columns have been deleted, but some of these elements

THEORY OF THE PARTITIONS OF NUMBERS. 53

must be rejected if they involve X or p. as coefficients, because by hypothesis we are only concerned with I X's and m //.'s, and these have already been accounted for.

Observe now that the columns which contain a symmetrical selected pair of X's only contain /t's which are in the same rows as these X's, and therefore the deletion of these columns cannot delete p.'s appertaining to any rows except those occupied by the selected pair of X's. Observe further that the column which contains an unsymmetrical X, say in the pih row, contains a /A in the 2n—p+Ia> row, and that therefore the disappearance of a p. in the 2n— p+lth row follows from the deletion of a column containing an unsymmetrical X in the pth row.

Hence of the 2n— I— m rows in question

l + m— %i— 2j rows contain '2n—l—m—l, a elements, and thence

2n— 2l— 2m+2i+2j rows contain '2ii—l—m—2, a. elements.

Accordingly if s is the sum of all the a elements except those which appear as coefficients of the selected X's and /A'S the co- factor of

n\ in-i\9,-2i in-l

l-2i contains l + m—2i—2j factors of type

and 2n— 2l— 2m + 2i + 2j factors of type

(*-«„-«,,), or ofn—l—m + i+j squared factors of type

(s-oiv -«„.)*,

since these factors occur in equal pairs.

Hence the co-factor is

n(*-a,,)ll(.s-a,.-a,,.)2,

wherein the quantities au, l + m—2i—2j in number, which appear in the first product, and the quantities «„ «„., 2n—2l-2m + 2i+2j in number, which appear in the second product, are all different.

Also (s— <*„— aK,)2 is effectively equal to

s2— 2 («„ + a,,.) s + 2ava.u,

since squares of the a's may be rejected.

Hence, by the reasoning employed in the first solution we may put the quantities a equal to unity, regard sp as equal to p ! symbolically, and say that the coefficient of

in the product

54 MAJOE P. A. MACMAHON: MEMOIR ON THE

has the symbolical expression

or, putting

s—l = o-j, s*-4s + 2 = o-2,

we obtain

/n\ n—i - _

for the number of squares such that —

(1) Sum associated with rows and columns is unity ;

(2) There are I units involving i symmetrical pairs in the dexter diagonal ;

(3) There are m units involving j symmetrical pairs in the sinister diagonal.

Giving i and j all possible values we find that the complete coefficient of

in the product, which we have already ascertained to have the expression

may be also expressed in the form

n~l H~

fn-l+ 1 \ m

THEOKY OF THE PARTITIONS OF NUMBERS.

55

Further simplification of this series cannot he effected because each term of the sum must be considered on its merits and does or does not add to the numerical result as may appear.

Art. 142. Writing the result for even order 2n

it appears that the result for uneven order 2»i+l may be written

For the squares of simple orders we have the results ORDER 2.

0 1 2
0	0	1
0	0	0
0	1	0

1 = 0 1 2 3	ORDER 3. 0123
0	2	0	0
2	0	0	1
0	0	0	0
0	1	0	0

ORDER 4.

1= 0 1

4 0	0 8	4	0	1
0	0	0
4	0	2	0	0
0	0	0	0	0
1	0	0	0	0

Z= 0 0

ORDER 5.

12345

II

s

16	16	8	4	0	0
16	20	4	4	0	1
8	4	8	0	0	0
4	4	0	2	0	0
0	0	0	0	0	0
0	1	0	0	0	0

56

MAJOK P. A. MACMAHON: MEMOIK ON THE

ORDER 6. 1= 0 1 2 3 4 5 6

0

80	96	60	16	12	0	1
96	96	48	24	0	0	0
60	48	24	0	3	0	0
16	24	0	0	0	0	0
12	0	3	0	0	0	0
0	0	0	0	0	0	0
1 0	0	0	0	0	0

Art. 143. I now proceed to consider the enumeration of the squares of even order 2n, such that every row and column contains two units, and the dexter and sinister diagonals / and m units respectively.

I form the product

n

CC-2

<2>

where «2<s) is the sum two together of the quantities

and I seek the coefficient, a function of X and /x, of

(ajOa^.o^)8

in the product.

The coefficient of X'/u,™ in the sought function of X and p. is the required number. Let pi, p2 he the sum and the sum two together of the quantities

then

whence

+ 2 (Xyx-1)

+ terms involving powers of «1; «2;1 above the second.

THEORY OF THE PARTITIONS OF NUMBERS. 57

The product of a2(l\ «2(2'0 is thus, after re-arrangement, effectively equivalent to pf+ (A + /A-2) (a, + 03.) #tf>,- (X + /4-2) (a^ + ag.8) p.

4)}a1a8( (a, +*,„)/,,

2!2. Regarded apart from £>2, ^ this expression is a function of a,, a3, ; the product

a^a/-1' is a function of «2, a2«-i> and generally the product

is a function of at,, aa,,+1_s, and all of these products are of similar form in regard to

Pa, Pi, ^, P-

Remembering that we desire the coefficients of

(«!«,... a2,)2 in the product

we must distinguish between p2 where it occurs as a multiplier of af' + a^ and where it occurs as a multiplier of a^a,,, and make a similar distinction in respect of ^,2. Put then

(tti' + as,,2) pi2 = (ai2+«a,,2) H^. Putting further the quantities a equal to unity and regarding a product

PaP\irfir^ as a symbol for the coefficient of symmetric function

/0«+rfli + -'-\

in the development of symmetric function

(i2)-+"(ir2',

I say that

>1-2 (X+ju-2)^3 + 2 (X-l) (/*-

is the symbolic expression of the required coefficient of

VOL. CCV. - A.

58 MAJOR P. A. MACMAHON: MEMOIR ON THE

This may be written

{o-4 + 2 (X + /*) 03+(X2 + /ts) <r,+ 2\pofa+2\p. (X + /t) 0-, + XV2}", 0-4 = pa-^paPi + 4 (p« + «j) + 2 (pj2 + 7T,2) - 8p, + 3,

0-3 = ^2^1 - ( P2 + 7I-2) - ( p^ + TTj2) + 5pi - 2,

o-2 =j>!2- 4^ + 2,

where

For the uneven order 2>i + l it is easy to show that the coefficient is symbolically

2X^ (X + ya) o-^XV}" x ( ^- It is easy to calculate the values of

KpiVV

for small values of <t, />, c, d.

Some results are, omitting the obvious result 2^1 — b ',

a.	b.	«•	d.	Value.
1				0
		1		1
			l	1
	1	1		3
	1		• 1	3
1		1		2
1			1	2
	2	1		12
1	2			5
		2		6
			2	6
		1	1	5

THEORY OF THE PARTITIONS OF NUMBERS. 59

enabling the verification of the results

o-4 = cr:j = o-2 = rr'2 = <r, = 0, o-/ = 4, <rso-, = 0, o-V = 2.

Hence for the even order 2 the whole coefficient is X2/na, corresponding to the only possible square

1	1
1	1

and I find for the uneven order 3

Art. 144. To find in general the number of squares which have two units iu each diagonal we find the coefficient of X2ja2 and obtain for even order 2n.

<r4-496er,4 ;

putting n = 2 we find for the order 4

2o-4 + 2cr/ + 4cr'/+l 60-30-1

and the verification of this number is easy.

For the uneven order 2n+l we obtain the number

'tr/^ 2<r>1+ (gJcrr^

^ (<r3ao-3+ 2o-3V8) + '

-

The general value of

may be obtained by means of the calculus of finite differences.

There is no theoretical difficulty in finding symbolical expressions for the enumera- tion of general magic squares associated with higher numbers, but the method does not lead to the determination of general magic squares. These must be regarded as arising from the generating function method of § 9.

I 2

III. Atmospheric Electricity in High Latitude.^.

By GEORGE C. SIMPSON, H.Sc. (18Z1 Exhibition, Scholar of tic I'niverxity

of Manclicxter).

Communicated by ARTHUR SCHUSTKI;, F.ft.S. Received February 17, — Read March -2, 190;").

INVESTIGATION into the problems of atmospheric electricity may be divided into two periods. The first period was devoted almost entirely to measurements of the normal potential gradient in the lower region of the earth's atmosphere, with the aim of finding its daily and yearly variations, its geographical distribution and its dependence on meteorological condition. To this period belongs the fine work of Lord KELVIN and Professor EXNEE.*

The second period commenced in 1899, when the interest in the problems of atmospheric electricity was at rather a low ebb, owing to the small real progress made during the latter few years. In that year the discovery that atmospheric air is always more or less ionized — made at about the same time by ELSTKR and GEiTEt,t in Germany and C. T. R. WlLSONj in England — had a completely revo- lutionizing influence on the theories held to account for the earth's normal field. This discovery has brought about a great revival of interest and opened a totally new field for investigation.

As long as air could be considered a perfect non-conductor EXXEK'S theory that the charge on the earth is a residual charge held a very strong position ; but with a conducting atmosphere it is untenable. An ionized atmosphere means a continual passage of electricity from the charged surface into the highest regions of the atmo- sphere, where only any residual charge could be held. The new discovery having proved conclusively that the charge on the earth is being continuously dissipated into the ionized air above, it became of prime importance to determine the rate at which the electricity is dissipated and the conditions under which the loss takes place.

The first serious attempt to do this was made by ELSTER and GEITEL.§ They designed an instrument consisting of a charged cylinder exposed to the air —

* For a good resumf of this period see EXNER, ' Terr. Mag.,' vol. 5, p. 167, 1900. t 'Phys. Zeit.,' 1, p. 245, 1899; 'Phys. Zeit.,' 2, p. 116, 1900. \ <-Roy. Soc. Proc.,' 68, p. 151, 1901.

§ 'Phys. Zeit.,' 1, p. 11, 1899; 'Terr. Mag. and Atm. Elect.,' 4, p. 213, 1899; ' DRUDE'S Aim.,' 2, p. 425, 1900.

VOL. COV. — A 389. 28.7.05

62 MR. GEORGE C. SIMPSON ON THE

protected from extraneous electrical fields — and so connected to an electroscope that the rate at which it lost its charge could be measured. By making certain assumptions it can be shown that the charge lost in a small interval of time from any charged body exposed to the air is always a definite fraction of the charge on the body. Thus, when ELSTER and GEITEL had found the charge lost by their cylinder in a minute, they were able to express the loss as a percentage of the charge on the cylinder, and then, by applying this percentage to the charge on the earth, were able to find the quantity of electricity being dissipated from every square metre of surface each minute.

Besides knowing the amount of electricity dissipated from the surface — which depends upon many factors — it became also of great importance to know to what extent the air is ionized at any moment. For this purpose EBEKT* designed an instrument which gives the amount of ionization independently of everything else. A known quantity of air is drawn through a cylinder condenser, the inner cylinder ol which is connected to an electroscope. As the air passes between the cylinders the charged inner one attracts t<» it all the ions of the opposite sign. These ions neutralize an equal amount of electricity, and so the charge lost by the inner cylinder is a measure of the number of ions contained in the known quantity of air which has been drawn through the instrument. In this way it is possible to find how many electrostatic units of each kind of electricity are free in a cubic metre of air.

These two instruments are very powerful weapons for attacking the new problems of atmospheric electricity, and have been used as such to a large extent on the Continent. Systematic observations of the dissipation were undertaken by ELSTER and GEITEL, and quite a number of other physicists have devoted themselves to finding the relations existing between meteorological conditions, ionization, the rate of dissipation and the potential gradient. As a result of this work the electrical conditions of the atmosphere are already fairly well known for lands lying within the temperate zone. With the idea of extending this knowledge to places within the Arctic Circle I was granted permission by the Commissioners of the 1851 Exhibition Scholarship to undertake a year's work on atmospheric electricity in Lapland. The work which I proposed to do was the following :—

1. By means of a Benndorf self-registering electrometer to obtain daily curves

of the potential gradient and from these to calculate the yearly and daily variation.

2. To make systematic observations of the dissipation by means of ELSTER

and GEITEL'S instrument.

3. To make corresponding measurements of the ionization with EBERT'S

apparatus.

* Short description, ' Phys. Zeit.,' 2, p. 662, 1901; fuller description, ' Aeronautische Mittheilungen ' p. 1, 1902.

ATMOSPHEKIC ELECTRICITY IN HI«H LATITUDES. 63

4. To measure the amount of radio-active emanation in the atmosphere.

5. To investigate as far as possible the influence of the Aurora on the

electrical conditions of the atmosphere.

In my choice of a station I decided to get as far north as possible without being actually on the sea coast, and found that the Lapp village of Karasjok (69° 17' N., 25° 35' E., 129 metres above sea-level) was very wfll suited for my purpose.

Meteorological Conditions.

Before going on to a discussion of the electrical results obtained, it will be as well to give a short account of the meteorological conditions experienced during the year's work. From its high latitude the north of Norway should be a very cold district ; but the presence of the open ocean on the north and West greatly modifies the temperature. The effect of the water is of course very much more marked on the sea coast than inland. As one recedes from the coast the mean temperature for the winter six months falls very rapidly, it being — 2°'3 C. at Gjesvoer, near the North Cape, and — 11°'7 at Karasjok. If there were no interchange of air between the ocean and the interior of the land the latter would of course have a very low temperature. This became very noticeable during periods of calm weather, for the temperature would then run down to very low values, reaching on several occasions —40° ('., while, on the contrary, whenever the wind rose the temperature rose also.

When there was no wind, a cap of very cold air would form over the land, causing a nearly permanent temperature inversion. Although 1 could not observe this inversion instrumentally — neither kites nor balloons forming part of my equipment — there could be little doubt as to its reality. On September 3()th, with an air-temperature of — G° C., a bright rainbow was observed. Then again, on descending the high banks of the river, one felt at once the cold air collected in the river basin, and the Lapps stated that it was seldom as cold on the hills as in the valleys. Then, again, the fact that a wind was always accompanied by mild weather also points to the cold of still weather being confined to a laver of air of no considerable depth lying over the surface. This condition of things almost entirely prevented the formation of ascending currents of air, so causing very small values of the amount of precipitation and almost entirely preventing the formation of low clouds during the winter. It also had a very marked effect on the electrical condition of the atmosphere, to which reference will be made later.

During the summer the weather conditions were very similar to those of England, with the exception that the precipitation was very much less and thunderstorms were scarce. On three days only was thunder heard and lightning was not seen once.

From November 26 to January 18 the sun did not rise above the horizon; never- theless, even in the darkest days there were two or three hours of twilight during which the sky was too bright for the stars to be seen. The period during which the sun did not go below the horizon extended from May 20 to July 22.

64 MK. GEORGE C. SIMPSON ON THE

Methods of Work*

Potential Gradient.— BENNDORF'S self-registering electrometer, with radium collector attached, was set in action on September 28, 1903, and produced a nearly continuous record of the potential gradient until October 1, 1904. Each day the curve for the previous day was measured and the mean potential gradient for each hour obtained. Tli is was done by first drawing a curve as smoothly as possible through the registered curve, then five equidistant ordinates in each hour were measured, and the mean of these five taken to represent the mean potential gradient during the hour. In discussing the potential gradient for any place it is usual to use only observations made during fine weather, neglecting all those which have been affected by any atmospheric disturbance. This plan I also followed during the summer months (April to end of September), for then the curves drawn by the instrument were exceedingly regular unless there was actually precipitation taking place in the neighbourhood. But during the winter the curves were so irregular, even on the finest days, that it was quite impossible to decide whether a particular curve ought to be neglected or not, so T used, during the winter, all the curves quite independently of the weather This caused irregularities in the final curves, but has not, I think, affected the

cl

conclusion to be drawn from them.

Iti^iitxtioii. — The value of the dissipation, as measured by ELSTER and GEITKL'S instrument, depends to a very great extent on the manner in which the instrument is exposed to the wind. This is as it should be, for the actual dissipation from the earth's surface (which the instrument is designed to measure) depends largely on the \vind strength. In order that the instrument should measure the amount of dissipation taking place from the earth's surface, it should be exposed to the same wind condition as the general surface. This fact has not been fully realized by most observers. It has been quite a common practice to shelter the instrument from the wind, either by erecting screens or by observing close to a building, and in several cases the instrument has been placed within a room close to an open window. Observations taken under such conditions are of very little value : they are certainly of no use in comparing the dissipation of one place with that of another, and at the best can only be used to compare variations from time to time at the same place. In order that the dissipation at one place may be compared with that of another, the instruments used should in both cases be exposed to the full force of the wind, for wind strength is just as much a factor in determining the dissipation as is the ionizatiou. For this reason my instrument was only used in a freely exposed situation, where it was in no way sheltered from the wind. This method also has its drawbacks, for with anything like a high wind the leaves of the electroscope were so blown about that they continually discharged the instrument by coming in contact with the case. Hence measurements could not be made in very high winds, and so the mean values of the dissipation

* For fuller particulars of methods of work and arrangement of apparatus see Appendix.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

found are slightly less than they would have been if observation could have been made in all winds ; but the final results are very little affected. Observations also could not be made during rain, owing to trouble with the insulation. Except when it was impossible to observe, owing to these two causes, measurements of the dissipation were made three times each day, between 7.30 and 9.30 in the morning, between 12 and 2 midday, and between 6 and 8 in the evening.

In expressing the dissipation, ELSTER and GEITEL'S example has been followed, i.e., the dissipation is expressed as the percentage of charge lost by a charged body in a minute. Thus a+ = TOO per cent, means that 1 per cent, of the positive charge on any body will be dissipated in a minute ; similarly, a_ expresses the dissipation of a negative charge. The ratio a_/a+ is written q. There are two methods of obtaining

the mean value of this ratio, either - 2 ( — ) or — - — ; in most cases these two are very

n \a+/ Sa+

nearly equal. In this paper q is always obtained by the latter method.

lonization. — EBERT'S instrument for measuring the ionization was used at the same time as the dissipation instrument. It was often possible, however, to use the Ebert apparatus on days when the wind made -it impossible to use the dissipation apparatus ; but, on the other hand, the insulation of the Ebert instrument would often fail, owing to high humidity of the atmosphere, when satisfactory measurements of the dissipation could be obtained. EBERT'S instrument also could not be used when the temperature fell below —20° C., for then the oil in the air turbine froze and prevented the clock- work running freely. EBERT'S method of expressing the ionization has been followed ; the positive ionization is expressed as the number of electrostatic units of free positive ions in a cubic metre of air ; similarly for negative ionization. The symbols used to denote positive and negative ionization are 1+ and I_ respectively. The ratio of positive ionization to negative, i.e., I+/I-, is written r, and the mean is obtained by the process 2I+/SI-.

RESULTS OF THE OBSERVATIONS.

Yearly Variations.

Potential Gradient. — Table I. gives the monthly values of the potential gradient.

TABLE I. — Potential Gradient.

Winter.	Volts/metre.	Summer.	Volts/metre.
October .	121	April	131
November	167	May	103
December	175	June	90
	199	July	98
	209		93
March	191	September ....	93

VOL. CCV. — A.

fiB MR. GEORGE C. SIMPSON ON THE

The yearly course of the potential gradient is shown in Curve I., fig. 1, on which each point gives the mean potential gradient for a week, the means for the months (as in Table I.) being shown by points enclosed within circles. It will at once be seen how irregular the potential gradient is during the winter when taken for such short time intervals as a week ; on the contrary the monthly means fall very nearly on a regular curve. It must be remembered that, as stated above, these values during the winter are obtained from both fine and disturbed days. If only fine days had been used not only would the curve have been more regular, but also the mean potential gradient would have been greater. The trend of the curve may be summed up as a regular rise in the potential gradient from. October to the middle of February, followed by a more rapid fall until the end of May, after which the potential gradient remains nearly constant during the summer months.

Dissipation. — The mean values of the dissipation for each month are shown in the following table. In order to find the effect of the seasons, and whether the total absence of the sun for nearly three months during the winter and its presence for an equal length of time during the summer influences the electrical conditions of the atmosphere, the observations have been grouped into periods of three months, the winter three months containing the period of no sun and the summer three months that of permanent sun.

TABLE II. — Dissipation.

Months.	a+.	a_. q.	a±.	Seasons.	«+.	«_.	?•	a±.
November	3-20	3-43 1-07	3-32	1
December	2-13	2-53 1-19	2-33	}• Winter .	2-44	2-76	1-13	2-61
January	1-98	2-33 1-18	2-17	J
February	1-37	1-47 1-08	1-42	}
March	2-79	3-74 1-34	3-27	> Spring .	2-65	3 '20	1-20	2-92
April.	3-78	4-38 1-16	4-07	J
May.	4-41	4-76 1-08	4-58
June.	4-24	4-68 1-10	4-45	Summer .	4-63	5-14	1-11	4-88
July .	5-25	5-97 1-13	5-61
August	4-32	4-94 1-14	4-63	]
September	4-28	4-89 1-14	4-58	> Autumn .	3-60	4-16	1-15	3-88
October .	2-21	2-65 1-20	2-43	J
				Whole year	3-33	3-82	1-15	3-57

On Curve II. these values of the dissipation (a±) have been plotted, also the weekly values. If no observations were made for a week a gap has been left in the curve. From the curve it will be seen that the yearly course of the dissipation is strikingly similar to that of the potential gradient when inverted, the one falling and

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

67

IDECEMSERI JANUARY

I FEBRUARY | MARCH | APRIL | MAY | JUNE | JULY | AUGUST

APRIL I MAY JUNE I JULY I AUGUST [SEPTEMBER

NOVEMBER IDECEMBER [JANUARY I FEBRUARY I MARCH

68

ME. GEORGE C. SIMPSON ON THE

rising at exactly the same time as the other rises and falls, and both remaining constant during the summer. These curves suggest that there is some relation between the two phenomena ; this relation will be discussed later in the paper.

The ratio of the negative dissipation to the positive (q) does not appear from these results to have a regular yearly course, but when they are considered in connection with the ionization it will be seen that it is very likely there is a yearly variation with a maximum in the winter and a minimum in the summer.

Ionization.— Table III. gives the monthly mean values of the ionization.

TABLE III. — Ionization.

Months.	I_.	I+.	T.	I±-	Seasons.	I_.	I+-	r.	I±.
November December	•25 •28	•35 •39	1-40 1-39	•30 •33	^ Winter .	•26	•33	1-28	•29
January .	•25	•26	1-04	•25	J
February March .	•20 •28	•24 •32	1-20 1-14	•21 •30	I Spring .	•26	•31	1-19	•28
April	•31	•38	1-22	•34	J
May .	•35	•40	1-18	•37	1
June .	•37	•41	1-09	•39	> Summer .	•38	•42	1-11	•40
July . .	•42	•46	1-10	•44	J
August . September October .	•45 •42 •34	•51 •46 •40	1-13 1-08 1-18	•48 •44 •36	> Autumn .	•40	•46	1-15	•43
					Whole year	•33	•38	1-17	•36

These results, together with the weekly means, have been plotted in Curve III. Here we have quite a different curve from either of the two previous ones. Instead of the rapid fall and rise in the winter followed by a constant period during the summer we have a six months' linear fall from August to February followed by a similar six months' linear rise from February to August.

That there should be such a great difference between the curves for the dissipation and the ionization was not to be expected, and at first one would be inclined to doubt the correctness of one or other of them. But this can be tested by the following considerations. The dissipation depends practically only on two factors : ionization and wind strength. If the effect of the latter could be eliminated, the course of the dissipation should then be the same as that of the ionization. In order to see if this were so, I took all my measurements of the dissipation and separated them according to the strength of the wind as estimated at the time of observation, then, using only one definite wind strength, took the means for each month and plotted them. ' The result is shown in fig. 2. Each curve represents one wind strength, and it will at once

ATMOSPHEEIC ELECTEICITY IN HIGH LATITUDES. 69

be seen that all four curves are practically parallel* and are similar in shape to that of the ionization. This shows at once that both the curves of the dissipation and ionization are correct, and that there is a real difference in the yearly course of the two, and also that there is a closer relation between potential gradient and dissipation than between potential gradient and ionization.

APRIL I MAY | JUNE | JULY | AUGUST [SEPTEMBER

OCTOBER | NOVEMBERIDECEMKRlJANUARY [FEBRUARY]

Fig. 2.

The value of the ratio !+/!_ shows a very distinct yearly period with a maximum in the winter and a minimum during the summer. Later it will be seen that very probably this ratio depends largely on the potential gradient, so that its yearly period might be expected on account of the yearly variations in the potential gradient.

Daily Variations.

Potential Gradient. — The daily course of the potential gradient varies greatly according to the season of the year. For this reason five curves of the daily course

* The lowness of the two curves for wind strengths 0-1 and 1-2 during the first part of the winter is due to the fact that, owing to the darkness at both the morning and evening observations then, it was impossible to see if the smoke of the village was drifting towards my place of observation or not. Nor was I quite aware then of the fact, which I found later, that with no wind the smoke of the village extended in an almost invisible haze over the whole valley, out of which it could not get. This smoke effect, of course, only acted when there was insufficient wind to drive the smoke away, and its effect is not at all visible on the two curves with wind strength greater than two, i.e., a steady breeze.

70

MR. GEORGE C. SIMPSON ON THE

are given : one each for the winter, spring, summer, and autumn three months and another for the year taken as a whole (fig. 3). It will at once be seen that the two

A.M.

11 12 1 MID-DAY.

Fig. 3.

10 11 12

curves for the winter and spring lie entirely above the curve for the year and those for the autumn and summer entirely below. The equations to the five curves are* : —

Winter three months, P = 180 + 64 sin (0+189) + 26 sin (20+155) + 4 sin (30+200), Spring „ „ P = 177 + 57 sin (0+176) + 37 sin (20+ 151) + 13 sin (30+ 195),

Summer „ „ P = 97 + 16 sin (0+141)+ 9 sin (2(9+144)+ 4 sin (3(9+ 126),

Autumn „ „ P = 103 + 23 sin (0+170)+19 sin (20+184)+ 2 sin (30+131),

Whole year . . . . P = 139 + 39 sin (0+177) + 23 sin (20+158)+ 5 sin (30+178).

From these equations we see that there are two periods which must be taken into account ; the amplitude of the third period falls without the limits of the accuracy of the instrument. Of these, the greater is a whole-day period and the lesser a half- day period. We also see that the phase of the main period undergoes a regular shift from a maximum in the winter to a minimum in the summer, which means that the evening maximum is earlier in the winter than the summer, thus following the sun. The phase of the second period does not vary regularly, and on account of its

* These equations are worked out to mean local time, taking 12 o'clock midnight as the zero and 15° to represent an hour. All other time used in this paper is mid-European, which is 42 minutes behind mean local time.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

71

smallness during the Bummer and autumn its position is not then well fixed. The ratio of the amplitude of the second period to the first is : winter '40, spring -65, summer '56, autumn '83, whole year '59. This shows no regular variation ; the large value for the autumn is due to the strengthening of the second period by the formation of mists over the river about the times of sunrise and sunset, which mists always give rise to high potential gradients.

The hourly values of the potential gradient corresponding to the five curves are given in the following table : —

TABLE IV. — Daily Course of the Potential Gradient.

		12tol.	1-2.	2-3.	3-4. 4-5.	5-6.	6-7.	7-8.	8-9. 9-10.	10-11.	11-12.
Winter j	A.M. P.M.	138 210	119 103 100 212 214 208	90 216	Ill 141 236 < 245	148 242	162 187 247 226	184 206	214 174
Spring 1	A.M. P.M.	134 193	i 122 : 107 201 i 190	99 196	99 177	109 198	135 238	146 262	164 175 260 247	187 233	187 185
									I
Summer i	A.M. P.M.	101 91	93 94	86 97	81 96	77 103	81 86 101 107	90 110	90 90 121 ! 131	89 122	87 108
Autumn <	A.M. P.M.	87 94	75 99	70 105	70 108	72 114	78 90 102 125 , 132 144	108 106 143 138	93 115	90 99
Whole f	A.M.	115	102	92 : 87 ! 84	95 113 j 121	131 140	138	144
year \	P.M.	147	151	151 152 153	165	180 189	194 185	169	142
				1

Dissipation. — As I had no self- registering instrument to record the dissipation and ionization, it is impossible to work out the daily course of these two as has been done for the potential gradient. Nevertheless, some idea of the course can be obtained by comparing the results according to the different times of observing. In Table V. the mean results from the morning, midday and evening observations are shown for each three months and then for the whole year.

MR. GEORGE C. SIMPSON ON THE

TABLE V. — Dissipation.

	Morning (8 to 9 A.M.).	Midday (12 to 1 P.M.).	Evening (6 to 7 P.M.).
a+.	a_.	<!•	a+.	a_.	?•	a+.	o_.	2-
Winter. "| Three months, 1 November, December, January J	2-11	2-71	1-02	2-02	2-47	1-23	1-92	2-37	1-23
Spring. 1 Three months, I February, | March, April J	3-00	3-58	1-19	2-84	3-29	1-16	2-08	2-55	1-23
Summer. ~| Three months, I May, f June, July J	4-54	4-97	1-10	4-96	5 31	1-07	4-45	5-07	1-14
Autumn. 1 Three months, | August, \ September, October J	3-51	4 '04	1-15	4-34	4-85	1-12	2-92	3-57	1-22
Whole year. .	3-43	3-83	1-12	3-54	3-98	1-12	2-84 3-39	1-20

During the winter and spring the morning observations show a slightly higher dissipation than the midday, while, on the contrary, during the summer and autumn the midday values are the higher. For the whole year the dissipation is slightly higher at midday than earlier in the morning, while the evening observations show the lowest dissipation of the three. The value of the ratio q for nine months shows a daily period, being lower at midday than at either the morning or evening observations. The difficulties of observing during the winter three months make the value of the ratio found then very doubtful.

The evening fall in the dissipation no doubt stands in some relation to the evening maximum of the potential gradient, while it is almost certain that the high evening value of q is directly caused by the high value of the potential gradient at that time.

lonization. — The results of the ionization observations are shown in Table VI. in the same way as those of the dissipation were in Table V.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. TABLE VI. — lonization.

73

	Morning (8 to 9 A.M.).	Midday (12 to 1 P.M.).	Evening (6 to 7 P.M.)
	I_.	I+.	r.	I_.	I+.	r.	I_.	I*.	r.
Winter. ")
Three months,
November, \-	•27	•32	1-20	•26	•35	1-35	•24	•32	1-35
December,
January J
Spring. 1
Three months, 1 February, [	•27	•34	1-26	•28	•31	I'll	•23	•30	1-30
March, April J
Summer. "|
Three months, 1 May, f	•39	•42	1-08	•36	•43	1-20	•36	•41	1-14
June, July J
Autumn. 1
Three months,
August,	•42	•46	1-11	•41	•46	1-13	•36 -44	1-23
September, October J
Whole year. .	•34	•39	1-15	•33 -39	1-18	•30 -37	1-23

The daily period of the ionizatiou is not so pronounced as that of the dissipation, but the ionization is slightly lower in the evening than in the morning or at midday during the whole year. There is practically no difference between the midday and morning ionizations. The daily period of the ratio q is a steady rise from the morning to the evening ; in this respect the ionization does not correspond with the dissipation.

Interrelation of the lonization, Dissipation and Potential Gradient.

Potential Gradient and Dissipation. — The relation between potential gradient and dissipation has been very closely studied by GOCKEL* and ZOLSS. t The latter shows that the potential gradient varies very considerably with the dissipation, high potential gradient being accompanied by low values of the dissipation, and vice versa; and both show very clearly that the ratio of negative dissipation to positive dissipation rises considerably as the potential gradient rises. Table VII. shows the

* 'Phys. Zeit.,'4, p. 871, 1903. t 'Phys. Zeit.,'5, p. 106, 1904.

VOL. OCV. — A.

74

MR. GEORGE C. SIMPSON ON THE

results of my observations of the dissipation tabulated according to the potential

gradient.

TABLE VII— Potential Gradient and Dissipation.

Potential gradient.	Winter.	Summer.	Year.
a+.	a_.	2-	a+.	a_.	2-	a+.	a_.	f-
volts/metre. 50 to 100 100 „ 150 150 „ 200 200 „ 300 300 „ 400 >400	3-94(5')* 2-34(03) l-75(») 1 • 32 (») •60(12) •51 H	4-14(60) 2-77 (64) 2 • 43 (24) l-54(41) •S5(13) • 64(20)	1-05 1-18 1-39 1-17 1-42 1-25	4-50(93) 4-18(81) 2-50 (!) 1-82 (5)	5-02 (93) 4-83 (8) 3-47 O 1-92 (5)	I'll 1-16 1-38 1-05	4-29 (15°) 3-38O 1-85 (26) 1-37 (46) •60 (12) •51 H	4-67 (15S) 3-93(«6) 2-58 (24) 1-58 (46) •85 (") •64 (*>)	1-09 1-16 1-40 1-16 1-42 1-25

It will be seen that here also there is the same marked relation between the potential gradient and the dissipation; but the relation between the potential gradient and the value of the ratio q does not appear so clearly. Nevertheless, the table does not disprove that the ratio rises with the potential gradient, there is rather some support given. In the first place there is a distinct rise in the ratio over the range from 50 to 200 volts/metre, and the highest value found falls between 300 and 400 volts/metre. When the whole year is taken into account there are only two out of the six divisions which do not conform to the rule.

Potential Gradient and lonization. — So far no results have been published showing the relation between potential gradient and ionization, so that the results given in the following table cannot be compared with previous work.

TABLE VIII. — Potential Gradient and lonization.

	1	Winter.		E	Summer.			Year.
Potential
gradient.
	I_.	I+.	r.	I_.	I+.	r.	I_.	I+.	r.
volts/metre.
50 to 100	•35(53)	•42 (52)	1-20	• 42 (84)	• 44(84)	1-07	• 39 (137)	• 43(136)	I'll
100 150	•29(52)	• 34(53)	ri5	•35 («)	• 42 (48)	1-18	•32(100)	.37(101)	1-15
150 200	• 28 (34)	• 33 (30)	1-26	•27 (4)	•37 (4)	1-41	•28 (38)	•36 (34)	1-28
200 300	•19(25)	•24(»)	1-26	•17 (»)	•30 (*)	1-74	•19 (3«)	•26 (26)	1-42
300 400	•15 C)	•15 C)	1-00			—	•15 (•)	•15 O	1-00
400 500	•12 (»)	•14 (*)	1-22	—	—	—	•12 («)	•14 (•)	1-22
>500	•12 («)	•10 (3)		—	—	—	•12 (•)	•10 (»)	—

The first striking fact which this table shows is the great dependence of the potential gradient on the ionization ; this we might have expected from the dissipation results already considered.

* These small numbers in brackets give the number of observations from which the mean is drawn.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

75

High values of the ionization accompany low values of the potential gradient and vice versa.

Here we find that the ratio between positive and negative ionization (r) does increase with the potential gradient over the range from 50 to 300 volts/metre. That there is not the same agreement higher is to be expected from the fact that for values of the potential gradient over 300 volts/metre the ionization is so small as to be only just within the power of the instrument to measure, and so one cannot expect the ratio of the observations to be given with any degree of accuracy ; also the number of observations with the potential gradient over 300 volts/metre is so small that better results could hardly be expected.

We may, then, take it that the ionization and dissipation have a great determining influence on the potential gradient, and that high values of the potential gradient are, on the whole, accompanied by high values of the ratio r and q.

Ionization and Dissipation. — It has already been stated that the values of the dissipation, as given by ELSTER and GEITEL'S instrument, depend mainly on the two factors ionization and wind strength. It would be of considerable interest to find how the dissipation varies with either of these factors, the other remaining constant.

When the greater part of my observations of the dissipation were made. EBERT'S instrument was also in use, and gave the true value of the ionization at the time when each observation of the dissipation was taken. In order to find how the dissipation varies with variations of the ionization, the wind strength being constant, I separated out all the results of the dissipation obtained with a given wind strength, then divided these again according to the values of the ionization observed at the same time. The results are given in Table IX., and have been plotted in fig. 4.

IO N IZ ATI. ON

Fig. 4.

L 2

76

ME. GEOKGE C. SIMPSON ON THE

TABLE IX. — lonization and Dissipation according to Wind.

	Dissipation.
lonization.	Wind 0-1.	Wind 1-2.	Wind 2-3.
•0--1	.45(12)	•65 («)
•1--2	•60(5a)	1-08(1(>)	—
•2--3	l-26(38)	l-85(20)	2-70(16)
•3--4	2-04 C28)	2-92(17)	3-88(47)
•4--5	3-03(44	3-83(33)	5-33(54)
•5--6	3 • 36 (24	4-48 (6)	5-90(14)
•6--7	3-56 (4		_

We see that, allowing a large margin for the uncertainties of such an investigation, the dissipation may be regarded as a linear function of the ionization for any given wind strength. It must be remembered that this agreement is only true when dealing with a large number of observations ; for the mobility of the ions affects the dissipation considerably. It would be interesting to compare individual observations of the ionization and the dissipation when the wind strength was accurately known. In that case the effect of the mobility of the ions would be very apparent. My observations do not allow of this being done, as the wind strengths were only roughly judged by the " feel " of the wind, and no doubt varied very much more amongst themselves than the mobility did. For the same reason it is of no use finding from my observations how the dissipation varied with the wind strength, the ionization being constant ; for my classification of the wind strengths, although based on the Beaufort scale, would almost certainly differ from a similar classification made by another observer.

Relation between the Meteorological and Electrical Conditions of the

Atmosphere.

dissipation and Wind. — After what has been already said about the method of estimating the wind strength, the following table cannot be regarded as final ; but as it shows the influence of the wind as found from all the observations it is printed here. It is of considerable interest to notice that the ratio q falls as the wind strength increases.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

77

TABLE X. — Dissipation and Wind.

Wind.	Winter.	Summer.	Year.
Beaufort
Scale,
0-12.	a+.	a_.	2-	a+.	a_.	2-	a+.	a_.	9-
0-1	•85(10«)	1-04(108)	1-22	2 -68 H	3 -16 (^	1-18	1-44(15«)	1-71 (15S)	1-19
1-2	1-84 (45)	2-21 (46)	1-20	3-71 C23)	4 • 30 (23)	1-16	2-48 (8S)	2-91 (6y)	1-17
2-3	3-64 (30)	3-99 (27)	1-09	4-62 (52)	5-20(50)	1-13	4-26 (82)	4-78 (")	1-12
3-4	4-45 (21)	4-85 (25)	1-09	5-11 (44)	5-58(43)	1-09	4-90 («5)	5-40 (°8)	1-10
>4	5-80 (20)	5-96 (2S)	1-03	6-05(36)	6-78 (33)	1-12	5-97 (56)	6-44 (M)	1-08

Dissipation and Relative Humidity. — GOCKEL* has gone very fully into the relation between dissipation and relative humidity, and his results, which have in the main been confirmed by ZoLSS,t show that the dissipation decreases with a rise in the relative humidity, and as the dissipation of the positive electricity decreases more rapidly than that of the negative, the ratio q increases as the relative humidity rises.

TABLE XI. — Dissipation and Relative Humidity.

	Winter.	Summer.	Year.
Relative
Humidity.
	a+.	a_.	?•	a+.	a	2-	«+.	«_.	*
per cent. 30 to 40	_	_.	_	4-61 ('")	4-97 (1C)	1-08	4-G1 (lt;)	4 • 97 (ll!)	1-08
40 „ 50	—	—	—	4-71 (U3)	5-23(03)	1-11	4-71 (63)	5-23 ((i3)	I'll
50 „ 60	—	—	—	4-68(52)	5-49f2)	1-17	4-68(S2)	5-49 (52)	1-17
60 „ 70	3-03(22)	3 '55 (22)	1-17	3-88(37)	4-53 (37)	1-17	3-56(MI)	4-16H	1-17
70 „ 80	2-61 (42)	3-01 (42)	1-16	2-90H	3-37H	1-16	2-73(71)	3-16 (71)	1-16
>80	l-37(51)	1-71 (")	1-25		.		1 • 37 (51)	1-71(«)	1-25

Table XI. shows that for relative humidities greater than 50 per cent, my results agree with GOCKEL'S, the decrease in the dissipation as the relative humidity rises being very marked, and the value of q also increases as the relative humidity increases. But it should be remarked that the fall in the dissipation as the relative humidity rises is not entirely due to the relative humidity, for the conditions in Karasjok were such that nearly all values of the relative humidity higher than 80 per cent, were accompanied by a calm atmosphere, and in the main low values of the relative humidity were accompanied by high wind.

Dissipation and Temperature. — ZOLSS (loc. cit.) has shown that the dissipation in the free air increases with the temperature, and he found that the variation was linear

* 'Phys. Zeit.,'4, p. 871, 1903. t 'Phys. Zeit.,' 5, p. 108, 1904.

78

MR. GEORGE C. SIMPSON ON THE

over the range he investigated. Later GOCKEL returns to this point * and throws out the suggestion that the increase in the dissipation is due to the increase which the ozone in the atmosphere undergoes as the temperature rises. In Karasjok the temperature fell so low during the winter that I was able to observe the influence of temperature on the dissipation at very much lower temperatures than had ever been done before, obtaining sixty observations with the temperature between - 40° and - 20° C. Table XII. shows the results, which, in the main, confirm ZOLSS'S

TABLE XII. — Dissipation and Temperature.

		Winter.		I	Summer.			Year.
Temperature.
	a+.	a	?•	a+.	a_.	<!•	0 +	a_.	2-
°C. <-20 - 20 to - 15 -15 -10	-76(28) • 99 (34) 1-51 M	•91 (31) l-22(34) l-73(39)	1-19 1-24 1-15	—	—	—	.76(28) •99(84) 1-51 (3!>)	•91 (»i) 1-22 (S4) l-73(39)	1-19 1-24 1-15
- 10 - 5	2 • 45 (44)	2-82 (44)	1-16		—	—	2 • 45 (44)	2-82(44)	1-16
- 5 0 0 5	3-17(63) 4 • 34 (18)	3-75(C4) 4- 66(20)	1-18 1-07	3-99(10) 3-71 (37)	4-71 H 3-73(37)	1-18 1-01	3 • 28 (73) 3-92 (56)	3 • 90 (74) 4-06(57)	1-19 1-03
5 10			.	4-41 H	4-99(so)	1-13	4-41 (80)	4- 99(80)	1-13
10 15	—	— •		4-68(66)	5-23(66)	1-12	4-68(6«)	5-23(06)	1-12

observations. The temperature has a great effect on the dissipation, for it rises from •83 with temperatures between — 40° and — 20° C. to 4'95 with temperatures between 10° and 15° C., and when the results for the whole year are considered the relation is practically linear. But here again attention must be called to the fact that the very low temperatures were always accompanied by calm weather ; and that there was very much more wind during the summer when the high temperatures were obtained than during the winter with its low temperatures. It is interesting to note that temperature has no apparent effect on the ratio q.

lonization and Relative Humidity. — It will be seen from Table XIII. that when the whole year is taken into account the effect of the relative humidity on the ionization is very similar to its effect on the dissipation. That is, the amount of ionization decreases with an increase in the relative humidity, while the ratio r increases. But it is very interesting to note that when the winter and summer results are taken separately this effect is hardly apparent at all. No definite effect of the relative humidity on the positive ionization can be detected during either the winter or summer six months. While the negative ionization is slightly affected during the winter, no effect can be seen during the summer. Nevertheless, during both winter and summer the value of the ratio r increases regularly with the relative humidity.

* ' Phys. Zeit.,' 5, p. 257, 1904.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

79

TABLE XIII. — lonization and Relative Humidity.

	Winter.	Summer.	Year.
Relative
humidity.
	I_.	!+•	r.	I_.	I+-	r.	I_.	IH-.	r.
per cent.
30 to 40	—	—	—	•45 (6)	•45 (•)	1-04	•45 (•)	•45 (6)	1-04
40 „ 50	—	—	—	•38(44)	•41 («)	1-10	•88(«)	•41(«)	1-10
50 „ 60	•32 (15)	• 35 (15)	1-09	•37(62)	•42 («)	1-14	•SSMj	• 40 (°7)	1-13
60 „ 70	•28(24)	•32 (24)	1-15	•39 (34)	•45 (34)	1-15	• 34(58)	• 40 (58)	1-15
70 „ 80 >80	•28(32) •23(")	•33(32) •32(17)	1-16 1-39	•48(8)	•55 (8)	1-20	•32(40) •28(»)	• 37 (40) •32 (17)	1-18 1-39

lonization and Temperature. — From Table XIV. it will be seen that temperature has a great effect on the ionization — the ionization at temperatures lower than — 20° C. being only a little greater than a third of those with temperatures between 10° and 15° C. No effect of temperature on the ratio r is apparent.

TABLE XIV. — lonization and Temperature.

Temperature.	Winter.	Summer.	Year.
	I_.	I+.	r.	I_.	I+.	r.	I_.	u.	;•.
0 C. <-20	•16(10)	•18 (»)	1-12		_	_	•16(10)	•18 (»)	1-12
-20 to -15	•18(26)	•22 (24)	1-23	—	—	—	' 18 (26)	•22(24)	1-23
-15 „ -10	•22 (27)	•26 (26)	1-18	—	—	—	•22(27)	•26(26)	1-18
-10 „ - 5	•30(41)	•36(38)	1-20	—	—	—	•30(41)	•36(38)	1-20
- 5 „ 0	• 32 (5«)	•39(53)	1-27	•31 (21)	•37H	1-19	•31 (")	• 39 (74)	1-24
0 „ 5	•36(31)	• 42 (29)	1-16	•36 (40)	• 39 (40)	1-07	•35('i)	• 40(69)	1-12
5 „ 10	—	—	—	•40(6(i)	• 45 (66)	1-13	• 40 (««)	.45(66)	1-13
10 „ 15	—	•	—	•43(28)	• 45(28)	1-06	•43(28)	• 45(28)	1-06

In discussing the effect of temperature on dissipation it was stated that the absence of wind at low temperatures might account for the decreased dissipation ; but we now see that the smallness of the dissipation is more likely caused by the low ionization at low temperatures.

Potential Gradient and Temperature. — It has already been shown that the potential gradient varies very greatly with the ionization and dissipation. As we have also seen that the ionization and dissipation depend greatly on the temperature, we should expect the temperature to have an effect on the potential gradient. That such is the case can be seen from Table XV. The potential gradient is high with low temperatures and low with high temperatures. This fact has often been noticed and recorded before.

80

ME. GEOKGE C. SIMPSON ON THE TABLE XV. — Potential Gradient and Temperature.

	Potential gradient, volts/metre.
Temperature.	Winter.	Summer.	Year.
- 40 to - 30	256 (29)	_	256 (»)
- 30 - 20	259 (65)	—	259 (65)
-20 -10	235 (»')	. — .	235 (117)
-10 0	158 (178)	126 (40)	152 (21S)
0 10	108 (45)	105 (m)	106 (21fl)
10 20	—	98 (79)	98 (97)

The Aurora and the. Electrical Conditions of the Atmosphere.

During the whole of my stay in Karasjok I could not detect the slightest effect of the aurora on any of the electrical conditions of the atmosphere, and most careful watching of the needle of the self-registering electrometer did not show any relation between potential gradient and the aurora. On first starting my observations I thought I found, as many other observers have done, an unsteadiness of the potential gradient during an aurora display, but longer experience showed that this unsteadiness had nothing to do with the aurora. In order for an aurora to be visible it must be a clear night, and a clear night is generally accompanied by low temperature and a high potential gradient. The high potential on clear cold nights was always unsteady and varied quite irrespective of the presence or absence of an aurora. When an aurora was visible naturally it often appeared as if a change in the aurora was coincident with a change in the potential gradient, but the attempt to connect changes in the potential gradient with changes in the aurora over any length of time always failed. Other observers have recorded negative potential gradient during an aurora display ; but during the whole winter my self-registering electrometer did not once record any such reversal.

CONCLUSIONS TO BE DRAWN FROM THE WORK.

The first and most important conclusion is that the difference in the electrical conditions of the atmosphere between mid-Europe and this northerly station can all be accounted for by the difference in the meteorological condition at the two places.

Dissipation.— For reasons which have been set out above, the actual numbers obtained for the dissipation cannot be compared directly with those of other observers, but one is quite safe in saying that they are of the same order as those obtained further south under the same meteorological conditions. They certainly do not show that great increase in dissipation and unipolarity which has been ascribed to places of

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. 81

high latitude by some writers, who base their general conclusions on a few observations made by ELSTEE.*

lonization. — At the time of writing no similar series of observations made with EBERT'S apparatus have been published, so it is impossible to compare the ionization in high latitudes with those in lower. But judging from my own experience, as with the dissipation there is no change in the ionization which cannot be explained by the meteorological conditions. There is certainly no abnormal ionization nor abnormal unipolarity, both the ratios q and r being in excellent agreement with those found in Germany.

The yearly course of the ionization is of great interest and of much importance. What causes the yearly variation is not at first obvious. The ionization of the air at any moment is determined by two factors : firstly, the rate at which ions are produced in the air, and secondly, by the rate at which they re-combine.t The yearly variation of the ionization must be caused by variation in either one or both of these factors. We do not yet know what the ionizing influences at work in the air are ; but possible ones are radio-emanation, the sun's light, and temperature. But none of these undergo a yearly change corresponding to that of the ionization. It will be shown later that the yearly course of the radio-active emanation in the air is exactly opposite to that of the ionization. The sun's light and the temperature both have a yearly course in some agreement with that of the ionization, but the maxima and minima do not agree : the maximum and minimum of the ionization fall two months behind those of the sun's light and one month behind those of the temperature. We should then rather expect to find a cause for the variation by assuming a constant ionizing factor and looking for a change in the conditions which affect the re-combination of the ions. One of the first things which ELSTER and GEITEL found when working at the ionization of the air was that the dissipation depends to a great extent on the clearness of the air. This factor in itself is capable of accounting for the yearly course of the ionization at Karasjok.

All who have travelled in Arctic regions know the peculiar haze which fills the air when the temperature falls very low and gives the " cold " aspect to Arctic scenes. Such a haze, which is not a mist or fog, was frequent during the winter at Karasjok. On the other hand, at the end of the summer the air reached a degree of transparency which I have never seen equalled in any other place. On going into the open air one was often struck with the great transparency of the atmosphere, giving sometimes the impression that the air between one and distant objects had been entirely removed. That it is the transparency of the air rather than the temperature which determines the ionization could often be seen from individual observation. On June 16 the temperature rose to the abnormal value of 247° C., the air being exceedingly hazy and oppressive ; the ionization was only '18, the mean

* 'Phys. Zeit.,' 2, p. 113, 1900.

t SCHUSTER, 'Proc. Man. Lit. and Phil. Soc.,' vol. 48, Part II., p. 1, 1904. VOL. CCV. — A. M

82 ME. GEOEGE C. SIMPSON ON THE

for the month of June being "39. On September 19 the temperature rose to 16-4° C., after having been below 5° for the previous few days; the air again was very hazy and sultry and the ionization went down to '24, the mean for the month being '44. On the contrary, a clear day in the winter would be accompanied by comparatively high value of the ionization : February 22 ionization '40, mean for month, -21. Much to my regret I cannot support this conclusion by actual figures, as Karasjok was so enclosed by low hills that it was impossible to obtain even a rough arbitrary scale of the clearness of the air by the visibility of distant objects. But there can be no doubt that the maximum of the transparency of the atmosphere corresponded with the maximum of the ionization.

Potential Gradient.—The yearly course of the potential gradient in Karasjok conforms to the general rule for the northern hemisphere formulated by HANN* in the following words : " The maximum of the potential gradient occurs in December, January or February ; it falls rapidly in the spring ; remains nearly at the same level during the summer and then rapidly rises again in October and November."

The fact that the potential gradient runs so exactly opposite to the dissipation makes it appear as though there were a constant charge of negative electricity being continually given to the surface of the earth during the whole year, and that the amount at any moment on the surface itself (measured, of course, by the potential gradient) is determined by the rate at which the charge is being dissipated. How this charge is supplied to the earth still remains, in spite of many theories, one of the unsolved problems of atmospheric electricity.

Two types of daily variation of the potential gradient are known, t The first is a double period, having a minimum between 3 and 5 A.M. and a second about midday, the corresponding maxima falling at about 8 A.M. and 8 P.M. Good examples ot this are Batavia and Paris. The other type consists of a single maximum and minimum, the former falling in the evening and the latter between 3 and 5 A.M. To this type belong the records made at high altitudes and at some places during the winter.

The daily course of the potential gradient for the whole year at Karasjok belongs to the latter class, there being only one maximum and one minimum. Taking the four seasons each by itself, we see that the winter and spring curves are of the same type, while that for the summer shows a slight tendency to form a minimum at midday, and the autumn curve has a distinct double period. As stated above, the morning and evening maxima of the autumn curve were considerably strengthened by the mists which formed over the river. The nearest place to Karasjok at which measurements have been made of the potential gradient is SodankylaJ in Finland, and the curves for the two places are in surprising agreement.

* ' Lehrbuch der Meteorologie,' p. 715.

t HANN, 'Lehrbuch der Meteorologie,' p. 716.

} ' Expedition polaire, 1882-83.'

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. 83

It would appear from these results as though the double daily period were confined to places having a large daily temperature variation. The daily variation of the temperature was much the greatest in Karasjok during the autumn three months : the sun not rising during the winter three months, not setting during the summer three months, and the snows still being on the ground during the spring, all tended to keep the daily temperature variations low during these seasons. In all places having a double daily period of the potential gradient the midday minimum is always greater during the summer than during the winter, which supports the same conclusion.

ATMOSPHERIC RADIO-ACTIVITY.

In 1901 ELSTER and GEITEL* made the very important discovery that the atmosphere always contains more or less radio-active emanation. Since the discovery several workers have repeated the observations and confirmed the results. During the whole of 1902 ELSTER and GEITEL t made daily observations of the radio-activity, and found that the amount of emanation in the atmosphere depends largely on some meteorological conditions, such as the rising or falling of the barometer and tempe- rature ; and, as a result of their work, made the suggestion that the emanation in the air is supplied entirely by the radium or radio-active emanation contained in the soil.

The method used by ELSTER and GEITEL to detect and measure the emanation in the air, which has been adopted by other observers, consisted of stretching a wire about 10 metres long between insulators in the open air. This wire was then charged to a negative potential of between 2000 and 2500 volts. After the wire had been exposed to the air at this potential for two hours, it was removed and wrapped round a net cylinder fitting inside the "protection cylinder" attached to their dissipation apparatus (specially closed at the bottom as well as the top for this measurement), and the rate at which the electroscope discharged was determined. When one metre of the wire discharged the electroscope one volt in one hour the atmospheric activity was said to be unity and written A = 1.

Using ELSTER and GEITEL'S method, I made observations of the atmospheric radio- activity in Karasjok. I first started by making odd observations every now and again, but found that the values obtained were so much higher than anything which had up to that time been recorded that I determined to make a thorough investi- gation of the matter. In December, 1903, 1 started a series of observations, observing three times each day. As each observation occupied over two hours, it was impossible to take them so frequently without interfering with my other work, therefore I decided to take three observations each day for a month, then wait a month, then

* ' Phys. Zeit.,' 2, p. 590, 1901.

t ' Phys. Zeit.,' 4, p. 526, 1903.

M 2

84

ME. GEORGE C. SIMPSON ON THE

repeat them the following month, and so on. This was done for the whole year with the exception of the summer months, when observations were made alternate weeks instead of alternate months. Besides the three observations during the day, for one week out of every four I continued the observations during the night, observing between the hours of 3 and 5 A.M.

In order not to interfere with my other observations, the observations of the radio- activity had to fit in between them, and the following times were chosen as being the most convenient : Night observation from 3 to 5 A.M. ; morning observation from 10 to 12 A. M. ; afternoon observation from 3 to 5 P.M. ; and evening observation from 8.30 to 10.30 P.M. In this way it proved possible to get a good idea of the yearly and daily course of the radio-activity. From the 420 separate observations the effect of the different meteorological conditions have been obtained.

As the value of the radio-activity varied very greatly from month to month, in all the following tables each month is treated by itself, and then the whole year treated in a separate column.

TABLE XVI. — Kadio-activity.

	Mean values.	Mean values.	Maximum values.
Early morning, 3-5 A.M.	Morning, 10-1 2 A.M.	After- noon, 3-5 P.M.	Evening, 8.30- 10.30P.M.	Early morning.	Morning.	After- noon.	Evening.
*November ~| and ^ December J	209 (")	87 (2<t)	88 (2*)	131 (22)	129	396	204	384	432
February .	221 («)	72 (23)	113 (2*)	101 (2-*)	127	366	234	342	228
April . . .	87 («)	41 (23)	37 (23)	55 (22)	55	210	120	90	120
May and "1 June j	79 («)	35 (20)	32 (20)	43 (20)	47	204	102	78	108
July and "1 August J	175 (6)	35 (2«)	32 (20)	76 (*>) 80	270	72	93	198
September . ;	201 (6)	81 (is)	70 (is)	142 (18) 123	390	156	122	264
Year . . .	162 (36)	58 (128)	62 (129)	92 (126) , 93	396	234	384	432

* For the observations of this month set out in full detail see 'Boy. Soc. Proc.,' vol. 73, p. 209, 1904.

ATMOSPHEEIC ELECTRICITY IN HIGH LATITUDES.

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Table XVI. gives the mean and maximum values of the activity for each month. From it the yearly course is seen to consist of two periods. During the first, extending from the beginning of September to the end of February, the radio-activity is constant and very high. During the other months the activity is much lower (less than half) and not quite so constant. The maximum falls in midwinter and the minimum in midsummer. A distinct daily period is also shown : the maximum falling in the early hours of the morning and the minimum about midday.

Table XVII. shows the effect of temperature on the radio-activity. It is interesting to notice that from the results for the whole year the temperature appears to have a very marked effect on the radio-activity ; but when each month is taken by itself, the effect is not apparent at all. It would appear from this that temperature only plays a secondary part in determining the amount of activity in the air.

TABLE XVII. — Radio-activity and Temperature.

Temperature.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
"0. <-30	127 (12)	98 («)					113 (23)
- 30 to - 20	166 (10)	126 (34)	—	—	—	135 («)
-20 „ -10	80 (i7)	96 (20)	—	—	—	88 («)
- 10 „ 0 82 (25)	66 (12)	51 (»)	—	—	271 (4) 78 («s)
0 „ 10 110 (8)	—	47 (4li)	33 (41)	62 (33)	100 (44) 63 ("2)
10 „ 20	—	—	—	56 (19)	56 (30)	83 (12) 61 (61)
>20	—	—	—	39 («)	65 (3)	48 O

The relative humidity appears to have a very large effect on the radio-activity, for not only can its influence be seen when the year is taken as a whole, but it is very apparent in each separate month with the exception of February.

TABLE XVIII. — Radio-activity and Relative Humidity.

Relative humidity.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
Per cent. <50		24 (7)	30 (2r)	38 (22)	53 («)	34 W
50 to 60 —	27 (2)	32 (n)	45 (10) 31 (8)	70 (w)	46 («)
60 „ 70 —	54 (ii)	39 (is)	43 (12) 32 (9)	86 (10)	50 (65)
70 „ 80 — 124 (*7)	48 (23)	43 (8) 51 («)	97 H	88 (102)
80 „ 90 >90	, —	90 (") 60 (i)	63 (n) 85 (»)	75 M 40 (2)	143 (9) 170 (8)	156 (n) 196 (8)	106 (6i) 132 (22)

8fi ME. GEOKGE C. SIMPSON ON THE

on

The wind strength has a most direct influence, which can not only be seen in the year and separate months, but can also be detected in nearly all the individual observations.

TABLE XIX.— Kadio-activity and Wind Strength.

Wind (Beaufort Scale).	November and December.	February.	April.	May and June.	July and August.	September.	Year.
0-2	116 (49)	110(68)	65 (32)	57 H	81 (»)	126 («)	98 C267)
3-4 5-6	79 H 63 («)	66 (7) 54 (2)	36 (22) 34 (16)	33 (20) 27 (»)	35 (22) 39 (2)	67 (13) 60 (4)	47 (97) 40 («)
>6	32 (3)	—	21 (3)	10 (•)	20 («)	114 (3)	33 (21)

The radio-activity is greater with a falling than with a rising barometer. The results show this every month without exception.

TABLE XX. — Radio-activity and Barometer.

Barometer.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
Rising .... Falling. . . .	95 (44) 115(23)	97 (34) 119 (34)	38 (29) 53 (40)	25 (26) 53 (40)	50 (42) 77 (23)	107 (27) 110(28)	71 (201) 85 (19°)

But this does not necessarily mean that the radio-activity is greater with a low than with a high barometer. Table XXI. shows that such is not the case. Out of the six separate periods only two, April and May and June, show a regular increase in the radio-activity as the height of the barometer decreases. In the other months, and for the year considered as a whole, no relation appears between the radio-activity and the height of the barometer.

TABLE XXI, — Radio-activity and Height of the Barometer.

Barometer.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
minims. >760 760 to 750	137 (20		73 (14) 104 (is)	30 (2) 39 (1S)	29 (25)	74 (9)	158 (5) 102 (33)	89 (2i; 81 (126	}
750 „ 740 740 „ 730	85 (23 109 ("		93 (29) 146 (2°)	42 (») 65 (23)	44 (32) 57 (9)	53 (86) 50 (9)	104 (28)	65 (174 93 (79
730 „ 720	66(i°		102 (i)	—			—	70 (»

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES.

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An uncertain result is obtained when the observations are divided according to the amount of cloud. For the whole year and for two of the separate months the clouds appear to have a direct influence on the radio-activity, but during the other four months there does not appear to be any relation between the two.

TABLE XXII. — Radio-activity and Clouds.

Clouds.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
0-3 4-7 8-10	130 (18) 107 (26) 76 (2r)	120 (») 124 (17) 96 (2r)	34 (») 61 («) 46 (5»)	30 (<0 49 (13) 39 (46)	117 O 102 (16) 42 («)	172 (7) 96 (2°) 102 (33)	114 ('*) 93 (107) 62 (22«)

The direction of the wind appears to have an influence on the radio-activity, for the latter is at a maximum with a south wind and a minimum with a north wind. It is very questionable if this is a real effect or only a re-statement of the relation between radio-activity and a rising or falling barometer, for every case of a north wind was accompanied by a rising barometer and nearly every case of a south wind by a falling barometer. That it is not the other way about is seen from the fact that observations taken with no wind show an unmistakable relation between the radio-activity and a rising or falling barometer.

TABLE XXIII. — Radio-activity and Wind Direction.

Wind strength.	N:	S.	E.	W.
Greater than 3 on the Beaufort scale . . .	25 (4S)	53 (57)	28 (4S1)	47 («)

No relation between the radio-activity and potential gradient can be detected either in the separate months or the whole year.

TABLE XXIV. — Radio-activity and Potential Gradient.

Potential Gradient.	November and December.	February.	April.	May and June.	July and August.	September.	Year.
Negative potential	33 (6)		41 (4)	24 (!)	42 (2)	137 (2)	49 (15)
0 to 100 volts/metre	106 (w)	148 (18)	51 (2-)	24 (42)	59 (35)	100 (40)	77 (m)
100 „ 200	143 (25)	119 (29)	51 (37)	34 («)	61 (28)	134 (15)	86 (166)
200 „ 300	90 (10)	90 (10)	32 (4)		—	—	81 (24)
300 „ 400	71 (10)	64 (•)	51 (2)	—	—	—	66 (18)
>400	83 (3)	61 (16)	31 («)	—	—	—	58 (24)

88 MR. GEORGE C. SIMPSON ON THE

I found it impossible to make observations of the ionization and dissipation at the same time as those of the radio-activity. This is much to be regretted, as it is very important to decide if the emanation in the atmosphere is the cause of the permanent ionization. That the ionization does not depend on the amount of emanation alone is quite clear from the yearly variations of the two, for the ionization is at a minimum during the winter, exactly the season when the activity is at its maximum. But that does not prove that the ionization is not due to the emanation ; we can only say that if it is, then the increase in the production of ions owing to the excess of emanation is overbalanced by the increased rate of recombination due to the winter conditions.

That all the relations shown by the above analysis should be as they are gives an exceedingly strong support to ELSTER and GEITEL'S theory of the origin of the atmospheric radio-active emanation. According to their theory, the air which is mixed up with the soil of the ground becomes highly charged with radium emanation.* When the barometer falls, this air passes out of the ground into the atmosphere, bringing with it its charge of emanation.

All the facts of the above analysis receive a very simple explanation by this theory if one extends it to include the fact that, as the emanation is a gas contained in the soil, it must constantly diffuse into the atmosphere above quite independently of the state of the barometer. Assuming this constant diffusion, we at once see that everything which tenth to reduce the atmospheric circulation, i.e., to keep the air stagnant, tends a/so to increase the quantity of emanation in the lower layers of the atmosphere.

Looking now at each of the tables in order, we see that the temperature does not have a direct, but an indirect influence on the radio-activity. This is explained by the fact that the low temperature of the winter produces a nearly permanent temperature inversion, as mentioned above, which entirely prevents ascending currents of air. Thus the emanation on leaving the ground in cold weather cannot rise, but collects in the lower atmosphere, causing the high winter values of the radio- activity.

The reason why the radio-activity is high with high relative humidity is easily found when one considers that each evening, as the temperature rapidly falls, two things happen : first there is a rapid rise in the relative humidity and secondly ascending currents of air are cut off. The latter fact gives rise to the high radio- activity. Also a mist or fog is always a sign of stagnant air.

A high wind is naturally accompanied by low activity, for it acts as a stirrer, and rapidly mixes the escaped emanation with a large volume of air.

ELSTER and GEITEL'S theory explains the relation found between radio-activity and a rising and falling barometer. If air stream out of the ground when the barometer falls, it must charge the atmosphere with its emanation.

* 'Phys. Zeit.,1 5, p. 11, 1904; 'Terr. Mag.,' vol. 9, p. 49, 1904.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. 89

The effect of the clouds is not easy to understand, and as the results do not show a pronounced dependence of the radio-activity on the clouds, perhaps it is not real, unless it is that clouds are usually associated with ascending currents of air. Other observers* have found a relation between radio-activity and clouds, but until more observations are made the question must be left unsettled.

Thus we see that the whole effect of the meteorological conditions on the radio- activity depends on whether those conditions tend to mix the emanation rapidly with a large volume of air, or to keep it near to the ground from which it is always escaping.

The same principles lead us to an explanation of the daily and yearly course of the radio-activity. During the daytime ascending currents are formed, while as the evening approaches these stop and the air lies cold and stagnant during the night. Thus we see why the minimum should be in the daytime and the maximum during the night.

The yearly period has a similar explanation. During the winter in Karasjok, when the snow is permanently on the ground, temperature inversion accompanied by stagnant air is the rule rather than the exception. On the contrary, during the summer when there is nearly permanent sunshine, ascending currents will be formed at all times of the day and night. This accounts for the winter maximum and the summer minimum of the radio-activity.

One strange fact is that the activity should be so high during the winter when the whole country is covered with snow. This at first led me,t with other observers,^ to doubt ELSTER and GEITEL'S theory, but the reason is not hard to find if it is remembered that the snow must form a very large reservoir to hold the emanation as it is escaping from the soil. It would be interesting to see if air, drawn from the snow in the way ELSTER and GEITEL drew it from the ground, would be charged with emanation. I wished to test this, but had no instruments with me which could be used for the experiment.

One would also expect high values of the radio-activity in Karasjok during the winter from another consideration. Karasjok is situated on the river, and just as the water from all the surrounding land flows down to the river, so when the temperature falls very low the cold air will also find its way into the river valley. This cold air flowing over the ground will sweep the emanation along with it, and so the valley will become filled with air highly charged with emanation.

In order to find if the minerals of Karasjok are particularly rich in radio-active constituents I sent samples of sand and rock to the Hon. R. J. STRUTT, who very kindly undertook to test them, and to whom my best thanks are due for the trouble he took in his investigation of them. In none of the specimens was he " able to

* GOCKEL, ' Phys. Zeit.' ; ZOLSS, ' Phys. Zeit.' t ' Roy. Soc. Proc.,' vol. 73, p. 209, 1904. I ' Phys. Zeit.,' 5, p. 591, 1904. VOL. CCV. — A. N

90 MR. GEORGE C. SIMPSON ON THE

detect the emanation with certainty, and none yielded more than a 100|000 part of •what the same quantity of pitchblende would give on heating." Thus the soil conditions of Karasjok do not appear to be abnormal, so that the high radio-activity found there during the winter must be due to the meteorological conditions being so favourable to the collection of the emanation in the lower atmosphere.

In order to compare the value of the radio-activity at Karasjok with that of other places, the only observations which can be used are ELSTER and GEITEL'S,* in Wolfenbiittel (mid-Germany), and GOCKEL'S,! in Freiburg (Switzerland) ; no other observer has extended his observations over a sufficiently long period to give good mean values. Neither ELSTER and GEITEL nor GOCKEL observed between 8 P.M. and 8 A.M., and as the values I found between those hours were very much the largest it is not right to compare my means with their means, so in what follows I use only the values which were obtained during the morning and afternoon observations in Karasjok.

The means for the whole year are Wolfenbiittel 18 '6, Freiburg 84 and Karasjok 60. Thus Freiburg is the highest and Wolfenbiittel the lowest. The absolute maxima (between 8 A.M. and 8 P.M.) are Wolfenbiittel 64, Freiburg 420, Karasjok 384, i.e., the same order as before.

It is a strange fact that the yearly period should be so marked in Karasjok, while no yearly period can be detected in either Wolfenbiittel or Freiburg. As stated above, neither ELSTER and GEITEL nor GOCKEL have observed after 8 P.M., so it is impossible to compare the daily periods. It would be exceedingly interesting to know if there is a large daily variation in mid-Europe, for if there is not, then the mean winter value of the radio-activity in Karasjok will be very high compared with mid-Europe, the mean for the winter, when night as well as day observations are taken into account, being 126 at Karasjok.

GOCKEL'S maximum observation of 420 was quite an exception, but even that was exceeded by my absolute maximum of 432 (observed between 8 and 10 P.M. on December 17). With this one exception the values found by GOCKEL did not exceed 170, while I found 200 quite a medium value during the winter in Karasjok. It would appear, from the results which have already been published, that high values of the radio-activity are much more common in Karasjok than in any place yet investigated.

ELSTER and GEITEL measured the radio-activity at Juist, an island in the North Sea, and found it only 6, while in the Bavarian Alps they found the high value of 137. From this, and their observations in Wolfenbiittel, they concluded that the radio-activity increased from the sea inland. In order to find if the same conditions held in the north, I stayed in Hammerfest on my way home, and made daily observa-

* ' Phys. Zeit.,' 4, p. 526, 1903. t ' Phys. Zeit,,' 5, p. 591, 1904.

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91

tions of the radio-activity there for four weeks (October 17 to November 12), in exactly the same way as I had done in Karasjok.

The result was in entire agreement with ELSTER and GEITEL'S observations. The mean for the month was only 58, which must be compared with the Karasjok winter value of 126, the numbers for the different times of observing being—

TABLE XXV. — Radio-activity in Hammerfest.

	Early morning, 3-5 A.M.	Morning, 10-12 A.M.	Afternoon, 3-5 P.M.	Evening, 8.30-10.30 P.M.	Mean.
Mean	97 (°)	33 (24)	50 (24)	52 f-'4)	58
Maximum . . .	204	156	252	-1-1 \ 1 150

But what is much more interesting and important is the great variation of the radio-activity with the wind direction. When it is remembered that Hammerfest is free to the open ocean on the north and west, while to the south lies the whole stretch of Norway and Sweden, the following table tells its own story : —

TABLE XXVI. — Radio-activity and Wind Direction in Hammerfest.

	North.	South.	West.
Mean	8 (10)	72 (30)	4 (io)
Maximum	20	250	10

It must be admitted that these results lend great support to ELSTER and GEITEL'S hypothesis.

OBSERVATIONS OF THE AURORA.

It was not my intention on going north to make a particular study of the aurora, but I naturally followed it with as much attention as possible. The necessity of making my regular observations during the daytime, beginning at 7.30 A.M., made it impossible to stay up to watch the aurora late into the night. Each evening I noted down the chief variations in the aurora's form and brilliancy, but did not go into minor details. I intend here to shortly record a few of the things which struck me, and which are rather of a general than particular interest.

During the year of my stay there were not many exceptionally fine auroras, and coloured auroras were very rare. From the one or two I did see the colours appeared to be of two distinct kinds (by colours in this connection I mean colours other than the greenish-white light of the ordinary aurora). There is first the mass of coloured

N 2

92 ME. GEORGE C. SIMPSON ON THE

light which retains its form and colour for a comparatively long time, and colours which flash out and disappear immediately. A very interesting fact struck me with regard to the latter class of colour. It is generally known that an aurora arch is often composed of a series of spear-like shafts of light arranged perpendicularly to the direction of the arch, and which appear to be in constant motion. A number of these spears will suddenly become brilliant and the lower ends shoot out of the arch into the black sky below. The brilliancy will then run along the arch like a wave of light, lighting up all the spears as it goes along. I noticed that the " front" of such a wave of brilliancy and the points of the spears when shooting out were bright red, but as soon as the motion stopped the colour disappeared, while the more violent the motion the purer and brighter the red. It appeared as if some physical process accompanied the passage of the aurora beam through the air and gave out a red light. For example, if the air had to be ionized before the discharge could pass through, then the process of ionization produced red light. If the motion was particularly violent, the production of red light would be followed by a production of brilliant green light, so that if a bright wave passed along an arch two waves of colour would appear to travel along, first a wave of red light, closely followed by a green wave, the two travelling so closely together as to appear one wave having a two-coloured crest. Similarly spears shooting out with a great velocity would appear to have red and green tips.

The question of the relation of clouds to auroras has been very often raised. Three of my observations bear on this point.

On the evening of October 11, 1903. after a fairly active display, the aurora disappeared ; but its place was taken by a system of narrow bands of cirrus clouds stretching right across the sky, which, being illuminated by the bright moon, had all the appearances of the aurora. That they did not form part of the aurora could only be decided at first owing to no line appearing in the spectroscope when pointed at them ; but later there could be no doubt, as they partly obscured the moon.

On October 26 a very similar phenomenon again appeared ; that which at first was taken to be aurora later turned out to be cloud.

On December 13 the most brilliant aurora display of my stay took place. The whole display reached a climax at 9.45, when a most brilliantly coloured corona shot out from the zenith. While this final brilliant display was taking place the sky suddenly became thinly overcast, and the aurora was only visible later as bright patches through the clouds.

It has long been a matter of controversy as to whether the aurora ever extends into the lower regions of the atmosphere. Several observers positively affirm that they have seen it quite close to the ground. This may be due to an optical illusion ; one evening I was, for a considerable time, in doubt as to whether the aurora was really under the clouds or not. All over the sky were detached clouds, the clouds and spaces between them being of about the same size and shape. Right across the

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. 93

sky a long narrow aurora beam stretched, showing bright and dark patches owing to the clouds. It looked exactly as if the aurora beam ran along under the clouds, brightly illuminating the patches of cloud which it met. In reality the bright patches were the openings and not the clouds. It took me a long time to make quite certain of this, and it was only by at last seeing a star in the middle of a bright patch that I could be quite certain.

LEMSTROM strongly supported the idea that the aurora often penetrates down to the earth's surface, and described how on one occasion the aurora line appeared in a spectroscope pointed at a black cloth only one or two metres away. I was able to repeat this observation on several occasions, and found that the line which then appeared in the spectroscope was not due to an aurora discharge in the air between the spectroscope and the black cloth, but was due to reflected light, which it was impossible to prevent entering the spectroscope, as the whole landscape was lit up with the monochromatic light of the aurora.

All the time I observed the aurora I could not detect the slightest noise accom- panying the discharge.

I cannot close this account of my work in Lapland without expressing my deepest thanks to each and every one of the small Norwegian colony in Karasjok — in particular to my host and hostess, Handlesmand arid Fru NIELSEN ; and to Lensmand and Fru HEGGE — all of whom did their very best to make my stay amongst them a source of the greatest pleasure and real enjoyment.

APPENDIX.

Potential Gradient. — The potential gradient was measured, as stated in the paper, by means of a Benndorf self-registering electrometer. The electrometer is of the quadrant type, the quadrants being kept at a constant voltage by means of small cells, and the needle itself connected to the collector. To the bifilar suspension of the needle a long aluminium arm is attached, which swings freely above a strip of paper drawn along by clockwork. Every two minutes an electrical contact is made which causes a bar to descend and to press the end of the aluminium arm down upon the paper, where a dot is left showing the position of the arm and so the potential gradient. The zero of the instrument was so arranged that on the normal side a potential gradient of 800 volts/metre could be registered. On the negative side only 100 volts/metre could be registered; but as all the days on which negative potential gradient occurred were disturbed days, and the results on such days not used, the range was quite great enough.

The collector was arranged in the following way : — My bed-sitting room in which

94 MR. GEOEGE C. SIMPSON ON THE

I had my instrument was a little hut near to my host's large <' handlesmand's " house. On the end of the large house was a flag staff, to the top of which I attached an insulator and from it took a wire through a window into my room. About a third of the way up the wire I attached two milligrams of radium bromide which acted as a collector. On the accompanying photograph, the insulator, wire and the position of the radium collector are shown. The height of the collector above the ground was 5| metres. This arrangement acted extremely well and, as far as I could judge, gave as good results as could be wished.

The potential gradient was reduced to that over a level surface by making simultaneous observations with a flame collector and leaf electroscope above the most level piece of ground I could find. The country was so rough that a good and accurate determination could not be made, but the error is certainly not 20 per cent. During the year this reduction was several times repeated, no change being found. Great attention was also paid to the insulations, which were never found defective. As the collector was situated between two houses over a much frequented road, no accumulation of snow took place under it, so corrections due to the height of the snow were not necessary.

Dissipation and lonization. — In order to observe the ionization and dissipation without being disturbed by the smoke of the village, two platforms (as shown in the photograph) were built at different parts of the village, but as both were to the north of a large part of the village, I could not observe when a south wind was blowing ; with all other winds one of the platforms was on the windward side of the houses. The platforms were about a metre above the ground and the instruments on a shelf about a metre and half over the platform ; above all was a roof to protect the instruments from rain and snow. By this arrangement the instruments were exposed to the full force of the wind. In order to read the dissipation electroscope in a high wind, a small screen was held to protect the instrument just at the moment of observation.

The usual method of observing the dissipation or ionization is to charge the electroscope, take a reading, then return in 15 minutes and take another reading. This method is open to great objections : first it is quite easy to make a false reading, and secondly in open-air work the leaves are not steady enough to allow of one reading being accurate. The method I adopted was to charge the two instruments, then take a reading of the dissipation instrument, half a minute later a reading of the ionization instrument, then at the minute take another reading of the dissipation instrument, at the next half minute a second reading of the ionization instrument, and so on for 5 minutes, when of course I had five readings on each instrument. Ten minutes later I started reading again, and at minute intervals read each of the instruments five times, then from a table found the value corresponding to each of the readings, took the mean of the first five, then that of the second five, and used these means as single values separated by an interval of 1 5 minutes. In this way

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96 ME. GEOEGE C. SIMPSON ON THE

errors of reading were avoided and errors due to the unsteadiness of the leaves

greatly diminished.

After having by this method obtained a measurement, using, say, a positive charge, the observation was repeated using a negative charge, and finally another observation with a positive charge. The mean of the two positive values, together with the negative value, were used as the result of the whole observation. This method I found to be absolutely necessary if reliable values of the ratios q and r were to be obtained, for both dissipation and ionization undergo great changes in the course of the time taken to make an observation. A whole observation when taken in this way occupied an hour and a quarter.

Long experience taught me to know when I could expect difficulties with the insulations. On such days, instead of the method sketched above, an observation was taken with one charge, and after that the insulation tested for 15 minutes, then an observation with the other charge, followed by a final insulation test for the same length of time, the whole observation taking about an hour and a half.

During the summer I had great difficulty in using the Ebert instrument owing to the mosquitoes being drawn into the instrument and so discharging the electroscope. In June the mosquitoes and other small flies were so numerous that it was quite impossible to use the Ebert instrument without some means of keeping the flies out, so I attached a funnel-shaped net to the front of the aspirator tube and used the instrument so protected. I expected that this net would cause some reduction in the value of the ionization as measured by the instrument, so as soon as the mosquitoes were sufficiently reduced in number to allow of observations being made with the unprotected tube I made a series of observations to find the effect of the net. Much to my regret and disappointment I found that the effect of the net varied very much according to the wind strength. In perfectly still air the net reduced the ionization by nearly a quarter, while with a stiff" breeze it had no effect. This made individual observation practically useless, and in all the above tables connecting ionization and the meteorological elements all the observations taken when the net was in use — from June 9 to August 12 — have been neglected. As the result of a long investigation I concluded that 10 per cent, added to the results in the bulk would just about correct for the effect of the net. Eesults so corrected are used in the curves and tables showing the yearly course of the ionization.

Radio-activity. — In my measurements of the radio-activity, as stated above, ELSTER and GEITEL'S method was used. In order to charge the wire to a negative potential of between 2000 and 2500 volts, I used a small influence machine, built on the principle of a Kelvin replenisher and driven by a falling weight. By means of a variable high resistance, consisting of a strip of ebonite, one side of which had been rubbed with a black-lead pencil and so mounted in a tube that an earth -connected pad could move along it, the potential of the wire could be very easily regulated.

ATMOSPHERIC ELECTRICITY IN HIGH LATITUDES. 97

The instrument worked splendidly, and I had very little trouble with it. Its only drawback was that it could not be left to itself for more than 20 minutes, for then the weight required winding up again, and sometimes the voltage would vary if not attended to.

For my insulators I used amber enclosed in a metal case, so designed that air had a long way to travel from the oiitside before it could reach the amber. The insulators acted very well even in rain and fog. It was seldom that I had any difficulty with them, and I was never compelled to give up an observation on account of insulation troubles.

Meteorological Measurements. — Karasjok is a second-order station of the Norwegian Meteorological Service, and 1 was granted full use of the observations made there. I depended solely on these observations for the height of the barometer. For temperature and relative humidity I used an instrument (the polymeter) made by LAMBKKUHT, Gottingen. The hygrometer consists of a bundle of human hairs mounted in such a way that a pointer is made to move over a very open scale showing directly the relative humidity. About once a month the zero was tested by painting the hairs with water in the way usual with such instruments. Treated in this way the instrument proved quite reliable. A mercury thermometer was attached to the metal frame of the hygrometer. The instrument was hung outside the window of a porch, the door of which always stood open. In this way the thermometer was not influenced by the radiation from a warm room, and. as the window looked north, the sun did not shine on it during the day time. The metal back of the instrument prevented the thermometer reading too low on a clear evening. I had no instrument for measuring the wind strength and had to estimate it as well as possible from the "feel." As the wind strength is only used qualitatively, the absolute values are of little importance.

VOL. ccv. — A. <>

[ 99 ]

IV. The Halogen Hydrides as Conducting Solvents. Part I. — The Vapour Pressures, Densities, Surface Energies and Viscosities of the Pure Solvents. Part II. — The Conductivity and Molecular Weights of Dissolved Substances. Part III. — The Transport Numbers of Certain Dissolved Substances. Part IV. — The Abnormal Variation of Molecular Conductivity, etc.

By B. D. STEELE, D.Sc., D. MC!NTOSH, M.A., D.Sc., and E. H. ARCHIBALD, M.A.,

Ph.D. (late 1851 Exhibition Scholars).

Communicated by Sir WILLIAM RAMSAY, K.C.B., F.lt.S. Received February 1, — Read February 16, 1905.

PART I.

The Vapour Pressures, Densities, Surface Energies and Viscosities of the Pure Solvents. By D. MC!NTOSH and B. D. STEELE.

ALTHOUGH our knowledge of the ionising power of non-aqueous solvents has been considerably increased during recent years by the investigations of WALDEN, FRANKLIN, KAHLENBERG, and others, the liquefied halogen hydrides and sulphuretted hydrogen have received little or no attention.

GORE (' Phil. Mag.' (4), 29, p. 54), who experimented at ordinary temperatures, found that the hydrides of chlorine, bromine, and iodine were very feeble conductors. BLECKRODE ('Pog. Ann.' (2), 23, p. 101) stated that hydrogen bromide conducts slightly; while HITTORF ('Pog. Ann.' (2), 3, p. 161, 4, p. 374, considered these substances to be non-conductors.

With regard to their behaviour as solvents, SKILLING ('Amer. Ch. Jl.,' 1901, 26, p. 383) found that at ordinary temperatures sulphuretted hydrogen dissolves potassium chloride freely ; but that the solution is a non-conductor of electricity.

HELBIG and FAUSTI (' Zeit. fur angewandte Chemie,' 1904, 17) state that stannic chloride is soluble in hydrogen chloride, but that this solution also is a non-conductor.

As it seemed highly improbable to us that sulphuretted hydrogen, which is analogous to water in so many ways, should be devoid of dissociating power, we decided to investigate its solvent action systematically, and at the same time to examine the hydrides of chlorine, bromine, iodine, and phosphorus.

VOL. CCV.— A 390. O 2 23.8.05

100 DR B. D. STEELE, DR. D. McINTOSH AND DE. E. H. ARCHIBALD

Preliminary Experiments.

It has been found, as a result of our preliminary experiments, that water and all the ordinary metallic salts which were tried are insoluble, or very sparingly soluble, in any of the solvents.

Hydrogen chloride and bromide are freely soluble in hydrogen sulphide, and hydrogen sulphide in hydrogen bromide.

The salts of the organic ammonium bases are soluble in hydrogen chloride, bromide, iodide, and sulphide, and the resulting solutions conduct the current. Certain ammonium salts also yield very feebly conducting solutions. Two metallic salts, namely, sodium acetate and potassium cyanide, were, at first, thought to be soluble, as their addition to the solvent greatly increased its conductivity. This has since been found to be due to decomposition of these salts into acetic acid and hydrocyanic acid respectively. Both of these acids are soluble in the foregoing solvents.

No substance has yet been found which will dissolve in phosphine and yield a conducting solution.

A few preliminary measurements of the conductivity were made, and in every case the molecular conductivity diminished considerably with dilution, instead of increasing as it does in aqueous solutions. The results of these measurements are given in Part II., which contains a detailed account of the measurements of solubility and of conductivity.

After we had ascertained that the hydrides of chlorine, bromine, iodine, and sulphur can act as conducting solvents, we proceeded to the measurement of the following physical constants of each of the pure substances : —

(1) The vapour-pressure curve ;

(2) The density and its temperature coefficient ;

(3) The surface energy and its temperature coefficient ;

(4) The viscosity and its temperature coefficient.

The results of these measurements are described in the following pages.

Preparation of Liquefied Gases.

Hydrogen chloride was prepared by the action ol sulphuric acid on pure sodium chloride. The gas was dried by passing it through two wash bottles containing sulphuric acid, and afterwards through a tube containing phosphoric anhydride. It was then led into a receiver which was maintained at -100°, by means of carbon dioxide and ether, under diminished pressure. At this temperature the gas liquefied rapidly, forming a colourless mobile liquid. This was re-distilled before being used for the measurements.

ON THE HALOGEN HYDKIDES AS CONDUCTING SOLVENTS. 101

The hydrogen bromide was prepared by the action of bromine on red phosphorus suspended in water. Traces of bromine were removed by passing the gas through a thin paste of amorphous phosphorus and a saturated solution of hydrogen bromide. The gas was then dried by passing it over about 40 centims. of phosphoric anhydride, and, in order to remove impurities other than water- vapour, it was passed through two U-tubes surrounded by solid carbon dioxide, in each of which a small quantity of liquefied gas soon collected. The gas bubbled through this liquid, which was thus submitted to a process of fractional distillation. It was finally condensed in a vessel surrounded by a mixture of carbon dioxide and ether.

The hydrogen iodide was made by the action of iodine and water on amorphous phosphorus, in a similar manner to that employed for the preparation of hydrogen bromide, and similar means were used to purify it. The liquid was invariably coloured, and it could not be obtained quite colourless even by repeated distillation.

The hydrogen sulphide was prepared by the action of dilute sulphuric acid on ferrous sulphide. The gas was washed by passing it through water, dried by passage over phosphoric anhydride, and condensed by means of carbon dioxide and ether. It was purified by distillation.

Phosphuretted hydrogen was prepared by the action of a solution ot potassium hydroxide on phosphonium iodide. It was dried by means of phosphoric anhydride and condensed in a receiver which was immersed in liquid air.

The Constant-temperature Bath.

The constant-temperature bath consisted of ether which was contained in a vacuum vessel and cooled by liquid air. The temperature was measured by a constant-volume hydrogen thermometer, similar to that described by TRAVERS, SENTER, and JAQUEROD ('Phil. Trans.,' 1902, A, 200, pp. 105-180). The arrangement of the apparatus is shown in fig. 1 , in which C represents the hydrogen thermometer, A the large vacuum vessel containing the ether, and B a large vacuum flask containing liquid air. The bulb, a, of the thermometer was connected to the dead space of the manometer by a fine capillary tube. A mercury reservoir was attached to the stop-cock k by rubber tubing, and by raising or lowering this reservoir the mercury in the dead space could be adjusted to the level of the glass point c.

The volume of the thermometer bulb and dead space was carefully determined by calibration with mercury. The constants were volume of—

(1) Bulb and portion of stem within the liquid = 17 '480 cub. centims. at 0° ;

(2) Stem from s to surface of ether = 0'1358 cub. centim. ;

(3) Dead space and stem to mark s = 0'5719 cub. centim.

It has been assumed, in making our calculations, that the average temperature of the section (2) was midway between that of the bath and that of the atmosphere ; an error of a few degrees in the temperature of this section is without influence on the bath temperatures, which are given only to the nearest tenth of a degree.

102

DE. B. D. STEELE, DK. D. McINTOSH AND DR. E. H. ARCHIBALD

Fig. 1.

The vacuum vessel A was closed by a large indiarubber stopper, through which holes had been cut to allow the passage of the stem of the thermometer, the apparatus containing the liquefied gas, and the tubes I), d, and /.

The closed tube b was about 7 millims. in diameter and long enough to reach nearly to the bottom of the vessel. The tube d was placed so that its open end came immediately under the tube b.

The large vacuum flask B was fitted with tubes as shown in the diagram, so that by blowing into / liquid air could be forced into the tube b.

In order to obtain any desired temperature between that of the room and the melting-point of ether ( — 117°) the vessel A was filled with ether, and the tube d connected to an air blast, by means of which the liquid was continuously and uniformly stirred, the air escaping through the tube I, which was provided for the purpose.

After mercury had been taken out of the manometer through k, the ether was cooled by blowing liquid air from B into b, where it rapidly boiled away. When the temperature of the bath had been adjusted, it could be kept constant for as long as was desired by blowing liquid air in very small quantities into b.

The deposition of dew on the walls of the vacuum vessel A was prevented by placing it inside a wider cylindrical glass vessel containing phosphoric anhydride.

ON THE HALOGEN HYDEIDES AS CONDUCTING SOLVENTS.

103

Ilie Vapour-pressure, Curves,

In order to measure the vapour-pressure, a tube containing the liquid was immersed in the bath and simultaneous observations of the temperature and corresponding vapour-pressure were taken.

This simple arrangement could not be used with hydrogen bromide and iodide on account of the action of these gases on the mercury of the manometer. The errors due to this action were avoided by the use of a special form of apparatus which is shown in fig. 2.

To use this apparatus, the tube in was attached to the pump, and the bulb a placed in the low-tempe- rature bath, after which the whole apparatus was exhausted to a pi*essure of about 60 millims. and the stop-cock h closed.

The outer portion of the apparatus was completely exhausted and the stop-cock cj closed. A vessel containing the liquefied gas was then attached to n by rubber tubing, the point I broken within the tube, and as soon as a sufficient quantity of liquid had distilled into the bulb a the apparatus was sealed off at k.

Before making any measurements the stop-cock g was opened for a few moments, and all traces of air were displaced from the tube by allowing a small quantity of liquid to evaporate into the pump.

The bath was cooled to the lowest temperature at which observations were to be taken, and the stop- cock h opened. As the vapour-pressure of hydrogen bromide and of hydrogen iodide, even at the lowest temperatures employed, was greater than 60 millims., a flow of gas from c into c' followed, and continued until the pressure in the manometer became equal to the vapour-pressure of the liquid. As soon as the pressure ceased to rise, the temperature and pressure were read and the stop-cock h was immediately closed. The temperature was then raised to the next point of observation, and the stop-cock h again opened, until the pressure in the manometer became constant, when readings were again taken and the stop-cock closed.

In this way a succession of readings was obtained without the hydrogen generated by the action of the gas on the mercury of the manometer finding its way into the bulb a. As a precaution against diffusion, the bulbs c, c and the capillary tube e were introduced, the stop-cock g being opened after each observation and the contents of the bulbs c and c withdrawn through the pump.

Fig. 2.

104 DB. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

The formation of hydrogen was reduced to a minimum by the device of leaving in the manometer a small quantity of air, which prevented the hydrogen bromide (or iodide) from reaching the surface of the mercury until a considerable time had elapsed.

-60

ZOO

400 600

FffESSURE //V MILLIMETRES OF AfffiCL/ffY

Fig. 3.

The results of the measurements are collected in Table I., which contains the experimental (a) and smoothed (b) values of the vapour-pressure for each of the liquefied gases.

In the case of hydrogen bromide and iodide the measurements have been continued considerably below the melting-point, and the vapour-pressure curve both for solid and for liquid are given in fig. 3. It will be noticed that the change in curvature at the melting-point is very slight both for hydrogen bromide and for hydrogen iodide.

The melting- and boiling-points found by us for the pure substances are given in Table II., together with recent measurements by other observers.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

TABLE I.

105

Temperature.	Vapour pressure.	Temperature.	Vapour pressure.
a.	b.	a.	b.
•0.	miUima.	millims. HYDROGEN	0 C. milliins. CHLORIDE.	inillima.
-80-0		896	- 95	_	363
-80-5	868	—	- 96		343
-81	—	851	- 97		323
-82	—	808	- 97-2	31G	—
-83	—	764	- 98	—	304
-83-2	748	—	- 99	. —	287
-84	. — _	718	-100	—	270
-85	—	673	-101		254
-85-9	648	— .	-101-3	245	—
-86	—	632	-102	—	238
-87	—	594	-103	—	225
-88	—	557	-104	—	210
-89	—	552	-104-5	198	—
-89-8	522	—	-105	—	19G
-90	—	493	-'106	—	184
-91	. —	463	-107	—	173
-92	—	435	-108	—	102
-92-9	430	—	-109	—	149
-93	—	410	-109-9	141	—
-94	—	385	-110	138
HYDROGEN IODIDE.
-35	_	783 - 56		274
-35-9	769	- 57	—	258
-36	—	750 - 58	—	244
-36-9	713	—	- 59	—	230
-37	—	718	- 59-5	224	—
-38	—	686	- 60	—	218
-39	—	657	- 61	—	206
-39-4	644		- 62	—	194
-40	—	628	- 63	—	183
-41	—	600	- 63-5	185	—
-41-7	578	—	- 64	—	173
-42	—	573	- 65	—	162
-43	—	547	- 66	—	152
-43-5	530	—	- 67	—	143
-44	—	519	- 68	—	134
-45	—	494	- 68-4	126	—
-46		471	- 69	—	126
-47	—	448	- 70	—	118
-47-7	438	—	- 71	—	111
-48		425	- 72	—	103
-49	.	404	73	—	97
-50	376	. .	- 73-5	92	—
-51		364	- 74	—	90
-52		343	- 75	—	84
-53		325	- 76	—	79
-54		307	- 77	—	73
-54-8	303		- 77-9	74	. —
-55		289 - 78	—	70

VOL. CCV. — A.

106

DE. B. D. STEELE, DR. D. McINTOSH AND DE. E. H. AECHIBALD

TABLE I. (continued).

Temperature.	Vapour pressure.	Temperature.	Vapour pressure.
a.	6.	a.	b.
°c.	milliiiis.	millims. " C. millims. HYDROGEN BROMIDE.	millims.
-65			- 87	_	283
-66	—	891	- 87-1	284
-67	—	835	- 88 —	266
-68	—	785	- 89		259
-68-4	775	—	- 89-3	245
-69	—	743	- 90		247
-70	—	704	- 91	239
-70-7	682	—	- 92 —	222
-71	—	671	- 92-8	214
-72	—	635	- 93		214
-73	609	- 94		204
-74	575	—	- 95		195
• -75	—	546	- 96		187
-76	—	519	- 96-3	185
-76-7	501	—	- 97		177
-77	—	483	- 98		167
-78	—	468	- 99		157
-79	—	445	-100		147
-79-3	431-5	—	-100-7	142
-SO	—	423	-101		136
-81	—	402	-102		125
-82	—	381	-103		114
-83	357	—	-104		102
-84	—	340	-104-2	96
-85	—	321	-105		90
-86	~	302
SULPHURETTED HYDROGEN.
-60	—	770	- 74		345
- 61 n f*	—	724	- 75		326
- 62	—	682	- 75-6	314
-62-2	676	—	- 76		309
- 63	—	644	- 77		292
-64	—	607	- 78		276
- 65	—	573	- 78-4	270
- 66 -66-1 nrj	538	541	- 79 - 80		261 246
- 67 -68 C(\	—	513 484	- 81 - 81-7	220	232
— 69 -69-1 -70 n"	456	458 432	- 82 - 83 - 84	193	218 205
-71-6 *7O	400	409	- 85 - 86	—	181 169
— IZ -73	—	384 364	- 87 - 88	—	158 148

ON THE HALOGEN HYDKIDES AS CONDUCTING SOLVENTS.

107

TABLE I. (continued).

	Vapour pressure.		Vapour pressure.
Temperature.		Temperature.
	a.	b.		a.	b.
°C.	million.	ruillinis. PHOSPHURETT	°C. ED HYDROGEN.	millims.	millims.
-86		770	- 97		403
-86-6	719	—	- 97-7	393	—
-87	— .	716	- 98	—	382
-88	—	668	- 99	—	362
-88-6	644	—	-100	—	342
-89	—	630	-101	—	324
-90	—	595	-101-2	319	—
-91	. —	563	-102	—	305
-92	—	531	-103	—	287
-93	—	503	-104	—	269
-93-1	498	—	-105	—	253
-94	—	473	-105-9	237	—
-95	—	448	-106	—	235
-96		425
	1

		TABLE	II.
	HCl.	HBr.		H,S.	PHS.
r Melting-point< I l§	-111-1 -111-3	-86 -88-5 -86-1	-50-8 -50-8 -51-5	-82-9	—
f* Boiling-point < I L§	- 82-9 - 83-7 - 83-1	-68-7 -64-9 -68-1	-35-7 34-1 36-7	-60-2 60-4	86-4 -85

* MclNTOSH and STEELE.

t ESTREICHER, ' Zeit. Phys. Chem.,' 1896, 20, p. 605. J LADENBERG and KRUGEL, ' B. B.,' 1900, 33, p. 637. § OLSZEWSKI, ' Monatshefte fiir Chemie,' 7, p. 371.

Heats of Evaporation.

CLAUSIUS has shown that the heat of vaporisation of a liquid can be calculated

from the equation

dp ^

~~

^

dT ~~ KT2 ' p 2

108

DE. B. D. STEELE, DE. D. McINTOSH AND DE. E. H. AECHIBALD

TO PUMP

in which -P- represents the change of vapour-pressure with temperature, P the

€(/ A

pressure, T the absolute temperature, R the constant of the gas equation, and W the latent heat of evaporation of one gram-molecule of the liquid.

The values of W at a pressure of 760 millims., as calculated from our vapour- pressure curves, are

for hydrogen chloride, 14'8 x 1010 ergs,

bromide, 17'4xl010 „

iodide, 207 xlO10 „

sulphide, 19'3xl010 „

„ phosphuretted hydrogen, 17 '2 x 1010 ,,

We can find no account of any direct determination of W for these substances.

The Measurement of Density.

The apparatus (fig. 4) employed for these measurements consisted of a bulb with a graduated capillary stem, to which a two-way stop-cock c was attached.

The bulb had a capacity of about 1*5 cub. centim., and its volume and that of each division of the stem was accurately determined by calibration with mercury. After the tube a had been sealed to the pump, and the apparatus exhausted, it was immersed in the constant-temperature bath. The tube b was then connected to a vessel con- taining the liquefied gas, which was distilled into the bulb until both bulb and stem were completely filled. The stop-cock was turned and the liquid allowed to evaporate into the pump until the meniscus had come to a definite position on the stem, when the stop-cock was turned so as to disconnect all the tubes. To obtain the volumes occupied by a constant weight of liquid, it was only necessary to read the position of the meniscus at different temperatures.

The weight of liquid was obtained by attaching to the Fig. 4. tube b a weighed set of GEISSLER'S bulbs containing

potassium hydrate solution.

On opening the stop-cock and raising the temperature of the bath the liquid evaporated and the gas was absorbed in the bulbs and weighed.

To prevent the potash solution sucking back, a little mercury was placed in the first bulb. The small amount of gas remaining in the apparatus was finally pumped out through a and measured.

<i

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

109

The numbers obtained by the foregoing method have been checked in the case of hydrogen chloride, hydrogen bromide, and hydrogen iodide by distilling each of them into a thick-walled bulb of known volume, and sealing it. After the bulb containing the liquid had been weighed it was cooled, the stem broken, and the empty bulb and stem again weighed.

The densities so obtained agreed to the 3rd decimal place with those obtained at the same temperature by the first method.

In the case of phosphine the density was determined by the second method and the temperature coefficient by the first method.

The results of the measurements are given in Table III., in which D' represents the experimental and D the smoothed value of the density. The density in each case is a linear function of the temperature and is given by the relation

DT, = DT[l + a(T-T')],

where T and T' represent the boiling-point and the temperature of observation respectively, both on the absolute scale ; DT and DT, being the corresponding densities.

The values of the coefficient a for the different substances are contained in the following table :—

HC1	Dr = 1-187	1+ • 000268 (T-T')]
HBr	DT- = 2-157	1+0-0041 (T-T')
' HI DT. =2-799	1+0-0043 (T-T')
H.,S Dr = 0 • 964	'1+0-00109 (T-T')
PH3	Dr =0-744	1+0-0008 (T-T')"

TABLE III. — Densities.

T (jibs.).	D.	D'.	T (abs.).	I).	D.
"aba.	1	°abs.
	HYDROGEN CHLORIDE.
164-0	1-257 180 1-213
166	1-251		180-1	1-2127
168	1-246	.	182	1-207	—
168-5		1-2438	183-2	—	1-2038
170	1-240	—	184	1-201	—
171-8		1-2347	186	1-196	—
172	1-234	—	187-2	—	1-1937
174	1-229	—	188	1-190	—
175-8	.	1-2242	189-9	—	1-1842
176	1-224	—	190	1-185	—
178	1-218	—	192	1-179	—

110 DR. B. D. STEELE, DK. D. McINTOSH AND DK. E. H. ARCHIBALD

TABLE II L — Densities (continued).

T (abs.).	D.	D'.	T (abs.).	D.	D'.
°abs.			"abs.
HYDROGEN BROMIDE.
182	2-245	_	195-3	2-1932
184	2-237	—	196	2-191	—
184-7		2-2337	198	2-183	—
180-0		2-2286	198-2	—	2-1823
186	2-229		200	2-176	—
188	2-222	.	200-4	—	2-1742
190	2-214		202	2-168	— •
192	2-206		203-8	—	2-1600
193-3		2-2047	204	2-160	—
194	2-199	—
HYDROGEN IODIDE.
222	2-863		232 2-822	—
223-3		2-8600	232-9	2-819
224	2-855	—	234	2-813	—
224-9		2-8496	236	2-805	—
'226	2-847		236-3	—	2-8034
227-0		2-8412	238	2-796	—
228	2-838		240	2-787	—
229-3		2-8330	240-4	—	2-7862
230	2-830	—	242	2-779
HYDROGEN SULPHIDE.
190 1-004		201-5	0-9846
191-3	1-0019	202	0-984	—
192	1-001	. —	203-9	—	0-9806
194	0-998	—	204	0-980	—
194-6	0-9968	206	0-976	—
196 0-994	—	206-9	—	0-9759
197-4		0-9925	208	0-973	—
198	0-991		210	0-970	—
199-7		0-9875	210-8	—	0-9692
200	0-987	— •	212	0-967	—
PHOSPHURETTED HYDROGEN.
166	0-761	_	180	0-750
167-1	—	0-7604	182	0-748	—
168	0-760		184	0-747	—
170	0-758		184-4	—	0-7465
171-8	—	0-7560	186	0-745	. —
172	0-756		186-5		0-7448
174	0-755		188	0-743	.
175	—	0-7534	190	0-742
176	0-753		192	0-740
178	0-751		192-8		0-7392
179-9		0-7504	194	0-739	—

ON THE HALOGEN HYDKIDES AS CONDUCTING SOLVENTS.

Ill

KOPP has shown that the molecular volume of a liquid at its boiling-point is an additive property, being equal to the sum of the atomic volumes of the component elements. Certain elements, however, such as oxygen, appear to possess two values for the atomic volume, depending on the nature of the linking of the oxygen to the other atoms in the molecule.

It has also been shown that, in the case of the elements chlorine, bromine, and sulphur, the atomic volumes calculated from the density of compounds containing them are the same as those obtained from the densities of the pure elements. We have calculated the atomic volumes of the elements chlorine, bromine, iodine, sulphur, and phosphorus from the densities of their respective hydrides, in order to see how the values so obtained agree with those given by KOPP and others.

The results of these calculations are given in Table IV., which contains D, the densities of the compounds at their respective boiling-points, the molecular volume M/D, and the atomic volume A' of the halogen elements.

The values of these are invariably higher than those of KOPP, which are given under A, in the fifth column. It is possible that this discrepancy is due to a variation in the atomic volume of hydrogen, which has accordingly been calculated from each compound by subtracting the figures in the fifth column from those in the third.

The values for A, so obtained, are given in the last column, and are uniformly higher than 5 '5, which is the number found from the study of organic compounds.

TABLE IV. — Molecular Volumes at Boiling-point.

Substance.	D.	M/D.	A'.	A, KOPP, &c.	AA.
HC1. . . . HBr . . .	1-185 2-158	30-8 37-4	25-3 31-9	22-8 27-9	8-0 9-5
HI .	2-799	45-7	40-2	37-8	7-9
H,S	0-964	35-2	24-2	22 -G	6-3
H3P . . .	0-743	45-7	29-2	/21-9*	8-0
				\2C-Ot	6-6

* MASSON. t THORPE.

The Molecular Surface Energies.

The molecular surface energies were measured by RAMSAY and SHIELDS' method, slightly modified in order that the measurements might -be made at low temperatures. The apparatus (fig. 5) consisted of a tube b, 6 centims. long and 1'3 centims. in diameter, which was provided with a small side tube d, and joined to a two-way stop-cock by a long tube a.

A small glass scale s, which had been very carefully calibrated, was securely fixed inside b. The capillary c was sealed to a long tube g, in the manner described by

112

DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

RAMSAY and SHIELDS. The tubeg, which enclosed a piece of soft iron, i, was selected so as to slide easily and smoothly in a. It was held in position by the two glass hooks e and /, which were placed so that when / was resting on e the bottom of the capillary was a few millimetres below the scale.

TO PUMP

Fig. 5.

A mark had been previously etched on the capillary, and when making the measurements the position of the tube g was adjusted by means of an electro-magnet, so that this mark always coincided with the meniscus inside the capillary. The radius of the capillary near the etched mark was determined by introducing a quantity of pure ether into the apparatus, and measuring the height of the column of liquid when the capillary was in different positions. The radius could then be calculated from KAMSAY and SHIELDS' values for the surface energy of ether. The following values were found : —

Position of meniscus.	Height of ether column.	Radius.
At the mark	millims. 34-28	•013767
1 • 9 millims. below mark . . 0-4 u * » » » • • 1 • 1 „ above „ . . 7 >Fi 1 u j» » » • •	34-29 34-32 34-31 34-57	•013763 •013751 •013755 •013652

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

113

To carry out the experiments, the apparatus was placed in the constant-temperature bath and exhausted through the tube L The stop-cock h was then turned, and an excess of the liquid to be measured was introduced through the tube I by distillation. The stop-cock was again turned, and all traces of air were displaced from the apparatus by allowing some of the liquid to evaporate into the pump, after which the stop-cock was closed. The bath was then maintained successively at different temperatures, and the mark on the capillary having first been brought into coincidence with the meniscus, the height of the column of liquid was accurately measured.

The tube d was attached to a manometer, and measurements of the vapour-pressure of the liquid were made during the experiment. This tube was removed during the measurements of hydrogen bromide and hydrogen iodide.

The results of the measurements are given in Tables V. to IX., in which the letters employed have the following meaning : —

T = the absolute temperature ; D = the density of the liquid ; (T = the density of the vapour ; V = the specific volume of the liquid ; V.P. = the vapour-pressure of the liquid ;

h = the height of the column of the liquid ; y = surface tension in dynes per centimetre = J ryh (D— er) ; y (MV)3 = the molecular surface energy in ergs ; g = the constant of gravity ; M = the molecular weight of the liquid.

TABLE V. — Hydrochloric Acid.

T (abs.).	D.	<T.	D-ov	V.P.	h.	y-	(MV)i.	y(MV)>.
163-1.	[1-2530]	0-00051	1-2525	miHims. 141	centime. 3-303	27-874	9-4600	263-68
168-5	1-2438	0-00069 1-2431	198	3-214	26-912	9-5073	255-87
171-7	1-2347	0-00083	1-2339	245	3-152	26-251	9-5537	250-80
175-8	1-2242	0-00105	1-2232	316	3-094	25-477	9-6080	244-78
180-1	1-2127	0-00139	1-2113	430	3-033	24-718	9-6690	239-00
183-2	1-2038	0-00167	1-2021	522	2-974	24-046	9-7167	233-65
187-2	1-1937	0-00202	1-1917	648	2-936	23-467	9-7725	229-30
189-9	1-1842	0-00230	1-1819	748	2-866	22-760	9-8233	223-57
192-6	1-1770	0-00263	1-1744	868	2-838	22-409	9-8634	221-03

VOL. CCV. — A.

114

DE. B. D. STEELE, DR. D. McINTOSH AND DR, E. H. ARCHIBALD TABLE VI. — Hydrobromic Acid.

T (abs.).	D.	<T.	D-o-.	V.P.	k.	y. (MV)I.	y(MV)I.
181-8 184-7 186-1 188-9 193-4 195-3 198-2 200-5 203-9	[2-2400] 2-2337 2-2286 [2-2185] 2-2047 2-1932 2-1823 2-1742 2-1600	0-0015 0-0018 0-0019 0-0023 0-0028 0-0031 0-0035 0-0039 0-0047	2-2385 2-2319 2-2267 2-2158 2-2019 2-1901 2-1788 2-1703 2-1553	millims. 210 250 275 327 410 430 525 600 730	centims. 2-015 1-990 1-958 1-926 1-887 1-830 1-800 1-790 1-740	30-191 10-932 29-728 10-953 29-182 10-970 28-570 11-014 27-812 11-049 27-019 11-087 26-440 11-124 26-201 11-152 25-399 11-201	330-1 325-6 320-1 314-6 307 • 30 299-6 294-8 292-2 284-5

TABLE VII. — Hydriodic Acid.

T (abs.).	D.	<T.	D-cr.
225-3	2-8523	0-0039	2-849
227-1	2-8401	0-0042	2 • 836
229-3	2-835	0-0045	2-831
230 • 9	2-829	0-0048	2-824
232-9	2-820	0 • 0053	2-815
235-0	2-812	0-0057	2-806
236-5	2-806	0-0061	2-800

V.P.	h.	'	(MY),.	y(MV),
inillims. 420	centims. 1-511	29-06	12-63	367-0
460	1-496	28-64	12-67	362-8
503	1-479	28-26 12-69	358-6
558	1-467	27-97 12-71	355-5
595	1-451	27-57	12-73	351-0
655	1-440	27-27	12-76	348-0
700	1-427	26-96	12-78	344-6

TABLE VIII. — Sulphuretted Hydrogen.

T (abs.).	D.	IT.	D-o-.	V.P.	h.	y-	(MV)i	y (MV)i.
189-0	[1-006]	0-00055	1-006	millims. 192	centims. 4-962	33-418	10-458	349-5
191-3	1-002	0-00063	1-001	219 ! 4-906	32-902 10-495	345-3
194-6	0-997	0-00076	0-996	269	4-816	32-126 10-522	338-0
197-4	0-992	0-00086	0-992	313	4-765	31-645 10-557	334-1
199-7	0-987	0-00098	0-986	363	4-695	31-020 10-584	328-3
201-5	0-985	0-00107	0-984	399	4-676	30-813 10-604	326-6
203-9	0-9806	0-00122	0-979	454 I 4-642	30-448	10-639	324-7
206-9	0-9759	0-00142	0-975	536 4-540	29-631 10-669	316-7
210-8	0-9692	0-00175	0-968	674	4-442	28-783 10-720	308-6

TABLE IX. — Phosphuretted Hydrogen.

T (abs.).	D.	or.	D-o-.	V.P.	h.	y-	(MV)5.	y(MV)',
167-1	0-760	0-00079	0-7592	millims. 237	centims. 4-484	22-783	12-605	287-2
171-8	0-756	0-00101	0-7550	319	4-372	22-095	12-654	279-6
175-4	0-753	0-00122	0-7522	393	4-282	21-553	12-683	273-4
179-9	0-746	0-00151	0-7450	498	4-171	20-798	12-761	265-4

ON THE HALOGEN HYDKIDES AS CONDUCTING SOLVENTS.

115

The foregoing results are represented graphically in fig. 6, in which the molecular surface energies are plotted against the absolute temperature.

^200

180

170

160

200

\

,^>

^

250 300

MOLECULAR SURFACE ENERGY

Fig. 6.

350

EF?GS.

The range of temperature over which measurements were made was small, and in the case of each substance the curve appears to be a straight line.

d

The temperature coefficients -j- y (MV)?i are given in Table X.

dt

TABLE X.

Q 2

Substance.	Temperature range.	|y(MV)I.
Hydrogen iodide. . . . „ bromide . . .	225-236 181-204	1-99 2-03
„ sulphide . . . „ phosphide . . „ chloride . . .	189-211 167-180 159-192	1-91 1-70 1-47

116 DE. B. D. STEELE, DR. D. McINTOSH AND DK. E. H. AECHIBALD

The average value of this coefficient is, according to EOTVOS, 2'27. From the experiments of RAMSAY and SHIELDS it is 2-12, while BALY and DONNAN have found that the liquefied gases oxygen, nitrogen and carbon monoxide give values very near to 2, and this number has also been found by us for the three substances hydrogen

bromide, iodide and sulphide.

RAMSAY and SHIELDS have shown that for normal liquids the relation between molecular surface energy and temperature is given by the equation

in which t represents the temperature measured from the critical point and d is a

small constant.

From this equation it follows that the surface energy disappears at a temperature d degrees below the critical point, and therefore the curve for a normal liquid, if produced, should cut the temperature axis at this point.

This is the case for hydrogen bromide, iodide and sulphide, for which, as will be seen from Table XL, the value of d is 16'3, 157 and 0'2 respectively.

HI-* 400 /ygj-> HB^ I 5300 I ?00 180

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The curves are shown in fig. 7, in which the critical temperature of each liquid is indicated by an arrow.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

TABLE XL

117

	Critical temp.	Temp, at which curve cuts T axis.	d.
Hydrogen chloride ,, bromide	°0. 52-3 91-3	"0. 65-2 75-0	+ 11-9 16-3
,, iodide	150-7	134-0	15-7
,, sulphide	100-2	100 -0	- 0-2
Phosphuretted hydrogen . . .		61-6

RAMSAY and SHIELDS also showed that another class of liquid exists, for which the above relation does not hold, inasmuch as the coefficient not only was less than 2 '12, but also varied with the temperature. If a tangent is drawn to the curve, for a liquid of this class, it will cut the temperature axis at a point above the critical temperature.

This abnormal behaviour is explained by the assumption that the molecules of such liquids are associated to form larger molecular complexes ; in other words, that their molecular weights are abnormally high. Hydrogen chloride and phosphide, from the magnitude of their temperature coefficients, must be classed with the abnormal or associated liquids, but the curves which we have obtained are too short to be distinguished from straight lines. These curves have, however, been produced and the results are shown in fig. 8 and Table XI.

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MOL SURFACE ENERGY. Fig. 8.

118

It will 65-2°, or

measured at 22°. In the

DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

be seen that the curve for hydrogen chloride cuts the temperature axis at

11-9° above the critical point.

We have not been able to find any record of measurements of the critical temperature of phosphuretted hydrogen.

The Measurement of Viscosity.

The viscosity apparatus (fig. 9) was of the usual form, but with some slight additions, which were designed to prevent access of water vapour. For this purpose the two ends of the apparatus were joined to the stop-cock h, and the two tubes c and d, containing phosphoric anhydride, were attached. After the apparatus had been placed in the bath, a definite quantity of the liquid was distilled into it through d.

In order to make the measurements, the stop- cock was closed and, by blowing into d, the liquid was forced into f until it reached a position about one centimetre above the mark a. The stop-cock was then opened and the time required for the liquid to fall from a to b with a stop-watch. The apparatus was calibrated with distilled water

tables, D refers to the density of the liquefied gas; 17 is the viscosity

compared with that of water at 22°, and -3- is the temperature coefficient of viscosity.

ctt

TABLE XII. — Hydrogen Chloride. Apparatus B. Time of flow for water = 7 5 '3 seconds at 22°.

					1
T (abs.).	Time.	D.	*?•	dr, dt '	Smoothed -2 . dt
				per cent.	per cent.
160-8	35-1	1-265	0-590	— «	—
166-7	34-3	1-249	0-569	0-61	—
171-7	32-3	1-236	0-530	0-03	—
177-0	31-7	1-221	0-514	0-91	—
183-2	30-8	1-204	0-493	0-88	—
188-2	30-2	1-189	0-477	0-86	0-88

ON THE HALOGEN HYDWDES AS CONDUCTING SOLVENTS.

119

TABLE XIII.— Hydrogen Bromide. Apparatus B.

T (abs.).	Time.	D.	'/•	dt\ ~di'	Smoothed -^ . dt
186-8	30-8	2-227	0-911	per cent.	per cent.
188-8	30-6 2-219	0-902	0-50
190-8	30-3 2-212	0-890	0-59
193-7	30-0	2-200	0-877	0-57
197-3	29-5	2-186	0-857	0-60
199-4	29-4	2-178	0-851	0-56	0-57

TABLE XIV.— Hydrogen Sulphide. Apparatus B.

				1
T (abs.).	Time.	D.	*	~ctt '	Smoothed "H . dt
		per cent.	per cent.
191-0	40-3	1-002 0-547	_
193-3	39-8	0-998 0-528	1-6
198-2	38-8 0-990 0-510	1-0
201-2	37-3 0-985 0-488	1-19	_
206-1	36-2 0-977 0-470	1-09
209-8	35-2 0-972	0-454	1-08	1-1

TABLE XV. — Hydrogen Iodide. Apparatus A. Time of flow for Water = 43 seconds

at 22°.

T (abs.).	Time.	D.	,.	dt'	Mean.
		1	per cent.	per cent.
223-3	22-3	2-858	1-479	—	—
225-6	22-0	2-849	1-454	0-75	—
227'2	21-8	2-842 1-437	0-75	—
229-6	21-7	2-832	1-426	0-59	—
231-5	21-4	2-824	1-402	0-67	—
233-9	21-1	2-813	1-377	0-70	—
236-4	20-8	2-802	1-353	0-71	0-70
SUMMARY of Tables XII. to XV.
			dn
	Substance.	•>! at B.P.	dt '
	HC1	0-47	0-90
	HBr	0-83	0-58
	HI	1-35	0-70
		H2S . .	0-45	1-10

120

DE. B. D. STEELE, DR. D. McINTOSH AND DK. E. H. ARCHIBALD

PAET II.

Tli e Conductivity and Molecular Weights of Dissolved Substances.

and E. H. AKCHIBALD.

By D. MCINTOSH

THE second part of this investigation deals with the solubilities of substances in the liquefied halogen hydrides and sulphuretted hydrogen, and with the conductivities of the resulting solutions ; the molecular weights of a few substances, when dissolved in each of these solvents, have also been determined.

The Temperature Bath.

As liquid air was not available in sufficient quantities to make use of the bath described in Part I., a mixture of carbon dioxide and ether, which under atmospheric pressure gives a very constant temperature of —81°, was used for the measurements of solutions in hydrogen bromide and sulphide.

The same mixture, under reduced pressure, was used for the measurement of solutions in hydrogen chloride. By carefully regulating the pressure over the mixture the temperature was maintained at —100°.

For the hydrogen iodide solutions a temperature of —50° was obtained by slowly running cold ether into the vacuum vessel and syphoning off the warmer upper layer.

The Determination of Solubilities.

The solubilities were measured by means of an apparatus (fig. 1) which consisted of a test-tube A, to the bottom of which a delivery tube B was sealed. The bottom of A was covered with a thick layer of asbestos which acted as a filter, and the whole was immersed in the constant-temperature bath.

The liquefied gas and the substance of which the solubility was under investigation were introduced into A, where they were vigorously stirred with a platinum rod. A portion of the liquid was then blown through the delivery tube into a weighed and

The volume of liquid in the test-tube was observed, the liquid

Fig. 1.

graduated test-tube.

allowed to evaporate, and the residue weighed. The solubility was calculated from

the data thus obtained.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 121

The results of the measurements may be summarised as follows :

(1) Inorganic substances insoluble, or soluble only in traces, in any of the solvents —

The chlorides, bromides, and iodides of the alkalis and alkaline earths : salts of -nickel, iron, lead, and mercury ; stannous chloride, potassium permanganate, and potassium bichromate.

(2) Inorganic substances soluble — in some cases with decomposition : —

(a) In hydrogen chloride—

*Stannic chloride, phosphorus pentabromide, phosphorus pentachloride, and phosphorus oxychloride ;

(b) In hydrogen bromide —

Phosphorus oxychloride, bromide, and sulphuretted hydrogen ;

(c) In hydrogen iodide-

Iodine and phosphorus oxychloride ;

(d) In sulphuretted hydrogen —

Sulphur, phosphorus oxychloride, hydrogen bromide, and hydrogen chloride.

(3 ) Inorganic substances soluble with decomposition in hydrogen chloride, potassium cyanide, ammonium sulpho-cyanate, sodium acetate.

(4) Organic substances. In addition to the organic ammonium bases, which, as stated in Part I., dissolve somewhat freely in all the solvents, we have found that a very large number of organic compounds are soluble, as, for example, the aldehydes, ketones, alcohols, ethereal salts of fatty and of aromatic acids, cyanides, and sulpho- cyanates, hydrocarbons, and nitro-compounds. Hydrogen sulphide is an excellent solvent for such bodies, but the solutions, as a rule, are non-conductors. The solutions hi the halogen hydrides, on the other hand, usually conduct the current.

77; e Measurement of Conductivity.

Although the investigation of the solubility of inorganic salts failed to indicate that these were soluble in more than traces in any of the solvents, we have tested the conductivity of the different solvents after the addition of certain inorganic substances. We find that an increase of conductivity was produced by adding the following substances to : —

* HELBIG and FAUSTI ('Atti E. Accad. Lincei,' 1904 (V), 13, p. 30) found that stannic chloride was soluble in hydrogen chloride. We regret that by an oversight we contradicted this statement in our Preliminary Note ('Roy. Soc. Proc.,' 1904, 73, p. 554), and we wish now to make the necessary correction.

VOL. CCV. — A. R

122 DR. B. D. STEELE, DR. D. McINTOSH AND DR. E H. ARCHIBALD

(a) Hydrogen chloride —

Bromine, potassium iodide, thionyl chloride, sulphuryl chloride, and uranium nitrate (very slight increase), phosphorus pentachloride, pentabromide, and oxychloride (considerable increase) ;

(b) Hydrogen bromide —

Phosphorus oxychloride (considerable increase) ;

(c) Hydrogen iodide-

Iodine, sulphuric acid, carbon disulphide, and phosphorus oxychloride (slight increase) ;

(d) Sulphuretted hydrogen-

Phosphorus pentachloride, and sulphuryl chloride (slight increase).

The following substances did not cause an increase in the conductivity of either of the solvents : —

Sodium, sodium sulphide, sodium biborate, sodium acid phosphate, sodium nitrate, sodium sulphide, sodium thiosulphate, sodium arsenate, chromic acid, the following salts of potassium : the nitrate, hydroxide, chromate, sulphide, acid sulphate, ferrocyanide, ferricyanide ; ammonium fluoride and carbonate ; rubidium and caesium chlorides ;

Magnesium sulphate, calcium fluoride, strontium chloride, barium chloride, oxide, nitrate and chromate ; copper sulphate, mercuric chloride, zinc sulphate, boron trichloride, aluminium chloride, and sulphate ; carbon dioxide, stannous chloride, lead peroxide, nitrate, and cyanide ; phosphorus tribromide, bismuth nitrate, tartar emetic, manganese chloride, ferric chloride, ferrous sulphate, nickel sulphate, and cadmium sulphate.

In addition to the organic ammonium bases, we have, in conjunction with Dr. J. W. WALKER,* examined the conductivity of solutions of about 80 organic substances in each of the foregoing solvents.

The only substances which form conducting solutions in H2S are the ammonium bases and a few alkaloids such as nicotine and pyridine. On the other hand, many ethers, ketones, esters, nitrites, and, generally speaking, substances containing oxygen or nitrogen, form conducting solutions in hydrogen chloride, bromide, and iodide. The hydrocarbons, although in some cases soluble in all proportions, do not conduct. We have noticed that the solution of those substances which conduct is accompanied by a considerable evolution of heat, while little or no heat is evolved in the case of other substances.

This indicates chemical interaction between the conducting solute and the solvent, and many of the resulting compounds have been isolated and analysed.! It has been

* 'Journal of the Chemical Society,' 1904, 85, p. 1098. t ARCHIBALD and MC!NTOSH, ' J. C. S.,' 1904, 85, p. 919.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

123

found, for example, that ether enters into combination with the three halogen acids, forming compounds which have the following formulae :

(C3H5)20, 5HC1, M.P. = -120, (C3H5)20, HBr, M.P. = - 42, (C2H5)2O, HI, M.P. = - 18.

We have explained the formation of these and similar compounds by assuming the existence of tetrad oxygen at these low temperatures.

The compound of ether and hydrogen bromide would thus have the formula

p2Tj /O\r>,> and might be expected to undergo electrolytic dissociation.

Quantitative Measurements of Conductivity.

The pure solvents are extremely poor conductors of electricity, their specific conductivity being as follows :

Hydrogen chloride about 0*2 x 10~",

bromide „ 0'05xlCr8,

iodide „ 0'2 x 1(T6,

,, sulphide ,, O'l x 10~li,

that of the purest water being 0-04 x 10~6.

The resistances are thus much greater than that of an ordinary sample of distilled water.

The majority of the measurements of conductivity were made in an apparatus (fig. 2) consisting of a graduated test-tube with fixed electrodes.

Fig. 2.

Fig. 3.

A sufficient quantity of the solvent was first placed in the conductivity vessel and a weighed quantity of the substance under investigation introduced by means of a

E 2

124 DR. B. D. STEELE, DE. D. McINTOSH AND DR. E. H. ARCHIBALD

cooled platinum spoon. The mixture was then stirred until the conductivity remained constant, after which the volume of the solution was observed. A further weighed quantity of the substance was then introduced and dissolved, and the conductivity again measured. The same series of operations was frequently repeated until a sufficient number of measurements had been made.

Other measurements were made in the apparatus shown in fig. 3. This consisted of a graduated test-tube A, provided with fixed electrodes, and with a delivery tube

B attached.

A saturated solution of the substance was made in the apparatus shown in fig. 1, a portion removed for analysis, and a sufficient amount put into the conductivity vessel, where its conductivity was measured and the volume noted. More of the solvent was then added, and the liquids were well mixed by blowing air through the delivery tube B. The volume was again read and the conductivity measured.

This succession of operations was repeated until the vessel became full of liquid, after which a measured volume of the solution was removed, and the operations were continued until a sufficient number of measurements had been obtained.

In all the measurements the electrodes were sufficiently immersed to give the maximum conductivity of the apparatus.

Our results are given in the following tables, which also contain the temperature coefficient of conductivity for those solutions which are marked with an asterisk.

The dilutions, which are given under V, represent the number of litres of solution which contain 1 gram-molecule of solute, and the molecular conductivities in reciprocal ohmsx 10~3 are given under p. The numbers are thus expressed in the same units as the molecular conductivity of aqueous solutions as given by KOHLRAUSCH and HOLBORN (' Leitvermb'gen der Elektrolyte ').

TABLE I. — Solvent : Hydrochloric Acid.

V.	p.	V.	/*•
HYDROCYANIC ACID.
41-4 21-3 14-1 10-2 9-2 7-35	0-51 0-91 0-98 1-08 1-34 1-48	4-90 4-08 3-12 *2-56 1-79 1-23	2-09 2-83 3-65 4-47 5-81 7-70
Temperature coefficient between - 99° and - 95° = - 2 • 0 per cent. -99" „ -90° = -1-8 „ -99° „ -85° = -1-3 „

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

125

TABLE 1. — Solvent : Hydrochloric Acid (continued).

V.

V.

TRIETHYLAMMONIUM CHLORIDE.

71-4 37-0 20-4 16-1 11-6

9-43

1-80 1-80 2-28 2-71 3-15 3-67

7-69	4-37
6-13	5-18
5-00	6-05
4-25	6-73
3-64	7-72
*2-99	8-51

Temperature coefficient between - 98° and - 89° = 0-39 per cent.

-98' „ -86° = 0-67 „

ETHYL OXIDE.

12-50	0-14
5-00	0-23
3-12	0-39
1-92	0-95
1-45	1-41

1-09	. 2-03
0-88	2-20
0-72	2-90
0-61	3-09

Temperature coefficient between - 99° and - 95° = 1 '9 per cent.

-99° „ -90° = 1-8 „ M ii ii — 99 „ — 85 = 1'7 ,,

ACETAMIDE.

29-4	1-59
12-8	3-12
8-62	4-27
4-65	6-39
4-15	6-92

2-86	8-20
2-13	9-41
1-54	10-8
0-95	12-1
0-51	12-3

Temperature coefficient between - 97° and - 92° = 1 -4 per cent.

-97° „ -86°= 1-2 „ -97° -83°= 1-2 „

ACETONITRILE.

21-7	1-51	2-17	6-82
8-33	2-44	1-54	6-25
4-76	3-89	1-09	8-OS
3-22	5-25	0-81	9-61

126

DR. B. D. STEELE, DE. D. McINTOSH AND DR. E. H. ARCHIBALD

TABLE II.- — Solvent : Hydrobromic Acidt

V.	,.	V.	P-
	TRIETHYLAMMO	VIUM CHLORIDE.
143 50 27-7 15-6 8-33	0-19 0-22 0-50 0-83 2-00	5-26 3-33 2-17 1-61	3-29 4-90 6-20 8-19
ETHYL OXIDE.
16-6 5-55 4-00 2-00 1-54	0-005 0-014 0-024 0-106 0-129	1-23 1-03 0-68 0-47	0-152 0-164 0-182 0-726
ACETONE.
8-33 5-00 3-23 2-00.	0-10 0-34 0-77 1-40	1-64 1 35 1-07 0-75	2-32 3-24 4-30 5-63
ACETAMIDE.
90-9 58-8 23-3 14-5 10-2	0-06 0-10 0-27 0-42 0-57	6-66 3-85 3-03 *2-08 1-41	0-94 1-47 1-80 2-37 3-15

Temperature coefficient between - 83° and - 77° = 0 • 94 per cent.

-83° „ -74° = 0-94

ACETONITRILE.

33-3	0-14	1-47	4-62
4-76	1-08	1-22	5-43
3-70	1-32	0-96	6-99
2-50	2-48	0-72	10-01
1-85	3-46	—
ETHYL PROPIONATE.
12-5	0-05	2-63	0-82
7-14	0-16	1-92	1-19
5-26	0-38	1-39	1-69
3-45	0-49
ORTHO-NITROTOLUENE.
25	0-04	3-85	0-21
16-6	0-07	2-38	0-45
12-5	0-07	1-50	0-67
11-1	0-10	0-92	1-02
8-33	0-11	0-66	1-28

t The solutions marked thus t were measured by McINTOSH and STEELE.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

127

TABLE II. — Solvent : Hydrobromic Acid (continued).

V.	p-	V.	11.
		1

200 21-3 14-1

166 62-5

62-5 34-5

tTETRAMETHYLAMMONIUM CHLORIDE.

5-40

8-94

10-56

10-5 9-42

tTETRAMETHYLAMMONIUM BROMIDE.

[12-6] 7-0

34-5 11-8

tTETRAMETHYLAMMONIUM IODIDE.

8-75 22-2

10-35

12-53 13-30

7-25 12-6

13-10

TABLE III. — Solvent : Hydriodic Acid.

V.	/*•	V.	p.
	TRI ETHYL AMMONIUM CHLORIDE.
27-8	0-07	5-55	1-15
21-7	0-11	4-50	1-48
15-4	0-23	3-85	1-91
10-4	0-43	3-13	2-37
8-55	0-65	2-50	2-97
7-14	0-80	2-17	3-58
ETHYL OXIDE.
10-0	0-02	1-49	1-11
5-88	0-07	1-25	1-40
3-33	0-22	1-06	1-79
2-46	0-61	*0-88	2-21
1-79	0-84	—	—

Temperature coefficient between - 50° and - 45° = 1 • 9 per cent.

-50° „ -40°= 1-8 „

ETHYL BENZOATE.

16-6	0-014	2-56	1-65
7-14	0-170	2-04	2-30
4-76	0-47	1-66	2-98
3-45	1-02	1-37	3-60

128 DK. B. D. STEELE, DK. D. McINTOSH AND DR. E. H. ARCHIBALD

TABLE IV.— Solvent : Sulphuretted Hydrogen.

V.

/*•

V.

TRIETHYLAMMONITJM CHLORIDE.

71-4 12-8 8-33

0-12 0-21 0-33

4-00 3-13

*2-50

0-87 1-17 1-58

Temperature coefficient between - 80° and - 75° = 0'88 per cent.

„ -80° „ -70° = 0-90 „ -80° , -65° = 0-85 „

NICOTINE.

66-7 14-3 6-67 4-00

9-09 1-18 0-90

0-03 0-04 0-06 0-16

0-02 0-29 0-39

2-27 1-92 1-03

0-38 0-50 0-76

PlPERIDINE.

0-75

0-64

*0'55

0-46 0-48 0'50

Temperature coefficient between - 80° and - 66° = 1 '82 per cent.

-80° and -63° = 1-84 „

TETRAMETHYLAMMONIUM CHLORIDE.

34-5 11-0 4-35

1-71 3-41

3-85

3-33 2-93

4-02 3-85

Temperature coefficient between - 70° and - 64 '7° = 0'95 per cent.

-70° „ -62-6J - 1-07 „ -70° -60-8°= 1-09

The foregoing results are shown graphically in the figs. 4 to 7, in which the molecular conductivities are plotted against the dilutions. It will be seen that in every case JJL decreases enormously with dilution, a variation which is exactly opposite to that which might be expected from analogy with aqueous solutions. These results indicate that, if conduction is due to ionisation, the degree of dissociation decreases with dilution, a result which is in opposition to the law of mass action.

This subject will be discussed fully in Part IV.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

129

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ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

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132 DE. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

As a general rule, the same substances conduct better when dissolved in hydrogen chloride than in the other solvents. Next, in the order given, come solutions in hydrogen bromide, iodide, and sulphide. An exception to this rule is found in the case of ether, which conducts best in hydrogen chloride, and worst in hydrogen

bromide.

The conductivity temperature coefficients do not appear to have anything in common with the viscosity temperature coefficient, so that the ions cannot be looked upon as being surrounded with an atmosphere of the solvent (KOHLRAUSCH, ' Roy. Soc. Proc.,' 71, 338, 1903).

The coefficients are for the most part positive, the conductivity increasing with rise of temperature, an exception occurring in the case of hydrocyanic acid dissolved in hydrochloric acid.

The Determination of Molecular Weights.

The molecular weights were determined by measuring the rise in boiling-point which was brought about by the addition of known quantities of the dissolved substance.

Considerable experimental difficulty has been experienced during the progress of the work, which has also proved expensive on account of the very large quantities of carbon dioxide which were required, and consequently only a few determinations have been made.

The accurate measurement of small differences of temperature at low temperatures has been successfully accomplished by the use of a differential method, in which two platinum resistance thermometers were employed to measure the temperatures, one of the thermometers being immersed in the pure boiling solvent and the other in the boiling solution.*

The thermometers were each made from about 2 metres of 6 mil wire, and had exactly the same resistance, which was of such a magnitude that a difference of 1° in the temperature of the two coils produced a displacement of 16 '7 millims. in the balance point on the bridge. The thermometers were supplied with compensation leads in the usual way. The bridge was of the Carey-Foster type. With the galvanometer used a difference in temperature of 0'03C could be detected with certainty.

The apparatus was tested by immersing the two coils in (a) boiling water, (b) melting ice, (c) boiling hydrogen sulphide, (d) boiling hydrogen bromide, and (e) boiling hydrogen chloride ; the same balance point was obtained in each case.

The measurements were made in the two pieces of apparatus shown in fig. 8,

f The bridge and resistance thermometers were lent to us by Dr. H. L. BARNES, through whose advice and assistance many difficulties have been avoided. We take this opportunity of expressing our thanks to him for his kindness and help.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

133

Fig. 8.

each of which consisted of two concentric tubes, A and B, which were sealed together at the top. The outer tube B was provided with the two side tubes C and D, to one of which, C, the condenser was attached, the other being closed by a well- fitting cork. The two thermometers were placed in A and A', and were held in position by waxed corks.

To carry out an experiment, a sufficient quantity of the liquefied gas was introduced through the side tubes D and D' into the vessels B and B', both of which contained beads to ensure steady boiling. In order to determine the quantity of solvent used in making the solutions, the volume of liquid in one of the vessels was measured at a definite temperature, or, if liquid air was available, the liquid was frozen and the

134 DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

vessel containing it weighed. The condensers were then filled with a mixture oi solid carbon dioxide and ether, maintained at atmospheric pressure for the experiments with hydrogen bromide and sulphide, and at reduced pressure for the experiments with hydrogen chloride.

After the liquid in the vessels had commenced to boil, the vessels were wrapped in natural wool, and the balance point on the bridge was determined. Weighed quantities of the substances whose molecular weight was to be determined were then successively introduced, and the displacement of the balance point was determined after each addition. From these displacements the corresponding rise in boiling-point was calculated. The loss by evaporation, due to the high vapour pressure of the solvents at the temperature of the condenser, was corrected for by means of a blank experiment.

Evaporation also occurred when the substances were introduced into the apparatus, but as this evaporation was proportional to the amount of substance added, a correction was easily applied. As a check on these corrections, the boiling-point apparatus was removed after each two or three determinations, and when liquid air was available, the apparatus was cooled and weighed. When liquid air could not be obtained, the tube was cooled to a definite temperature, the volume of solution measured, and the amount of solvent calculated on the assumption that no volume change occurred on mixing.

From the data thus obtained the molecular weight constant was calculated by means of the formula

M =

r< A > G A

in which the molecular weight of the dissolved substance is expressed in terms of y, its weight in grammes dissolved in G-gramme of the solvent, and of the corresponding rise in boiling-point A, K being a constant in the case of a solute which is neither associated nor dissociated. The values of y, G, A, and K for the various substances investigated are given in the first four columns of Tables V, VI, and VII. The following example will show the method of making the calculations :—

Toluene in Hydrochloric Acid.

Apparent volume of liquid +0'39# toluene. . 38'3 cub. centims.

). ,, ,, beads ....... 17'0 „

Real volume of liquid ......... 21 -3

Volume of toluene ......... . . 0'4

„ „ hydrochloric acid ...... 20 '9 „

Weight (Part I.) .......... 25%.

Rise of boiling-point ......... 0'42°.

Constant ........ = 2480.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

135

VAN 'T HOFF has shown that the molecular rise of boiling-point can be calculated by means of the formula

,T _ 0-02T2 ~W~'

where T is the absolute temperature and W is the latent heat of evaporation of 1 gramme of the solvent.

We have calculated the values for the molecular rise of boiling-point of the various solvents by means of the latent heats which are given in Part I.

The molecular weight of the dissolved substance has been calculated from the values so obtained for the molecular rise.

The molecular weights are given under M' in the sixth column of the tables, and in the fifth column the concentrations of the solutions are expressed in grammes of solute dissolved in 100 grammes of solvent.

0-09T2 TABLE V. — Molecular Weights in Hydrogen Chloride, IT = 720.

ff-	G.	A.	K.	C.	M'.
TOLUENE.
0-39	25-0	0-42	2480	1-56	26-6
0-88	24-5	0-93	2380	3-59	27-8
2-25	24-5	2-25	2250	9-18	29-4
3-25	23-9	3-35	2270	13-60	29-2
4-28	23-9	4-31	2210	17-91	29-9
ETHER.
1-54	38-4	0-27	500	4-01	107
2-99	36-3	0-96	860	8-24	61-8
3-62	33-9	1-56	1080	10-68	49-3
3-62	30-4	1-98	1230	11-91	43-3
4-29	29-2	2-95	1485	14-69	35-9
4-69	28-4	3-63	1630	16-02	32-8
4-69	25-5	4-82	1940	18-40	27-5

136

DE. B. D. STEELE, DR. D. MoINTOSH AND DR. E. H. ARCHIBALD

TABLE VI.— Hydrogen Bromide,

-02T2

= 1770.

9-	G.	A.	K.	C.	M'.
TOLUENE.
1-82	51-3	0-77	2000	3'54	81-4
2-15	50-3	1-02	2200	4-27	74-1
2-71	49-3	1-27	2120	5-50	76-7
;i • 45	48-3	1-62	2090	7-14	78-1
ETHER.
0-39	38-8	0-24	1745	1-02	75-1
0-79	38-1	0-48	1700	2-09	77-0
1-15	37-4	0-78	1870	3-08	69-9
1-69	36-7	1-27	2045	4-60	64-1
2-18	35-5	1-95	2350	6-14	- 53-7
2-64	34-0	2-94	2800	7-76	46-7
3-05	32 '6	4-18	3300	9-37	39-7
ACETONE.
0'55	45-4	0-22	1060	1-21	97-1
1-23	42-4	0-53	1060	2-91	97-1
1-89	39-4	1-20	1450	4 '80	70-7
2-76	37-4	2-94	2310	7-38	4-44

A-09T2

TABLE VII.— Sulphuretted Hydrogen, ^=jL = 620.

9-	G.	A.	K.	C.	M'.
TOLUENE.
1-25 2-25 3-29 4-22	22-1 22-1 22-1 22-1	0-44 0-72 1-01 1-16	670 650 625 560	5-64 11-51 14-89 19-10	79-5 87-8 91-4 102-1
TRIETHYLAMMONIUM CHLORIDE.
0-76 1-30 1-62 1-98	22-6 22-3 21-9 21-6	0-23 0-44 0-60 0-69	940 1040 1115 1035	3-36 5-83 7-40 9-17	90-7 82-2 76-5 82-4

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 137

These results may be briefly summarised as follows : —

Toluene, which is a non-conductor in each of the solvents, has an average molecular weight of about 30 in hydrogen chloride, 78 in hydrogen bromide, and about 90 in sulphuretted hydrogen, and therefore appears to be dissociated when dissolved in hydrogen chloride and hydrogen bromide and to a greater extent in the former solvent.

KAHLENBERG ('Jour, of Phys. Chem.,' 1901, v., 344; 1902, vi., 48) has noticed a similar anomaly in the case of a solution of diphenylamine in methyl cyanide.

Ether in hydrogen chloride and hydrogen bromide, and acetone in hydrogen bromide have molecular weights which indicate association in the more dilute solvents and dissociation in the more concentrated.

Triethylamine hydrochloride appears to be dissociated when dissolved in sulphur- etted hydrogen, the dissociation being greater in the more concentrated solutions.

VOL. COV. — A.

138 DR. B. D. STEELE, DR. D. MoINTOSH AND DR. E. H. ARCHIBALD

PART III.

The Transport Numbers of Certain Dissolved Substances. By B. D. STEELE.

THE strikingly abnormal variation ot molecular conductivity with dilution that we have found to occur in solutions in the liquefied halogen hydrides finds a possible explanation in the assumption that it is the solvent and not the solute which is ionised. As the transport number of the dissolved substance might be expected to yield information not only as to the correctness of this assumption, but also as to the constitution of the electrolyte, the transport numbers of a few substances have been measured, and the results are given in the following pages.

The only measurements of the migration ratio which have hitherto been made in solvents other than water are those of a few salts in methyl and ethyl alcohol, and ol silver nitrate in pyridine and in acetonitrile.

Direct measurements of the velocities of certain ions in liquefied ammonia have recently been made by FRANKLIN and CADY ('Journal of Amer. Chem. Soc.,' 1904, vol. 26, p. 499), who used a modification of MASSON'S method (' Phil. Trans.,' A, 1902, vol. 192, p. 331).

Method of Measurement.

It has been shown by the author (STEELE, 'Phil. Trans.,' 1902, A, vol. 198, p. 105) that the direct method of measurement gives trustworthy results only when the salt under examination is of the simplest type. Now HITTORF has shown that in alcoholic solution cadmium iodide and certain other salts are dissociated into ions which are much more complicated than those occurring in aqueous solutions of the same concentration.

The only substances which we have found to be capable of forming conducting solutions in any of the solvents which we have been investigating are certain organic compounds, and although the nature of the ions into which these dissociate is entirely unknown, it is probable that the ionisation is even more complicated than that of cadmium iodide dissolved in alcoholic solution.

From these considerations it was decided to use HITTORF'S method, notwithstanding the fact that it is much more tedious and presents greater experimental difficulties than the alternative method of direct measurement.

HITTORF'S method consists in the analysis, after electrolysis, of the solution which surrounds one of the electrodes. The original concentration being known, the actual

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 139

amount of substance which has been carried to the electrode by the current can then be calculated. The current is usually measured by a silver voltameter placed in the same circuit as the electrolytic cell. The calculation, neglecting certain small corrections, is as follows : —

If x grammes of the substance, whose equivalent weight is n, be transported by the current which deposits y grammes of silver on the cathode of the voltameter,

then the transport number of the cation is given by p = — — . p represents the

u fraction of the total current which is carried by the cation, on the assumption that

one unit charge of electricity is associated with one equivalent of the dissolved substance. It is probable that this condition is fulfilled only in solutions of salts of the simplest type (STEELE, loc. cit.).

Preparation of Solutions.

The most convenient refrigerant which was available was a mixture of carbon dioxide and acetone, and as at the temperature of this mixture hydrogen chloride is a gas and hydrogen iodide a solid, the choice of solvent was limited to hydrogen sulphide and hydrogen bromide. Solutions in the former solvent are very much more difficult to analyse than those in the latter, and accordingly hydrogen bromide only has been used as solvent during the investigation.

The hydrogen bromide was prepared and purified by the method described in Part I. In order to make the solutions for electrolysis, the gas was condensed in a graduated vessel in which a sufficient quantity of the substance under examination had been placed, the condensation being stopped as soon as the desired volume of solution had accumulated.

A quantity was usually made sufficient for two experiments, and by placing the receiver in a good silvered vacuum vessel with a stiff paste of the carbon dioxide and ether, the solution could be kept for a period of twenty hours without renewal of the refrigerant. The apparatus in which the electrolysis was carried out was immersed in a bath of solid carbon dioxide and acetone contained in a large cylindrical silvered vacuum vessel. This mixture can be maintained at a practically constant tempera- ture by blowing a steady stream of air through it ; the temperature, moreover, may be varied within certain limits by altering the rapidity of the air current.

The Validity of FAEADAY'S Law.

The measurement of the transport number depends on FARADAY'S law, and although this is known to hold rigidly for aqueous solutions, there is no evidence as to its validity for solutions such as those under investigation. Experiments were therefore undertaken with the object of testing the law.

This was accomplished by comparing the weight of silver deposited in a voltameter

T '2

140 DR. B. D. STEELE, DE. D. McINTOSH AND DE. E. H. ARCHIBALD

with the volume of the hydrogen evolved at the cathode during the electrolysis of solutions in hydrogen bromide. The apparatus used (fig. 1) consisted of a tube A with a coiled platinum wire p sealed through the bottom and projecting about an inch

into the tube, the stem of the projecting part being covered with blue enamel glass. Electrical contact with this electrode, which was used as cathode, was made by means of mercury contained in the tube b. The inner cell C was provided with a long capillary d, which passed through the rubber cork e, and served for delivering the hydrogen into a measuring tube. The anode g consisted of a ring of platinum wire, which was attached to the tube /.

In carrying out the experiments the tube A was first immersed in a bath of carbon dioxide and ether ; the solution to be electrolysed was then run in, and C, which had been previously cooled, placed in position. A silver voltameter was then placed in the circuit, and current from a battery of about 60 volts was passed through the cell. The hydrogen evolved in A escaped through d, and was collected and measured.

Two experiments of this nature were carried out, the details of which are as follows : —

Experiment 1. Solution of diethylamine in hydrogen bromide. E.M.F. = 50 volts. Current = 0'091 ampere. Silver deposited in voltameter = 0'1894 gramme,

equivalent to 19 '8 cub. centims. hydrogen at 0° and 760 millims. Hydrogen evolved = 19'7 cub. centims. at 0° and 760 millims.

Experiment 2. Solution of acetophenone in hydrogen bromide. E.M.F. = 60 volts. Current = 0'190 ampere. Silver deposited in voltameter = 0'1661 gramme. Hydrogen equivalent = 17 '31 cub. centims. at 0° and 760 millims. Hydrogen evolved = 17 '3 8 cub. centims. at 0° and 760 millims.

These experiments were considered sufficient to show that FARADAY'S law is valid for solutions of organic substances in hydrogen bromide.

Fig. 1.

The Method of Analysis.

As the total increase in concentration which had to be measured amounted to only a few centigrammes, it was necessary to carry out the analysis with a high degree of accuracy. This was found to be extremely difficult on account of the very high vapour-pressure of hydrogen bromide even at temperatures near its freezing-point,

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

141

and it was only after months of failure that an apparatus was designed, by means of which sufficiently accurate analyses were obtained.

The apparatus consisted of two parts, the transferrer and the absorber.

The transferrer (T, fig. 5, see p. 144) consisted of a wide H-shaped tube, with a capillary tube a passing through three of its branches, the two tubes being sealed together at b and &', as shown in the figure. By filling the space between the tubes with a mixture of carbon dioxide and ether, the capillary between 6 and I' could be cooled to —81°. In fig. 5 the transferrer is shown when placed in the electrolytic cell.

The absorber (fig. 2) consisted of a stoppered tube A, connected by C with the

-e

Fig. 2.

Fig. 3.

bubbler B ; this bubbler was so constructed that it was impossible for water to be either ejected from the apparatus or sucked back into A. A smaller vessel b of the same type was contained within the apparatus. The calcium chloride tube d, which was provided with a stop-cock, was attached to B by a ground joint.

The method of using the apparatus was as follows : — A quantity of moist garnets were first placed in the tube A, and a quantity of glass beads in the section c of the absorber. The requisite amount of distilled water was then placed in the bubblers B and b, and the tube d and stop-cock f were replaced. The apparatus was then weighed, a glass counterpoise of approximately the same size, shape, and weight being used. The tube A was next immersed in a mixture of solid carbon dioxide and ether, and after the cap k, fig. 3, had been placed in position by means of the rubber cork m,

142

DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

the limb of a transferrer was passed through k, and the whole apparatus made air- tight by means of a piece of rubber tubing, e.

The object of the cap k was to prevent the limb of the transferrer from coming into contact with the rubber grease, with which the stoppers were lubricated, and it was constructed so that when in position its narrow portion was exactly in the axis of the tube g.

The transferrer was then packed with the carbon dioxide and ether mixture, and the hydrogen bromide blown into A by means of a small indiarubber bellows. The rubber tube e was then cut away, after which the transferrer and then the cap k were removed, and the stop-cock f was re-inserted.

The tube A was finally removed from the cold bath and the hydrogen bromide allowed to boil off. This always took place steadily, provided the vessel contained moist beads or garnets ; in the absence of these, or if they were dry, the violent bumping which resulted was liable to blow out one or other of the stoppers. The hydrogen bromide as it boiled off passed through C, and was almost completely absorbed at the surface of the water in the outer portion of B ; a small quantity ot gas bubbled through the hole h and was absorbed inside ; very occasionally a few bubbles passed through b, where any traces of acid which might have passed through the larger portion of the apparatus were absorbed. After all the hydrogen bromide

had evaporated the stop-cock e was closed, the apparatus immersed in distilled water, carefully wiped and again weighed, the necessary correction being made for the increase of volume of the liquid contained in it. The increase in weight gave the amount of solution that had been used. The contents of the absorber were next washed into a large beaker and the hydrogen bromide determined by titration with a twice normal alkali solution, which had been carefully standardised and was free from carbonate. The difference between the amount of acid found in this manner and the amount of solution actually weighed gave the weight of the dissolved substance. The alkali was contained in a weighing burette of the pattern shown in fig. 4. In order to deliver from this burette, the cap 6 was removed and the stop-cock opened, when by blowing through the side tube d the liquid was forced through the tube c. The burette was weighed to 0-002 gramme, a glass counterpoise of approximately the same volume being used.

Phenolphthalein was used as indicator, and an excess of one or two drops of alkali added, the exact amount of excess being determined by titration with a twentieth normal solution of hydrobromic acid.

d.

Fig. 4.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 143

In order that the determination of acid by titration should be strictly comparable with the weighing of the solution, the alkali was standardised by direct comparison with about 60 grammes of pure hydrogen bromide which was weighed in the absorber. Duplicate standardisations of the same alkali solution gave the following figures for the amount of hydrobromic acid equivalent to 1 gramme of alkali solution :

(1) 1 gramme solution = 0'149293 HBr,

(2) 1 „ 0-149301 HBr.

The Electrolysis.

In designing an electrolytic cell it was necessary to consider the changes of density which were brought about during electrolysis, and to construct an apparatus in which the lighter solution would be formed at the top and the heavier at the bottom. Preliminary experiments were undertaken to ascertain the influence of an increase in concentration of the dissolved substance on the density of the solution, and it was found that in all the cases examined the less concentrated solution was the heavier.

The experiments which had been conducted to test FARADAY'S law having shown that the bromine is carried to the anode, the apparatus was designed to enable the cathode solution to be analysed. The anode solution was neglected, as its analysis was complicated by the presence of bromine.

The apparatus which has been employed is shown diagrammatically in fig. 5. It consisted of a U-tube, both arms of which were provided with side tubes. The anode a consisted of a platinum wire, which was sealed through the bottom of the side tube 0, the other end of the wire projecting into the glass tube h, by means of which connection with the battery could be made. A small side tube n was also attached to the same arm of the U-tube, and there was a constriction at r into which the hollow stopper s was ground to fit tightly. The stopper A- was sealed to a branch of the transferrer T', and a hole was bored in its shoulder to allow the free passage of liquid through the transferrer.

It was found in the preliminary experiments that a considerable amount of mixing was occasioned by the escape of hydrogen at the cathode, and in order to reduce this to a minimum, the side tube P was attached to the apparatus. In the centre of P a narrow tube u was fastened, inside which the cathode was placed, so that the escaping bubbles of hydrogen were confined to this tube and very little mixing took place outside P. A small hole had been blown in the wall of the tube u so that the pressure should be equal at all parts of the surface of the liquid. The tube P was made long enough to be held in the clamp outside the vacuum vessel in which the apparatus was placed and its end was closed by a stopper w, through which passed the platinum wire which was used as cathode. One end of the transferrer T was made long enough and bent so as to reach to the bottom of the U-tube, and both the

144 DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

transferrers were attached to the open ends of the main apparatus by pieces oi india- rubber tubing, t and t'.

Fig. 5.

In carrying out an experiment the method of procedure was as follows : — The apparatus was placed in the vacuum vessel and held rigidly by a clamp which grasped the tube P. The open ends of the transferrers T and T' were then closed by rubber tubing and glass plugs, and a tube filled with phosphoric oxide was attached to n. The stop-cock w with the cathode attached was then removed and a third transferrer (not shown in the figure) inserted in the tube P. The acetone was then placed in the vacuum vessel and solid carbon dioxide added till the temperature had fallen to the desired point. The outside arm of the third transferrer was provided with a filter of glass wool and reached to the bottom of a vessel containing 10 cub. centims. more of the solution than was necessary for the electrolysis. After the whole of this liquid had been forced into the apparatus by blowing air from a rubber bellows into the vessel containing the liquid, the transferrer was removed and the stopper w replaced.

Before commencing the experiment three absorbers had been filled with distilled water and weighed. The transferrer T was next attached to one of these in the

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 145

manner previously described and 10 cub. centims. of the solution were taken out and analysed. This analysis gave the concentration of the solution before electrolysis. A silver voltameter and a milliampermeter having been placed in the circuit, the electrolysis was started by connecting the electrodes to the terminals of a battery, the voltage of which could be varied within wide limits, and the voltage was adjusted so as to give a current of not more than 12 or 14 milliamperes. If a larger current than this was employed, the heating effect was found to cause mixing of the liquid by convection. After the electrolysis had been continued for about two hours it was stopped by removing the cathode from the voltameter, and the liquid in the cell was separated into two portions by inserting the stop-cock ,s into its socket. An absorber was then attached to the transferrer T' and the column of liquid contained within the dotted lines was blown into the absorber, which was then removed. The third absorber was next attached to the transferrer T and the solution contained in the apparatus forced into it by blowing through w, the tube r acting as a syphon to remove the liquid contained in P, and finally the third absorber was removed and weighed. Three solutions were thus obtained for analysis, namely: (1) the 10 cub. centims. which had been removed before electrolysis and gave the original concen- tration ; (2) the small quantity taken out in the second absorber (which should be of the same concentration as (1)); and (3) the solution surrounding the cathode, which gave the change of concentration brought about by the electrolysis.

In order that an experiment should be successful it was necessary that the foregoing procedure should be strictly followed. Identical values for the original concentration and that of the middle portion have never been obtained unless the whole solution was first placed in the apparatus and all three portions were taken from it.

At least half-a-dozen other methods have been tried without success. Fortunately it is easy at the close of an experiment to see if any mixing has taken place, from the fact that bromine is liberated at the anode, where it forms a deep red solution in the hydrogen bromide. If the experiment has been successful, this solution remains as a very clearly defined layer surrounding the anode, and the coloration does not extend more than about 1'5 centims. up the tube. On the other hand, if mixing has taken place, as may happen either if the current is too large or if the temperature of the bath is allowed to vary, the bromine is distributed throughout the solution and no clearly defined layer is seen at the anode.

Experimental Results.

The results of the experiments are contained in the following table, in which the concentration of the various solutions is expressed under N, which gives the number of gramme equivalents of dissolved substance per litre of solution, the percentage also being given in the 3rd column. The 4th column contains the weight of silver deposited on the cathode of the silver voltameter, the 5th column gives the weight of substance transported, the 6th column the cation transport numbers.

VOL. cov. — A. u

146

DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

Number of experiment.	N.	Percentage of composition.	Deposited in cathode.	Transported.	f
1 SERIES 1. ETHER. N = 1 • 0 approximately.
2 3 7	1-07 1-14 1-06	3-96 4-26 3-92	•0995 •0918 •1120	•0567 •0530 •0607 Mean =	•83 •82 •79
•82
SERIES 2. ETHER. N = 1-7 to 2-04.
6 8 17	1-72 1-80 2-04	6-61 6-85 8-03	•0988 •0997 •0918	•0489 •0212 •0366	•73 •31 •58
SERIES 3. TRIETHYI.AMMONIUM BROMIDE. N=-5to-75.
5 9 10	•75 •515 •622	6-54 4-47 5-37	•1064 •0921 •0817	0-382 0-288 0-300 Mean =	•21 •18 •22
•20
SERIES 4.	TRIETHYLAMMONIUM BROMIDE. N = 1-04.
11 12	1-04 1-05	8-98 9-01	•0907 •1051	•0473 •0700 Mean =	•31 •39
•35
	SERIES 5. ACETONE. N = TO.
15 16	1-05 1-01	2-98 2-87	•0863 •0938	•0151 •0141 Mean =	•41 •36
•38
	SERIES 6. ACETONE. N = 1-8.
13 14	1-83 1-82	5-37 5-32	•0931 •0878	•0361 •0372 Mean =	•91 •99
•95
SERIES 7. METHYLHEXYLKETONE.
18 19	•90 •90	5-34 5-36	•0965 •0880	•0398 •0384 Mean =	•38 •41
•39
SERIES 8. METHYLHEXYLKETONE.
20 21	1-80 1-80	11-83 11-87	•1015 •0943	•0815 •0830 Mean =	•75 •82
•77

ON THE HALOGEN HYDEIDES AS CONDUCTING SOLVENTS.

147

SUMMARY.

Substance.	N.	Mean value of p.
Ether	1 -0	•82
Triethylammonium bromide .... » » .... » !> .... Acetone	0-5 0-62-0-75 1-04 1-0	•18 •2-2 •35 •38
	1 -82	•95
Methylhexylketone	0-9	•39
	1-8	•77
:

With the exception of Nos. I and 4, which unfortunately were lost, all the experiments which have been made are given in the tables. The transport number of each substance has been measured at two concentrations, the more conceTitrated solution usually containing about twice as much solute as the other. It will be seen that the cation transport number is always increased by increase of concentration and that the amount of disagreement between parallel experiments, although in some cases approaching 10 per cent., is never sufficient to leave any doubt as to the influence of change of concentration. This change from analogy with aqueous solutions indicates an increase in the complexity of the cation as the solution becomes stronger, but the measurements of conductivity and of the molecular weight, which are given in Part II., do not appear to confirm this conclusion. The significance of the change will be discussed in Part IV.

A special significance is to be attached to the results of Series 2, for the following reasons : — In the foregoing description of the method of analysis it has been explained that after the solution was transferred to the absorber the hydrogen bromide was allowed to evaporate and to become dissolved in the water. During the evaporation the temperature of the liquid in A gradually rose until finally it reached that of the atmosphere, when the liquid which remained was a saturated solution of hydrogen bromide in ether.

At this stage little or no decomposition occurred in a solution which had not been electrolysed, or in a dilute solution which had, but in the case of the cathode portion of a concentrated electrolysed solution the decomposition which occurred was sufficient to give results so discordant as those tabulated in Series 2. This behaviour seems to indicate the formation at the cathode of some extremely unstable substance during the electrolysis of these solutions, and it is possible that a compound is formed by the union of two or more discharged cations by a reaction similar to that by which persulphuric acid results from the electrolysis of sulphuric acid.

u 2

148 DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

PART IV.

The Abnormal Variation of Molecular Conductivity, etc.

•i

By B. D. STEELE, D. MC!NTOSH and E. H. ARCHIBALD.

IN discussing the nature of those inorganic liquids which are able to act as " ionising" solvents, WALDEN (' Zeit. fur anorg. Chemie,' 1900, 25, p. 209) states that "a measurable dissociation (ionisation) occurs only in combinations of the elements of the 5th and 6th groups of the periodic table and in compounds of these elements with hydrogen and the halogens." We have shown in Part I. of this investigation that the hydrides of the halogen elements and of sulphur belong to the class of " ionising " solvents, so that this class consists of compounds of the elements of the 5th, 6th, and 7th groups amongst themselves and with hydrogen. Attempts have been frequently made to arrive at some generalisation connecting the so-called " ionising" power with . certain physical constants of the pure solvents.

Thus, according to NERNST and THOMSON, a close relationship exists between the dissociating power and the dielectric constant. These investigators were led to look for this relation by the consideration that the force with which two electrically charged bodies attract or repel each other depends on the magnitude of the dielectric constant of the separating medium, and as the ions are to be regarded as electrically charged bodies, the force attracting two unlike ions will be more weakened, and dissociation aided, in a solvent of high than in one of low dielectric constant.

This expectation is only partially realised in the parallelism which exists for a great number of solvents between the two properties in question ; thus liquefied ammonia which possesses a low dielectric constant is a better dissociating solvent for some substances than water which has a high dielectric constant ; moreover, the majority of electrolytes are far more dissociated in water than in hydrocyanic acid or in hydrogen peroxide, although the dielectric constant of water is less than that of either of these liquids. No measurements of the dielectric constant of the halogen hydrides or of sulphuretted hydrogen have yet been made.

DUTOIT and ASTON (' C. R.,' 1897, 125, p. 240) have attempted to show that ionic dissociation occurs only in solvents in which the molecules are associated, but, although numerous instances occur in which this parallelism obtains, it is by no means a general rule. Thus, although both ammonia and sulphur dioxide are unassociated liquids, both are able to form conducting solutions ; and although the hydrides of bromine, iodine, and sulphur are unassociated, and hydrogen chloride is

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 149

associated, all four compounds are equally able to act as conducting solvents, whilst this property is not possessed by hydrogen phosphide, which is associated.

BRUHL (' Zeit. Phys. Chem.,' 1898, 27, p. 319) has pointed out that unsaturated compounds, as a rule, are good conducting solvents. A consideration of the non- conducting unsaturated solvents phosphorus hydride and trichloride, and of the conducting saturated solvent phosphorus oxychloride, is sufficient to show that this is not a general rule.

The heat of vaporisation (OBACH, 'Phil. Mag.,' 1891, (5), 32, p. 113) is a fourth property which has been suggested as being intimately connected with the dissociating power of the solvent. In this case, as in that of the others considered, the connection is very obscure and many exceptions occur.

The temperature coefficients of conductivity and of viscosity are approximately equal in the case of aqueous solutions. This is not so in solutions in the solvents examined by us, although in these, also, a rise of temperature conditions an increase of conductivity and a decrease of viscosity. It is interesting to note that the increase of conductivity is, in nearly all cases, greater than the decrease of viscosity.

The foregoing summary shows that failure has attended every attempt which has been made to express the power of forming conducting solutions as a function of the solvent only.

As a matter of fact, every solvent exhibits a very marked selective action an regards the nature of the conducting solute. Thus water dissolves the majority of salts to form solutions which conduct the current ; organic bodies also, other than salts, are in many cases soluble, but the solutions are not conductors. Hydrocyanic acid behaves similarly to water, but only a few salts are appreciably soluble in this solvent. Ammonia, sulphur dioxide, and some other solvents form conducting solu- tions, not only with many salts, but also with a few organic substances not usually classed as electrolytes. The halogen hydrides, on the other hand, form conducting solutions with non-saline organic substances, as well as with salts of the ammonium bases, but such solutions are not formed with metallic salts.

It is evident, therefore, that the ability to form a conducting solution is a function of both the solute and the solvent, and this has been recognised in the various attempts that have been made to connect the ionising power of a solvent with its tendency to form compounds with the solute. Indeed CADY ('Jour. Phys. Chem.,' 1897, 1, p. 707) was led to investigate the conductivity of solutions of substances in ammonia from the analogy between the water and the ammonia compounds of copper sulphate.

KAHLENBERG and SCHLUNDT (' Jour. Phys. Chem.,' 1902, 6, p. 447) express the opinion that conductivity is due to mutual action between the solute and the solvent ; and an attempt to obtain experimental evidence in support of this view has been made by PATTEN (' Jour. Phys. Chem.,' 1902, 6, p. 554).

150 DR. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

For many solvents the substances which dissolve to form conducting solutions may be broadly designated as those which enter into combination with the solvent.

Thus the metallic salts as a class are characterised by their tendency to form compounds with water, while non-saline organic bodies as a class are not able to form such compounds.* Many, but not all, salts which form ammonia compounds dissolve in ammonia to form conducting solutions.

Compounds of the solute with the solvent are also clearly indicated in the case of many conducting solutions in sulphur dioxide, and WALDEN and CENTNERSZWER (' Zeit. Phys. Chem.,' 1903, 42, p. 432) have isolated and investigated two such compounds containing potassium iodide and sulphur dioxide.

In the halogen hydrides we find that the only substances which conduct are the amines, alcohols, ethers, ketones, &c., all of which are able to enter into combination with the solvents. Many similar cases have been observed amongst organic solvents, and as an example of these reference may be made to solutions in amylamine (KAHLENBERG and RUHOFF, ' Journ. Phys. Ch.,' 1903, 7, p. 254).

The study of the behaviour of aqueous solutions has led to ARRHENIUS' theory of ionic dissociation and to VAN 'T HOFF'S theory of solutions ; and numerous investigations have been undertaken with the object of testing these theories, when applied to solutions in non-aqueous solvents.

As a result it has been found that, as required by the theory, most substances, when dissolved in ammonia, sulphur dioxide, hydrocyanic acid, and some other solvents, show an increase of the molecular conductivity, /j., with dilution, but that the opposite change occurs in solutions of a few substances in the same solvents. This difference in behaviour cannot therefore be conditioned by the nature of the solvent only, although if we consider the inorganic hydrides as solvents, we find that n varies normally, that is to say, increases with dilution, in solutions in water and ammonia, hydrides, namely, of elements in the first series of the periodic table, whereas /A decreases with dilution in solutions in the remaining hydrides, the variation therefore being abnormal, t

The results of the molecular weight determinations in non-aqueous solvents are, as a general rule, not concordant with the conductivity results, many conducting solutions being known in which, contrary to expectations, the dissolved substance appears to be associated.

* The view that compounds of the solute and the solvent exist also in solutions appears to be steadily gaining ground, see MORGAN and KANOLT ('Jour. Amer. Chem. Soc.,' 1904, 26, p. 635) and JONES and GETMAN (' Zeit. Phys. Chem.,' 1904, 49, p. 390).

t Amongst others the following cases have been observed of solutions in which the molecular conductivity decreases with dilution : Silver nitrate, cadmium iodide, and ferric chloride in amylamine (KAHLENBERG and RUHOFF, 'Jour. Phys. Chem.,' 1903, 1, p. 284); Antimony bromide and phosphorus pentabromine in bromine (PLOTNIKOFF, 'Jour. Russ. Phys. Chem. Soc.,' 1902, 34, p. 466 ; 1903, 35, p. 794) ; Hydrogen chloride in ether and in amyl alcohol (KABLUKOFF, 'Zeit. Phys. Chem.,' 1889, 4, p. 429); Hydrogen chloride in cineol (SACKUR, 'Ber. D. Chem. Ges.,' 1902, 35, p. 1242), &c.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 151

The consideration of these abnormalities has led KAHLENBERG to conclude that the theory of ionic dissociation is not applicable to the majority of conducting solutions.

It is our object to show that the abnormal behaviour of solutions in the solvents examined by us can be simply and consistently explained in terms of the theory of ARRHENIUS, if the assumption is made that the original dissolved substance, being itself incapable of undergoing ionic dissociation, either polymerises or combines with the solvent to form a compound containing more than one molecule of the solute, and that the polymer, or compound, as the case may be, then acts as the electrolyte.

Those non-saline organic substances which are able to conduct the current when dissolved in certain solvents are considered by WALDEN (' Zeit. Phys. Chem.,' 1903, 43, p. 385) to be abnormal, in view of their usually well-known constitutions and their behaviour in aqueous solutions, but if the foregoing assumption of the formation of compounds be made, these substances are not more abnormal electrolytes than ammonia, which with water forms the compound ammonium hydroxide.

It has been suggested that the existence of compounds of the solute with the solvent is proved by the abnormal variation of the molecular conductivity, to which reference has been made ; but the following considerations will show that an increase of fi with dilution furnishes no evidence for or against the occurrence of such compounds.

Let us suppose that a reaction between the solute, AB, and the solvent, CD, takes place according to the equation

AB + CD^±AB, CD, and let

a b c

be the active masses of the three substances.

Now, provided that moderate dilutions are used, we are justified in regarding b as

constant, when from the law of mass action - = constant.

a

Now if conduction is due to the dissociation of the compound ABCD, the number of dissociated molecules is given by etc, where a is the degree of ionic dissociation ; but c = Ka ; therefore the concentration of the ions is equal to poiKa if p is the number of ions formed from one molecule of solute.

But the specific conductivity K of the solution is proportional to the ionic

concentration, and therefore

K = paJcKa = a.K'a (l),

and since the molecular conductivity

K

= - = aK', a

it must vary with a, that is to say, it must increase with dilution even when a compound of the solute with the solvent is formed.

152 DR. B. D. STEELE, DR. D. McINTOSH AND DR. F, H. ARCHIBALD

This is the case, for example, in an aqueous solution of ammonia, to which reference will be made later.

If, however, we assume that two or more molecules of AB unite to form a compound which undergoes ionic dissociation, AB itself being unable to conduct the current, then the molecular conductivity may decrease with dilution whether the solvent enters into the composition of the electrolytic compound or not.

If we consider the two cases :—

(1) A compound of n molecules of solute with m molecules of solvent is formed according to the equation

n (AB) +m (CD) ^± (AB). (CU)mf the active masses being

a b and c.

Then, if we again consider sufficiently dilute solutions, b may be regarded as

constant, and

kan = k'c or c = Ka".

If ionic dissociation occurs so that a* of the compound is ionised, then, as before, the ionic concentration = pa.c = paKa*. The specific conductivity

K = pVaKa" = aK'a" (2).

The molecular conductivity

a = - = aKV-1.

a

and since the dilution

V = -

a'

K = aK'V-", or

»cV- = aK'.

* In the development of this relation no .assumption has been made us to the nature of the ionisation of the electrolyte.

If we consider the second case, for example, there are a number of ways in which the compound A,,B,, can ionise.

Thus 0 Q

(1) AnB,t 7-*- A»Bn-i + B.

«©

(2) AA-^Att + and generally «0

(3) AnBn-^-»AA

If dissociation takes place according to the first of these equations, 2 ions result from the dissociation of 1 molecule of the electrolyte.

If according to the second equation, the number of ions is (n+1), and, generally, the number is (r+l).

Now whatever value r may have, the number of ions present is given by a (r+l) and is therefore proportional to a.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 153

(2) Combination between the solute and the solvent does not occur, but an ionising polymer of the solute is formed. In this case the equation is

ABB,,, the active masses are

a and c, and from the law of mass action

lea* = k'c and c - Ka", which leads to the same expression as before, namely

K = «KV, p = aKV-', and *V" = «K'.

Now, since the number of molecules always increases with dissociation, « must increase with dilution in whatever manner dissociation takes place ; but unless the increase in a is greater than the diminution in a""1 which is brought about by dilution, the sum of the effects due to the variation of a and of a"'1 in the equation

/,„»-!

p = aK'a*

must produce a diminution in p. with increasing dilution.

It follows, therefore, that the molecular conductivity may decrease with dilution in the case of any conducting solution in which the electrolyte is a compound of two or more molecules of the dissolved substance, whether it is a simple polymer or a compound containing one or more molecules of the solvent.

It follows, also, from the equation

K = aK'an

that in the case of a solution in which the ionic dissociation was approaching completion, in which a therefore varied but slightly, the specific conductivity of the solution should be very nearly proportional to the nth power of the concentration of the dissolved substance ; for such solutions we therefore have the relation

JL = ,cV" = K'.

Although we have shown that it is not necessary that union with the solvent should occur in order to bring about an abnormal variation of p, we nevertheless consider that the formation of such compounds* affords the best explanation of the behaviour of solutions of organic substances in the halogen hydrides and in sulphuretted hydrogen.

* See also WALKER, 'J. C. S.,' 1904, vol. 85, p. 1082, and WALKER, MC!NTOSH, and ARCHIBALD, 'J. C. S.,' 1904, vol. 85, p. 1098.

VOL, CCV, — A, X

154 DE. B. D. STEELE, DE. D. McINTOSH AND DE, E. H. AECHIBALD

We are led to this conclusion by a consideration of the following facts :—

(1) Large quantities of heat are evolved when conducting solutes are added to either of these solvents. This heat evolution we take to indicate chemical union.

(2) Compounds containing a varying number of molecules of solvent have been isolated (ARCHIBALD and MC!NTOSH, 'Jour. Chem. Soc.,' 1904, vol. 85, p. 919).

(3) The ionisation of a compound such as ((CH3)2CO),lHBr is much easier to understand than that of a simple polymer such as ((CH3)2CO)n.

In order to apply the foregoing conclusions to a specific case, we will consider a solution of acetone in hydrogen bromide.

According to our hypothesis, such a solution contains a compound of acetone and hydrogen bromide, the formula of which we will assume to be Ac3(HBr)Bl.

This compound dissociates simultaneously in two different ways, a certain number of molecules being dissociated into acetone and hydrogen bromide, other molecules being dissociated into ions, and the ratio of the number of molecules undergoing each dissociation will be constant.

Applying the equation ^ = «K'as, we see that the molecular conductivity will increase with increasing concentration of the acetone, the increase — neglecting variation of a — being proportional to the square of the concentration. Similarly, we see that if a is nearly constant, the specific conductivity will be proportional to the cube of the acetone concentration. If, however, a is not constant, then K/a will be proportional to the cube of the acetone concentration.

This conception of an intermediate compound which is able to break up in different ways is by no means new to chemists, and the solution of ammonia, which we have already referred to, furnishes an example of such a case, which, in many ways, is analogous to the preceding.

The compound that is formed in this solution is ammonium hydroxide, and the dissociations are

(1) NH4OH:z±NH3+H20;

(2) NH4OH -± NH4 + OH.

The relation between specific conductity and concentration for such a solution has been already developed in equation (1) K = aK'a, which is a special case of equation (2). Here again

K T7" /

M---.K',

so that

a = £ ; but a = JL K ^

so that K' is simply the molecular conductivity at infinite dilution.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 155

In order to test the conclusions that have been arrived at, we require to know the concentration of the unassociated solute for each dilution.

If A gram-molecules of the solute are dissolved in one litre of solvent, and if there are formed c gram-molecules of the electrolytic compound, then, if n molecules are required to form one molecule of the compound, nc molecules of solute will have been used, when equilibrium is established, so that the equilibrium equation is

k(A-nc)a = k'c, or

c = K (A-wc)",

and the general expression for the specific conductivity becomes

ic = aK'(A-wc)".

No solution of any substance in any one of the halogen hydrides lias yet been found with high value for the conductivity, a fact which may be assigned to one of three causes, namely, either (1) the concentration, or (2) the coefficient of ionisation of the electrolyte is small, or (3) the ionic velocities may be very small. If we assume the first to be the most probable cause, A— nc will not differ much from A, and we may without sensible error make use of the values for the total concentration in applying the above equation to our results.

This has been done, the equation being used in the form of /cV" = aK', and the results of the calculations are given in Tables I. and II.

Table I. contains the values of V and of /cV" (or aK7) for those solutions in which n = 2, that is, in which two molecules of solute combine with the solvent to form one molecule of the electrolyte. Table II. contains the similar value for those solutions for which n = 3.

It will be noticed that in some cases the figures exhibit considerable irregular variation. This is to be expected from the fact that the measurements of conductivity and of concentration are subject to considerable experimental error. These errors were not specially guarded against, as our object was to establish beyond question the nature of the variation of p. with V rather than to obtain accurate measurements, which, in the present state of our knowledge, would not possess any special value.

The figures for K at very high dilution are, in some cases, quite valueless as a test of our hypothesis, on account of the enormous influence of very slight errors of observation at these dilutions.

The results contained in Tables I. and II. are shown graphically in figs. 1 and 2 respectively.

x 2

156 DE. B. D. STEELE, DE. D. McINTOSH AtfD DE. E. H. AECHIBALD

TABLE I.

V

3.K = «V2

ACETONITRILE IN HYDROGEN CHLORIDE.

= 21-7, 8-33, 4-73, 3-22, 2-17, 1-09, 0-81 = 32-8, 20-3, 18-5, 17-0, 14-8, 8-8, 8-3

ACETAMIDE IN HYDROGEN CHLORIDE.

= 29-4, 12-8, 8-62, 4-65, 4-15, 2-86, 2-13, 1-54, 0-95, 0-51 = 46-7, 40-0, 36-8, 29-6, 28-7, 23'4, 20-0, 16-6, 11'5, 6'42

TRIETHYLAMMONIUM CHLORIDE IN HYDROGEN CHLORIDE.

= 71-4, 37-0, 20-4, 16-1, 11-6, 9-43, 7 -69, 6-13, 5-00, 4-25, 3-64, 2-99 ZK = K-V2 = 128-6, 667, 465, 437, 3G5, 343, 334, 316, 302, 285, 281, 254

HYDROCYANIC ACID IN HYDROGEN CHLORIDE.

V = 41-4, 21-3, 14-1, 10-2, 9-2, 7 -35, 4-90, 4-10, 3-12, 2 -56, 1-79, 1-23

=>-V2 = 21-6, 19-6, 14-0, 11-0, 12-0, 10-9, 10-.3, 11-5, 11-5, 11-5, 10'4, 9-4

ETHER IN HYDROGEN CHLORIDE.

12-5, 5-0, 3-12, 1-92, 1-45, 1-09, 0'88, 0-72, 0-61 1-75, 1-15, 1 22, 1-82, 2-05, 2-22, 1-93, 2-08, 1-88

TETRAMETHYLAMMONIUM CHLORIDE IN HYDROGEN BROMIDE.

V 200, 21-3, 14-1, 10-5, 9'4

= K-V2 = 1080, 190-0, 149-0, 131-0, 125-0

TETRAMETHYLAMMONIUM BROMIDK IN HYDROGEN BROMIDE.

= 62-3, 34-5, 11-8 2 | = 43-7, 25-0, 14-8

TETRAMETHYLAMMONIUM IODIDE IN HYDROGEN BROMIDE.

V I = 62-5, 34-5, 22-1 aK = «V2 = 546, 360, 290

TRIETHYLAMMONIUM CHLORIDE IN HYDROGEN BROMIDE.

143-0, 50-0, 27-7, 15-6, 8-33, 5-26, 3-33, 2-17, 16-1 27-4, 11-0, 13-8, 12-9, 16-6, 17-3, 16-4, 13-5, 13-3

ETHER IN HYDROGEN BROMIDE.

16-6, 5-55, 4-00, 2-00, 1-54, 1-23, 1-03, 0-68, 0-47 •083, -078, -096, -212, -199, -187, -170, -124, -341

ON THE HALOGEN HYDEIDES AS CONDUCTING SOLVENTS. TABLE I. (continued).

157

V

O.K = *

ACETAMIDE IN HYDROGEN BROMIDE.

90-9, 58-8, 23-3, 14-5, 10'2, 6-66, 3-85, 3'03, 2-08, 1-41 5-94, 5-88, 6-3, 6-1, 5-8, 6-25, 5-65, 5-45, 4-92, 4-45

ACETONITRILE IN HYDROGEN BROMIDE.

33-3, 4-76, 3-70, 2-50, 1-85, 1-47, 1-22, 0-96, 0-72 4-61, 5-15, 4-80, 6-2, 6-4, 6-8, G'7, G-7, 7-2

ETHYL PROPIONATE IN HYDROGEN BROMIDE.

12-15, 7-14, 5-26, 3-45, 2-63, 1-92, 1-39 •62, 1-14, 2-00, 1-7, 2-16, 2-30, 2-28

ORTIIO-NITROTOI.UENE IN HYDROGEN BROMIDE.

25, 16-6, 12-5, 11-1, 8-33, 3-85, 2-38, 1-50, 0-92, 0-66

1-00, 1-1G, 0-87, 1-11, 0-92, 0-71, 1-OG, I'OO, 0-94, 0-845

PIPERIDINE IN SULPHURETTED HYDROGEN.

9-09, 1-18, 0-90, 0-75, 0-G4, 0-55 •18, -34, -35, -34, -31, -28

TETRAETHYLAMMONIUM CHLORIDE IN SULPHURETTED HYDROGEN.

= 34-5, 11-0, 4-25, 3-32, 2-93 = 59-4, 37-G, 16-4, 13-4, 11-3

158 DR. B. D. STEELE, DE. D. McINTOSH AND DE. E. H. AECHIBALD

TABLE II.

V. «V3.	II V.	KV3.
1 ACETONE IN HYDROGEN BROMIDE.
8-33 6-94 5-00 8-50 3-23 8-03 2-00 5-6	1-64 1-35 1-07 0-75			6-2 5-9 4-9 3-2
V. *V2.	aK-«V3.	V.	(cV*.	aK = «V3.
TRIETHYLAMMONIUM CHLORIDE IN HYDROGEN IODIDE.
27-8 1-95 21-7 2-4. 15-4 3-56 .10-4 4-47 8-55 5-56 7-14 5-71	54 52 55 47 47-5 41	5-55 4-50 3-85 3-13 2-50 2-17	G-38 6-16 7-35 7-42 7-42 7-77	35-4 30-0 28-3 23-4 18-6 16-9
ETHER IN HYDROGEN IODIDE.
10-0 -2 5-88 -41 3-33 -73 2-46 1-50 1-79 1-50	2-0 2-42 2-44 3-69 2-69	1-49 1-25 1-06 0-88	1-65 1-75 1-90 1-94	2-46 2-19 2-01 1-70
ETHYL BENZOATE IN HYDROGEN IODIDE.
16-6 -233 7-14 1-22 4-76 2-23 3-45 3-54	3-8G 8-67 10-6 12-1	2 • 56 2-04 1-66 1-37	4-22 4 -69 4-94 4-93	10-8 9-6 8-2 0-68
TRIETHYLAMMONIUM CHLORIDE IN SULPHURETTED HYDROGEN.
71-4 8-G 12-8 2-69 8-33 2-75	61-3 34-4 22-9	4-00 3-13 2-50	3-48 3-66 3-95	13-9 11-5 y-88
NICOTINE IN SULPHURETTED HYDROGEN.
66-7 2-0 14-3 -57 6-67 -40 4-00 -64	2-67 2-56	2-27 1-92 1-03	•86 •96 •79	1-96 1-84 0-81

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

159

o i

Fig. 1.

(1) Acctiimide in hydrogen bromide.

(2) Tetramethylammonium bromide in hydrogen bromide.

(3) „ iodide „ „

(4) Acetonitrile in hydrogen chloride.

(5) Tetramethylammonium chloride in hydrogen sulphide.

(6) Acetamide in hydrogen chloride.

(7) Triethylammonium chloride in hydrogen chloride.

(8) Tetramethylammonium chloride in hydrogen bromide.

(9) Hydrocyanic acid in hydrogen chloride.

(10) Ether in hydrogen chloride.

(11) Triethylammonium bromide in hydrogen bromide.

(12) Orthonitrotoluol in hydrogen bromide.

(13) Piperidine in hydrogen sulphide.

160

DR. B. D. STEELE, DE. D. McINTOSH AND DR. E. H. ARCHIBALD

5*

i

10

V

Fig. 2.

(1) Triethylammonium chloride in hydrogen iodide.

(2) Nicotine in hydrogen sulphide.

(3) Ether in hydrogen iodide.

(4) Triethylammonium chloride in hydrogen sulphide.

(5) Ethyl benzoate in hydrogen iodide.

(6) Acetone in hydrogen bromide.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. Fig. 3 contains typical curves showing the variation of *V = p. with V for

(1) Substances dissolved in halogen hydrides ;

(2) Potassium chloride in water ;

(3) Sodium carbonate in water ;

(4) A solution of ammonia in water.

161

10 I? 14 16

I/

18 ?0

Fig. 3.

The similarity between the variation with dilution of /cV2 (or /cV") for solutions in the halogen hydrides and that of «:V = p. for aqueous solutions is at once apparent.

Since in the former case *V2 = «K', and in the latter p. = /cV = «/*, and since both p.^ and K' are constants, it is evident that both sets of curves represent a variation in a and that K' represents the value of the molecular conductivity at infinite dilution of the electrolytic compound.

Although the majority of the curves in figs. 1 and 2 are analogous to those for water solutions, some of them exhibit a maximum value for /cV", whilst others are extremely steep, thus indicating a very rapid increase in the value of «.

These irregularities are to be expected, since, as already stated, we have been

VOL. CCV. — A. Y

162 DB. B. D. STEELE, DE. D. McINTOSH AND DE. E. H. AECHIBALD

compelled to use, in the calculation of *V", the total concentration instead of that of the unassociated substance.

It is also possible that more than one type of electrolytic compound is formed in a given solution, as, for example, the compounds ABCD and (AB)2CD ; in which case the total conductivity will be the sum of the conductivities due to the ionisation of each of these compounds. In such a case as this extremely complicated curves might result. Moreover, we cannot strictly apply the equation to the concentrated solutions, since for these the active mass (b) of the solvent is no longer constant.

The fact that the curves, as a whole, are so analogous to those for a simple electrolyte in aqueous solution appears to indicate that, as a general rule, the main effect is due to the ionisation of a single substance.

The equation «:V" = aK' should also be applicable to abnormal solutions in other solvents. This is the case for the solutions investigated by PLOTNIKOFF (' Zeit. Phys. ('hem.,' 1904, 48, p. 224), who found very abnormal variations of /u, for antimony tribromide and phosphorus pentabromide in bromine. The experimental figures for antimony tribromide lead to the following values for V and /cV" : —

V KV3

251, 154,

312, 178,

356, 171,

418, 174,

445, 164,

552, 98,

918 168 !

The molecular conductivity of phosphorus pentabromide in bromine varies so irregularly as to suggest that some disturbing effect is at work rendering the figures valueless.

Another solvent in which p. increases with concentration is amylamine (KAHLEN- BERG and HUHOFF, ' Jour. Phys. Chem.,' 1903, 7, p. 254), and the equation has been applied to the measurements of conductivity for cadmium iodide, ferric chloride and silver nitrate dissolved in this solvent. The results of the calculations when n = 2, 3 and 4 are given in Table III. It will be noticed that maxima are shown in each case.

Passing on now to the consideration of the molecular weight determinations which are recorded in Part II., we find that some of these afford confirmation of our hypothesis, inasmuch as ether and acetone in dilute solutions possess a greater molecular weight than the theoretical.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS.

TABLE III.

163

V.	*V.	*V2.	K-V3.
	SILVER NITRATE	IN AMYLAMINE.
•4001	•530	•212	0-085
•4351	•639	•278	0-121
•5096	•870	•443	0-226
•6206	1-128	•700	0-434
•8629	1-402	1-21	0-021
1-158	1-476	1-71	1-98
1-685	1-376	2-32	3-91
2-302	1-144	2-63	6-06
2-850	0-908	2-59	7-37
3-261	0-744	2-43 7-91
6-330	0-168	1-06	6 • 73
11-45	0-038 0-44	4-98
31-07	0-008 0-24	7-72
81-63	0-002 0-16	1 • 33
CADMIUM IODIDE IN AMYLAMINE.
0-7810 -465	•363	•284
•8909	•534	•476	•424
1-095	•542	•594	•650
1-237	•480	•594	•735
1-450	•346	•502	•728
1-738	0-187	•325	•565
2-473	0-034	•084	•208
5-482	0-002	•Oil	•055
V. A. AV.
FERRIC CHLORIDE IN AMYLAMINE.
5-021 0-217 1 09
13-43 0-158 2-12
18-34 0-138 2-53
27-05 0-086 2-32

We have been unable to ascertain whether the molecular weight reaches a limiting value with dilution, as the experimental errors incidental to measurements at the low temperatures involved prevented the examination of the more dilute solutions.

KAHLENBERG, WALDEN, and others have called attention to many solutions in which, although p. varies normally, the solute is associated.

Y 2

164 DE. B. D. STEELE, DR. D. McINTOSH AND DR. E. H. ARCHIBALD

Thus WALDEN and CENTNERSZWER ( 'Zeit. Phys. Chem.,' 1902, 39, p. 513) found that the molecular weight of potassium iodide dissolved in hydrocyanic acid is twice as large as the normal. ABEGG ('Die Theorie der electrolytischen Dissociation,' p. 103) has pointed out that this can be explained by the assumption that the undissociated substance is polymerised ; in which case a high average molecular weight might occur even with considerable ionisation.

A compound such as (KI)4, for example, if it were completely dissociated into two ions, would have an average molecular weight of 332. We find that in most cases the molecular weight increases with increasing concentration, and although the opposite change occurs in dilute aqueous solution, this variation is the same as that which takes place in more concentrated aqueous solutions.

This will be seen from the following comparison of the figures for acetone dissolved in hydrogen bromide with those for lithium bromide dissolved in water, the latter figures being taken from a recent paper by JOXES and GETMAN ('Zeit. Phys. Chem.,' 1904, 49, p. 390).

(a) Acetone in hydrogen bromide—

c = concentration in gram-molecules per litre = 0'51, 1'17, 1'85, 2'5G ;

- = molecular depression = 4 '5, 4 '5, G'5, 11 '5.

C-

(&) Lithium bromide in water —

c = 0'48, 0-97, 1-94, 3'88 ; A = 4'07, 4'41, 5'31, 7'86.

JONES and GETMAN attribute the apparent increase in the number of molecules in more concentrated solution to the formation of hydrates in solution.

The low molecular weight which we have found for triethylammoniam chloride in sulphuretted hydrogen, although at first sight difficult to reconcile with the hypothesis of association, is not inconsistent with it.

Thus if the compound formation and subsequent dissociation takes place according to the general scheme

nAB+mCD ^± (AB),, (CD)W and mj.0

(AB), (CD)m — (AB),, (CD)m+mD,

and if dissociation were nearly complete, it is evident that if m is equal to or greater than n, a larger number of molecules than n would be formed, and therefore the average would be less than the theoretical molecular weight.

We can offer no suggestion as to why toluene, when dissolved in hydrogen chloride, although it absolutely fails to conduct the current, possesses such an extremely low molecular weight. Similar cases have been observed by KAHLENBERG, but no explanation has been suggested.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 165

A possible explanation of the abnormal variation of molecular conductivity might be found in the hypothesis that when acetone or ether is added to hydrogen bromide the acetone or ether acts as an ionising solvent, and the hydrogen bromide is ionised. When looked at from this point of view, the variation of p. which actually occurs appears as a normal one. This explanation is, however, shown to be incorrect when we come to consider the transport number experiments.

Thus, during the electrolysis of ether in hydrogen bromide, the deposition of 1 gram-molecule of silver by the current is accompanied by a transport of -8 gram- molecule of ether to the cathode. But, if the ether did not take part in the electrolysis, the same result would be obtained by the transport of a sufficient quantity of bromine as anion from the cathode to the anode.

A simple calculation, however, shows that in order to bring this about no less than 23 gram -molecules of bromine must be transported for every gram-molecule of silver.

Now we have shown that FARADAY'S law is valid for solutions in hydrogen bromide, and accordingly we conclude that ether takes part in the carriage of the current, and that conduction is not due to ionisation of the hydrogen bromide.

Information regarding the constitution of the electrolyte is also afforded by the transport number. If we again consider the case of ether dissolved in hydrogen bromide, there is in solution an electrolyte of the formula ((C2H5)20)2(HBr)n, which can ionise either

+ (1) into H ions and a complex anion ((C3H5)20)2BrB

or (2) „ Br „ „ cation ((C2H,)2O)oH.

If the former, the ether will be transported to the anode as a component of a complex anion ; if the latter, it will be carried to the cathode as a component of a complex cation. Experiment has proved that the latter is the case not only for ether but also for the other substances which have been examined.

It has been found that the cation transport number increases considerably with concentration. This increase can be easily explained if we assume, with JONES and GETMAN (loc. cit.), that the number of molecules of solvent in combination with one molecule of solute is greater in the more dilute solution.

According to the theory of ABEGG and BODLANDER (' Zeit. fur Anorg. Chem.,' 1899, 20, p. 453), the resulting change of constitution of the electrolyte would be conditioned as follows :

Any salt in which one ion is much weaker than the other manifests a tendency to form complex ions by the addition of a neutral molecule to the weaker ion. In the solutions under discussion the weaker ion would undoubtedly be the complex cation, which, when the active mass of the solvent (neutral molecules) was increased by dilution, would tend to become still more complex by the addition of more solvent molecules.

166 DE. B. D. STEELE, DE. D. McINTOSH AND DE. E. H. ARCHIBALD

The effect of this increased complexity would be that the velocity of the ion would be diminished without altering the ionic change, and also that the concentration change at the cathode would be lessened, owing to the carriage of extra solvent molecules to the cathode. Both of these effects would cause a diminution of the cation transport number as the solution was diluted.

It will be noticed that this explanation involves a change in the active mass of the solvent, and, as a matter of fact, it was not possible to measure the transport number except in solutions which were so concentrated, that the assumption of a constant active mass for the solvent was no longer justified.

We have not been able to calculate, even approximately, the velocity of the various ions, as we had no means of determining the actual nature, concentration, or degree of dissociation of the corresponding electrolytes.

Summary.

The foregoing pages contain an account of measurements of the vapour pressures, densities, surface energies, and viscosities of the liquefied hydrides of chlorine, bromine, iodine, sulphur, and phosphorus.

The solvent action of these substances has also been investigated, and we have shown that, with the exception of phosphuretted hydrogen, they are all able to act as ionising solvents, and the conductivity, molecular weight, and transport number of certain dissolved substances have been measured.

The results of the measurements, although abnormal, are not inconsistent with the ionic theory ; since we have shown that —

(1) If in a given solution the electrolyte is a compound containing n molecules ot the dissolved substance, the concentration of this compound will be proportional to the nth power of the concentration of the dissolved substance, and therefore the expression for the molecular conductivity of the electrolyte becomes icV" instead of /cV. We have also shown that /cV" = aK', and therefore the molecular conductivity of the electrolyte increases with dilution in these solutions in the same manner as in aqueous solutions.

The variation of the molecular conductivity of the electrolyte with dilution is probably complicated by the occurrence of compounds which contain a different number of solvent molecules at different dilutions.

(2) The want of agreement between conductivity and cryoscopic measurements is a necessary consequence of the occurrence of polymers or compounds in solution, and may be taken as evidence of the existence of such compounds.

(3) The conduction of organic substances when dissolved in the halogen hydrides is best explained by the occurrence of electrolytic compounds of the organic substance with the solvent. Transport number measurements have shown that the organic substance is carried to the cathode as a component of the complex cation.

ON THE HALOGEN HYDRIDES AS CONDUCTING SOLVENTS. 167

In conclusion, we wish to express our thanks to Professor B. J. HARRINGTON, Director of the McGill University Chemical Laboratory, and to Professor JOHN GIBSON, of the Heriot Watt College, for placing facilities at our disposal and for kindly interest taken in the work. Our thanks are also due to Professor JOHN Cox, Director of the McGill Physical Laboratory, for the use of apparatus, and for kindly supplying us with large quantities of liquid air. We also wish to express our thanks to the Research Grant Committee of the Chemical Society for a grant made to one of us, by means of which a large portion of the expense of the work of Part III. has been met.

[ 169 ]

V. The Atomic Weight of Chlorine: An Attempt to determine the Equivalent of Chlorine by Direct Burning with Hydrogen.

By HAROLD B. DIXON, M.A., F.R.S. (late Fellotv of Balliol College, Oxford),

Professor of Chemistry, and E. C. EDGAR, JB.Sc,, Dalton Scholar

of the University of Manchester.

Received May 18,— Head May 18, 1905.

CONTENTS. PART I. — GENERAL.

PART II. — DETAILS OF EXPERIMENTS.

Page

1. Preparation of hydrogen 172

2. The palladium bulb 175

3. Preparation of chlorine 177

4. The chlorine bulb 180

5. Preparation of reagents 181

6. Weighing the bulbs 185

7. Method of carrying out the combustion 189

8. Results of the experiments 195

Appendix ... .198

PART I.- — GENERAL.

SOME apology seems needed in presenting a new research on the atomic weight of an element already measured with a precision which the highest living critic has emphasised as " the magnificent accuracy of STAS' determination."1* Moreover, the present experiments cannot claim an accuracy to be compared with any individual series of STAS' ratios. But, on the other hand, STAS' atomic weight of chlorine is derived indirectly from oxygen by a series of operations which include the deter- mination of (1) the oxygen in potassium chlorate, (2) the silver equivalent to the molecule of potassium chloride, and (3) the composition of silver chloride. STAS himself has assigned different values to these ratios at different times ; e.g., in 1860 he found that 100 parts of silver were equal to 69'103 of potassium chloride, in 1882 he

* F. W. CLARKE, 'A Recalculation of the Atomic Weights.' New edition. 1897, p. 57. VOL. CCV.— A 391. Z 24.8.05

170- PEOFESSOE H. B. DIXON AND ME. E. C. EDGAE

found 100 of silver equal to 69'119, and in his latest work to 69'123 of potassium chloride. Therefore, although STAS' value 35'457 (0 = 16) is in satisfactory agreement with CLARKE'S value 35 '447 re-calculated from all the best determinations, it is possible that some constant error may occur in some part of the long chain connecting the value of hydrogen with that of chlorine, an error which would be repeated from link to link, and would become evident only when the two ends of the chain were connected up.

A direct comparison between hydrogen and chlorine might not only serve to detect any systematic error in this chain of ratios, but such a comparison, inasmuch as it does not involve the probable error of other ratios, would be cceteris paribus more exact. Again, the closing of the chain between hydrogen and chlorine with reasonable accuracy would permit the accidental errors to be distributed and prevent their accumulation at the unconnected end. The accumulated "probable error" in CLARKE'S recalculated value for chlorine is ±'0048 ; the " probable error" of our nine experiments is ±'0019.

The suggestion to carry out this work was made to us by Professor EDWARD W. MORLEY, who happened to visit our laboratories when pure chlorine was being prepared by the electrolysis of fused silver chloride. He suggested that we should burn weighed hydrogen and chlorine in a closed vessel, just as he had burnt weighed hydrogen and oxygen. After some discussion we decided to make the attempt — an attempt which was rendered possible by the fact that one of us was enabled, by a research scholarship, to devote his whole time to the investigation.

A year was spent in designing, making and testing the several parts of the apparatus. In the second year we put together the pieces and carried through preliminary experiments, which led to some modifications and further trials. In the third year the apparatus was got into Avorking order and the determinations made. After the three years' work we are painfully aware how far our attempt falls short of the precision of Professor MORLEY'S own determination, but the relation we have found between hydrogen and chlorine seems worthy of record on account of the directness of the method of comparison.

Our method was, briefly, as follows : — Chlorine prepared by the electrolysis of fused silver chloride (with purified carbon poles in a Jena-glass vessel) was condensed and weighed as a licpuid in a sealed glass bulb. This was attached to a vacuous " combustion globe " and the chlorine allowed to evaporate slowly .nto the globe. The hydrogen prepared by the electrolysis of barium hydrate was dried and absorbed by palladium in a weighed vessel. The palladium on being warmed gave off the hydrogen, which was ignited by a spark and burnt at a jet in the combustion globe previously filled with chlorine. The gases were regulated so as to maintain the hydrogen flame until nearly all the chlorine had been combined ; then the palladium was allowed to cool and the hydrogen was turned off just before the flame died out. The hydrogen chloride, as it was formed in the flame, was dissolved by water standing

ON THE ATOMIC WEIGHT OF CHLORINE.

171

in the globe, which was kept cool by ice. A little hydrogen chloride was formed by the action of the water-vapour on the chlorine in the flame, a corresponding amount of oxygen being liberated. This oxygen was determined in the analysis of the residual gases, which contained, besides traces of air, the small quantity of hydrogen which filled the capillary tube between the tap and the jet when the flame was extinguished, and any that might escape unburnt from the flame.

The chlorine remaining in the globe unburnt, as gas and in solution, was determined by breaking a thin glass bulb containing potassium iodide. The residual gases having been pumped out (and any iodine vapour caught by a wash-bottle), the liberated iodine was determined by standard thiosulphate in an atmosphere of carbonic acid. In calculating the unburnt chlorine from the iodine, the atomic weight of chlorine was assumed to be 35'195 and the atomic weight of iodine 126'015.* In each experiment we burnt about 11 litres of hydrogen and 11 litres of chlorine. The volume of chlorine left unburnt was about 2 per cent, of the volume burnt.

The balance (by OERTLING) was fixed on a stone pedestal in an underground cellar. The vibrations of the pointer were read by a telescope, GAUSS' method of reversals being used. The chlorine and the hydrogen bulbs were counterpoised on the balance by bulbs of the same glass and of nearly the same displacement, and the small weights used in the weighings were reduced to a vacuum standard.

In the following table are given the corrected weights of hydrogen and of chlorine burnt in the several experiments — the weights of hydrogen being rounded off to

•1 milligramme :

TABLE I.

	Hydrogen burnt,	Chlorine burnt,	Atomic Weight of
	in grammes.	in grammes.	Chlorine.
1	•9993	35-1666	35-191
2	1-0218	35-9621	35-195
3	•9960	35-0662	35-207
4	1-0243	36-0403	35-185
5	1-0060	35-4144	35-203
6	•988V	34-8005	35-198
7	1-0159	35-7639	35-204
8	1-1134	39-1736	35-184
9	1-0132	35-6527	35-188
Mean ....	35-195 ±-0019

In the whole of these experiments 9'1786 grammes of hydrogen combined with 323 "0403 grammes of chlorine ; hence the atomic weight of chlorine, calculated in mass, is 35-195.

* G. P. BAXTER, 'Proc. Amer. Acad.,' xl., 419. z 2

172 PKOFESSOK H. B. DIXON AND ME. E. C. EDGAR

The percentage composition of hydrochloric acid according to these deter- minations is : —

Chlorine 97-237

Hydrogen 2 -763

100-000

The number we have obtained for the atomic weight of chlorine is appreciably hio-her than that calculated by F. W. CLARKE from the previous determinations, and is slightly higher than STAS' value :—

CLARKE'S calculation.	STAS.	Dixox and EDGAR.
35-179	35-189	35-195	H = l
35-447	35 '457	35-463	0 = 16

After our experiments were completed, we heard that Professor T. W. RICHARDS was engaged on a revision of STAS' work on the composition of silver chloride. G. P. BAXTER quotes the value 35 '467 as having been obtained by RICHARDS and WELLS for the atomic weight of chlorine, a number slightly higher than our own.*

It would not be difficult to extend our experiments, using larger quantities of the gases, if in the judgment of chemists it were thought desirable, t

PART II. — -DETAILS OF EXPERIMENTS. 1. Preparation of Hydrogen.

For the preparation of hydrogen we employed the electrolysis of a solution of barium hydrate, first proposed by BRERETON BAKER,! as a means °f preparing hydrogen free from traces of hydro-carbons.

Since barium carbonate is quite insoluble in a solution of barium hydrate, any slight action of the carbonic acid of the air ' on the dissolved hydrate during its unavoidable exposure while filling the electrolytic apparatus might be safely neglected. We have to thank Mr. BRERETON BAKER for kindly supplying us with some of his highly purified barium hydrate. It had been re-crystallised fifteen times and was not radio-active. It still contained a very small trace of barium carbonate.

The arrangement of the hydrogen apparatus is shown in fig. 1. Three preliminary

* Professor RICHARDS writes (February 13, 1905) that he finds 100.00Q parts of silver yield 132,867 of silver chloride, whereas STAS considered 132,850 the most probable result. This new determination, combined with our value for chlorine, would give silver an atomic weight 107 '90.

t As further experiments have shown that chlorine can conveniently be burnt in an atmosphere of hydrogen, one of us proposes to make a fresh set of determinations in this way and to condense and weigh the hydrochloric acid formed. — July, 1905.

I ' Jl. Chem. Soc.,' 1902, vol. 81, p. 400.

ON THE ATOMIC WEIGHT OF CHLORINE.

173

C-™maEi.'..:i4&?='-?w>j&?i6 S23^23SSEngEj

1

tf\

I

§"

I

bo

S

174 PEOFESSOR H. B. DIXON AND MR. E. C. EDGAR

drying tubes were employed, each 1 metre in length and 2 '5 centims. in diameter, filled with small pieces of purified potassium hydrate. The gas then passed through a U-tube containing platinised pumice, kept at a temperature of 220° C., in order to remove any oxygen diffusing from the + electrode, and then through a short horizontal tube and three long U -tubes filled with pure phosphorus pentoxide. As recommended by COOKE, the phosphorus pentoxide was packed closely into the drying tubes and was alternated at frequent intervals with plugs of clean glass-wool. To ensure efficient drying, the current of hydrogen was passed through these drying tubes at a rate not greater than 2 litres an hour. At the end of our experiments the last layers of phosphorus pentoxide had picked up so little moisture that a slight tapping of the tube threw the powder into a cloud.

The drying tubes, when filled, were fused together, and to the last phosphorus pentoxide tube was fused the tap C. This, in turn, was fused to one limb of a T-piece, to the other two limbs of which were fused the bulb containing the palladium foil and the Toepler pump. To the first potassium hydrate drying tube at the other end of the apparatus was fused a three-way tap. The U-tube E. in which the electrolysis of barium hydrate was carried out, was also fused to D, whilst the third arm of D opened into the air.

When the current was passed through the warm barium-hydrate solution between the platinum electrodes G and G,, the evolved gases were allowed to escape into the atmosphere until the air, which was originally contained in the two arms of the U-tube, had been replaced by hydrogen and oxygen respectively. Connection with the atmosphere was then cut oft' by closing the tap H and by fusing off the capillary portion of the opening F. During our experiments the solution showed no signs of milkiness and no precipitate settled at the bottom of the tube ; we believe, therefore, that no carbonate was present.

The preparation and occlusion of hydrogen was carried out as follows : — Before the fusion of the palladium bulb to the apparatus, the tubes on the right-hand side of the tap D were exhausted as far as possible by means of the pump. The U-tube, containing the solution of barium hydrate, was raised to a temperature of 60° C. in a water-bath (in order to dissolve the hydrate which had crystallised out from the solution) and the electrolysis commenced. The evolved oxygen escaped into the air through the tap H and the tube K, filled with a dried mixture of CaO and Na2SO4, while the hydrogen was admitted, by very cautiously opening the tap D, to the evacuated part of the apparatus. The stream of hydrogen was continued until the bubbling of the gas through the manometer tube showed that the previously evacuated portion of the apparatus was now full. The electrolysis was discontinued, the tap D closed and the drying tubes again evacuated. This operation of filling and exhausting was repeated twelve times in order to get rid of all traces of air. The taps C and D were then closed and the electrolysis stopped. •

The bulb A containing the thin palladium foil was then fused to the apparatus.

ON THE ATOMIC WEIGHT OF CHLOEINE. 175

The palladium was raised to a very low red heat and the apparatus on the right-hand side of the tap C evacuated, arid then allowed to cool to the ordinary temperature. The tap C was cautiously opened, the electrolysis resumed, and hydrogen admitted to the palladium until it was saturated, care being taken that the pressure on the left- hand side of the tap C was always kept slightly above the atmospheric. This last precaution was easily effected by opening the tap D fully and regulating the admission of the gas to the palladium by means of the tap C. C was now closed, the temperature raised to a very low red heat, and the evolved gas sucked out by the pump. The operation of alternately filling the palladium with hydrogen and evacuating the bulb at a high temperature was repeated four times, when it was considered that all traces of nitrogen or other gases had been removed from the palladium bulb and the connecting tubes on the right-hand side of C. The palladium after the final exhaustion was maintained at a low red heat whilst hydrogen was admitted to it through C.

When the pressure throughout the apparatus had become a little more than atmospheric — this was easily attained by so adjusting the tap H of the electrolysis tube that the rate of escape of the evolved oxygen through it was slightly less than its actual rate of evolution — the tap B was slightly opened and the current of hydrogen was passed through the palladium, the gas finally escaping through a capillary tube dipping under mercury. The palladium was now allowed to cool very slowly, the current of hydrogen passing through it all the time. Great care was taken that the rate of entry of hydrogen to the palladium was always greater than its rate of occlusion, or, in other words, that an excess of hydrogen was constantly escaping through the capillary tubing during the occlusion.

At the temperature of maximum absorption of hydrogen from 07° to 100° C., the cooling of the palladium was interrupted and the temperature kept constant for one hour. The cooling was then allowed to continue, hydrogen passing through the apparatus all the while, until the temperature of the room was reached. The taps B, C, D, and H were then finally closed and the electrolysis discontinued. The palladium bulb was fused off from the rest of the apparatus, the outside cleaned and dried, and the whole was then ready for weighing.

2. The Palladium Bull}.

The palladium vessel A (fig. 2) was a bulb of hard Jena glass of about 180 cub. centims. capacity, fitted on the one side with a tap B, the inner portion of a ground glass joint M, and a glass jet J, at which the combustion of hydrogen in chlorine was carried out ; and on the other with a capillary tube by which it could be attached to the rest of the hydrogen apparatus and afterwards separated by fusion with the blowpipe flame.

Since the date when Professor E. W. MOELEY defined a tap as "a contrivance for

176

PKOFESSOR H. B. DIXON AND MR. E. C. EDGAR

lessening the flow of -gas through a tube," improvements have been made which seemed to us to justify the use of one for regulating the flow of hydrogen from the palladium bulb. The tap B, fig. 2, was made with a long barrel with its bearings

B

Fig 2. The palladium vessel.

ground to the sheath for a length of 30 millims. The barrel forms a portion of an elongated cone, its diameter at the wider end being 10 millims., and at the narrower end 8 millims. The bore of the tap is inclined so that one opening is 10 millims. above the other. The sheath of the tap ends in a closed bulb below and a cup above.

The method of lubricating and fixing the tap was as follows : — After thorough cleaning and drying, the bulb b of the tap was filled with dry mercury to such a height that the barrel of the tap, when placed in position, just touched its surface. The mercury was then gently heated until it filled the whole bulb. The lubricant glacial phosphoric acid was melted and carefully rubbed over the barrel, which was placed in position, turned several times to ensure equal distribution of the lubricant, and then pressed firmly into the sheath whilst the bulb b containing the mercury was cooled. The bulb now contained no air, but the cooling resulted in the production of a partial vacuum, which kept the tap firmly fixed. We have tested this tap by a pump and found it to remain perfectly gas-tight. All the other taps used in the apparatus with the exception of the chlorine tap were made and used in the same way. We are indebted to the skill of the University glass-blower, OTTO BAUMBACH, for the accurate grinding of these taps, and for the joints by which he succeeded in fusing hard Jena to soft glass.

It was of course essential that the weight of the palladium bulb should be most accurately determined before and after the combustion of its charge of hydrogen. To avoid change of volume the bulb was made of a hard Jena glass which preliminary hydrostatic weighings showed not to alter when heated to dull redness and cooled. The charged palladium bulb was heated and cooled alternately to

ON THE ATOMIC WEIGHT OF CHLORINE. 177

determine its change of volume, if any. When immersed in water to a mark on the stem the bulb weighed—

Before heating 230-314 grammes.

After heating for two hours to a dull red heat .... 230 '311 „

After a second heating 230-307 „

After a third heating 230-308 „

The alteration in volume of the bulb, after heating to dull redness, was there- fore so slight that the difference in its displacement of air was negligible. In the actual experiments the bulb was never heated beyond 550° C.

The palladium was used in the form of thin foil. We are indebted to Messrs. JOHNSON and MATTHEY for kindly supplementing our stock for the purpose of this investigation. The bulb contained sufficient palladium foil (360 grammes) to absorb about li grammes of hydrogen. When the bulb A had been detached from the rest of the hydrogen apparatus and had been cleaned, it was suspended by platinum wire from one arm of the balance, from the other was suspended a counterpoise (of the same Jena glass) which had nearly the same displacement as the palladium bulb. It was then weighed by GAUSS' method of reversals. The balance case, after each reversal, remained closed at least half-an-hour before a new weighing was started. The air displaced by the small weights added to secure equilibrium was allowed for. By equalising the volumes of the systems suspended from the arms of the balance, errors due to variations of temperature in the balance case and to any deposition of moisture on the bulb were avoided.

3. Preparation of Chlorine.

SHENSTONE,* in 1893, first proposed the electrolysis of fused silver chloride in a vacuum as the best means of obtaining pure chlorine. He stated that his chief difficulty was the rapid formation of silver trees, which eventually made contact between the electrodes and thus prevented any further decomposition of the fused chloride. In 1901, MELLOR and KussELLf substituted for SHENSTONE'S tube a V-tube of the hardest Jena glass, so that the silver tree had to travel along the two limbs of the V before making contact, and thus the decomposition of silver chloride could be carried on to a greater extent than in SHENSTONE'S apparatus. They fastened their carbons to glass tubes (ground into the necks of the V) by means of a plaster of Paris joint.

We have modified their apparatus by drawing out the upper portion of each arm of the U, and melting it on to the carbon electrode for a length of about 2-5 centims. We fused a little silver chloride round the top junction of glass and carbon. A mercury cup completed the joint, and served for making electrical contact with the carbon. Such a joint, even with a vacuum in the interior of the tubing, is quite

impervious.

* ' Journ. Chem. Soc.,' 71, 471 (1897).

t 'Journ. Chem. Soc.,' 82, 1272 (1902). VOL. CCV. — A. 2 A

178 PROFESSOR H. B. DIXON AND MR. E. C. EDGAR

The arrangement of the apparatus is shown in fig. 3. A was the U-tube of Jena glass, having two delivery tubes B and Bt which united at C ; its capacity was such as to'admit of the fusion of 800 grammes of silver chloride introduced through the side tube F. The carbon electrodes D and D1} of 2 millims. diameter, were specially made for us by the Acheson Graphite Company, Niagara. Before being fixed in position they were heated to redness for twelve hours in a current of chlorine, and

Fig. 3. Chlorine apparatus.

were then kept in vacuo in a porcelain tube for three hours at a bright red heat. A special glass joint at G permitted the junction of the U-tube to the other portion of the apparatus, which was constructed of soft glass. This joint was made by fusing together a series of twelve very short pieces of tubing which varied by small gradations from hard Jena to soft glass. H was a small drying tube containing pure phosphorus pentoxide, which was kept in position by two plugs of clean glass-wool. K was a glass tube (capacity 25 cub. centime.) in which a sample of the prepared chlorine could be collected to test its purity. L was a T-piece, one limb of which was fused to the chlorine generator ; another led, via the absorption tubes M and N, to the mercury pump, whilst the third was fused to the " chlorine bulb."

To prevent any residual chlorine reaching the pump, it was passed through the tube M (which could be filled with mercury to a suitable height by raising a reservoir), and then through a tube N, 1 metre in length and 4 centims. in diameter, packed closely with pure potassium hydrate. A little mercury, contained in the cavity 0, acted as a tenioin. The dulling of its bright surface would have indicated

ON THE ATOMIC WEIGHT OF CHLORINE. 179

that the absorption of chlorine had not been complete, but, at the end of our experiments, its lustre was unimpaired.

Preparation of Silver Chloride. — Commercial silver nitrate was purified by re-crys- tallisation twice from water. A strong solution of hydrochloric acid was prepared by • cautiously distilling the pure concentrated acid, washing the evolved gas with a little water and then dissolving it in re-distilled water, kept cool by means of an ice and salt freezing mixture.

A strong solution of re-crystallised sodium chloride was prepared, and into it was passed the acid gas evolved by heating the solution of hydrochloric acid previously made. The precipitated sodium chloride was washed with a little ice-cold water, dissolved in fresh re-distilled water and again re-precipitated by means of hydrochloric acid gas. This re-precipitation was carried out three times. Silver chloride was then prepared by adding a dilute solution of the re-crystallised silver nitrate to excess of a dilute solution of the purified sodium chloride.

The precipitation of the silver" chloride and all subsequent operations were carried out in the absence (as far as possible) of actinic light. The supernatant liquors were decanted as soon as possible and the silver chloride washed repeatedly with boiling distilled water, until a test portion of the washings gave no cloudiness with silver nitrate. The silver chloride was then frequently agitated with more hot distilled water and allowed to stand in contact with it for some time. Then the final washings were decanted and the silver chloride was dried as completely as possible in large porcelain dishes on a water-bath. It was then cautiously fused in deep porcelain crucibles and kept in the molten state for twenty-four hours, care being taken to prevent contact, during the prolonged heating, between the acid gases of the flame and the molten chloride. The chloride was then poured into a clean silver trough so as to form thin sheets. These, on cooling, were easily detachable, and were cut into small fragments. The silver chloride prepared in this way was a colourless, horn-like, translucent substance, which could be easily broken or cut into small pieces.

The operation of fusing the requisite amount of silver chloride in the U-tube A was carried out as follows: — The U-tube (filled through the side tube F with the solid pieces of silver chloride up to the level of the carbon electrodes) was gradually raised in temperature by heating the cast-iron box in which it was closely packed round with asbestos. A high-range thermometer, with its bulb resting on the bend of the U-tube, indicated the temperature of the chloride. When the contained silver chloride had fused, more was slowly added until the calculated amount, 800 grammes, had been completely reduced to the molten state. The side tube F was then sealed and the whole apparatus was exhausted by the pump. When the tubes were thoroughly evacuated the tap P was closed, and the current from one storage cell was passed through the silver chloride for a short time. The current was then increased by the addition of another cell, and chlorine was steadily evolved until the whole of the apparatus on the left-hand side of the tap was filled with chlorine at a

2 A 2

PKOFESSOE H. B. DIXON AND ME. E. C. EDGAR

pressure slightly above atmospheric. Then the current was discontinued, the tap P cautiously opened, and the gas allowed to escape, first through mercury contained in the tube M and then through solid potassium hydrate. Any gas other than chlorine was then sucked out by the automatic pump, which, during 'the first part of the electrolysis, was kept constantly working.

The operation of filling the apparatus with chlorine and exhausting was repeated four times. The gas from the first two fillings was not completely absorbed. In preparing chlorine for our determinations we filled the apparatus five times, and tested the fifth by fusing-off the side tube K and opening it under mercury. The absorption was so complete as to leave no visible gas residue. This test assured us that no air was left in our chlorine. The fact that the chlorine first evolved was allowed to escape was a safeguard against the possible presence of bromine or iodine, for any bromide in the silver chloride would have been decomposed by the chlorine, and the evolved bromine would have been carried over with the chlorine first escaping.

The chlorine bulb (immersed in a cooling mixture of solid carbonic acid and ether contained in a silvered Dewar tube) was then filled with liquid chlorine. The current was increased and the condensation allowed to proceed until the liquid reached the level of a circular line etched on the bulb, when the current was stopped. About 37 grammes of liquid chlorine were collected in each experiment. Finally the chlorine bulb was separated by fusion.

Irregularities, arising in the electrolytic cell, were shown by an ammeter placed in the electrical circuit. We found it advisable, as SHENSTONE says, to prevent these irregularities by frequently reversing the current for a short interval of time, thus shattering any incipient silver tree.

4. The Chlorine Bulb.

Chlorine, prepared by the electrolysis of fused silver chloride in vacuo, and dried by phosphorus pentoxide, was condensed by means of a freezing mixture of solid carbonic acid and ether, or by liquid air, in an apparatus shown in fig. 4. The chlorine vessel, which was made of soft glass, consisted of a stout glass bulb, A, holding about 40 cub. centims. To this was attached one limb of a T-piece, made of capillary tubing ; another limb could be fused to the source of chlorine, whilst the third ended in a cul-de-sac (B).

B was a contrivance by means of which we got over a difficulty, which threatened at one time to bring our work to a premature end. For a long time we were unable to discover any means by which liquid chlorine could be safely weighed, and, at the same time, be under such complete control as to admit of its subsequent regular entry to the combustion globe. The pressure of liquid chlorine at ordinary temperatures is from 6 to 8 atmospheres, and the difficulties of successfully controlling such a pressure by means of a tap were found very great.

ON THE ATOMIC WEIGHT OF CHLORINE.

181

After many failures we finally designed the vessel shown in fig. 4. The chlorine weighed in the bulb A could only reach the tap when the sealed end of the inner tube B was broken off by the rod of glass C falling on it. The tap D was an inversion of the ordinary form of tap, that is, its smallest diameter is at the top of the tap ; so that instead of the key having to be pushed into its socket, it has to be pulled into it to fit. Internal pressure, instead of tending to loosen the key, only made it fit more tightly. Of course, if the internal pressure became too great, the key was so firmly driven into its socket that it stuck, and then became useless. However, the taps we used, when lubri- cated with viscid phosphoric acid, with- stood a pressure of four atmospheres without sticking. Their chief disadvan- tages lay in the difficulties of cleaning and lubricating them, and in the fact that it was necessary to affix to them weights, suspended from a pulley, when carrying out exhaustions of vessels to which they were attached. We are not .aware that such taps have been used before in scientific research work ; they were made for us by the University glass-blower.

The small space E (less than -5 cub. centim.) immediately below the key of the special tap D, and the glass tubes connected with it, were first evacuated and then filled with pure chlorine from the silver chloride through the tube F, which was sealed off while the apparatus was cooled by immersion in a freezing mixture. On the removal of the freezing mixture, the gas trapped between E and F (about 4 cub. centims.) tended to expand, and thus held the tap D firmly in position.

The chlorine condensation bulb, filled with approximately 37 grammes of liquid chlorine, was weighed in a precisely similar manner to that detailed for the palladium bulb.

5. Preparation of Reagents.

Iodine. — Pure iodine was prepared by the first of the two methods proposed by STAS. A strong solution of potassium iodide was saturated with resublimed commercial iodine. To this, sufficient water was added to precipitate one half of the dissolved iodine. The supernatant liquid was decanted and the precipitated iodine repeatedly washed with small quantities of distilled water. It was then divided into two portions. The iodine, in the first, was distilled in steam, the solid distillate

Fig. 4. Chlorine bulb.

182 PEOFESSOR H. B. DIXON AND MR. E. C. EDGAR

collected and dried in vacua over solid calcium nitrate, which was frequently changed. The iodine was then intimately mixed with 5 per cent, of its weight of purified barium oxide, and distilled to remove the last traces of water and hydrogen iodide. The wet iodine, in the second portion, was dissolved in a strong, cold solution of purified potassium hydrate until the solution had acquired a per- manent light yellow tinge. The solution was then evaporated to dryness on a water bath. The mixture of potassium iodide and iodate so obtained was then placed in a large platinum crucible, fitted with a platinum hood, and heated to dull redness for six hours. The resultant potassium iodide was re- crystallised five times from water and dried in vacuo over calcium nitrate, which was frequently changed. It was pure white in colour, and contained no trace of potassium iodate ; its solution in water was neutral and remained colourless when

exposed to light.

Standard Solution of Iodine in Potassium Iodide.— In a small weighing bottle, carefully cleaned and dried, iodine, purified as described, was placed. This was kept in a desiccator until ready for weighing. The details of the weighing are given below : —

Temperature at start 16°'5 C., Barometer at start 759 -8 millims.,

Temperature at end, 15°'5 C., Barometer at end 757 '0 millims.,

Weight of bottle and iodine 52-28137 grammes.

Weight of bottle 23-70084

28-58053 Vacuum correction + -00279

28-58332

The weight of iodine dissolved was therefore 28 '58332 grammes.

This iodine having been dissolved in a solution of potassium iodide, the iodine solution was brought into a 2-litre flask through a drawn out funnel, and the residual solution carefully washed in.

The flask was calibrated by means of a burette previously calibrated, the neck of the flask being drawn out in the blowpipe flame. After cleaning and drying, the flask was filled with pure water from the burette, at the same temperature as that at which the burette had been calibrated. The last drops were allowed to run into the flask by contact with the glass surface immediately above the water, which stood in the constricted part of the neck. A circular line was etched on the glass to mark the exact level of the liquid in the constriction.

The iodine solution was brought up to the etched mark by slowly adding pure water, the solution being shaken after each addition of water. The final tempera- ture of the solution was almost identical with the temperature at which the volume

ON THE ATOMIC WEIGHT OF CHLORINE. 183

of the 2-litre flask was determined. It was assumed that no loss in weight of the iodine had occurred during its solution in the potassium iodide solution. We had then 28'58332 grammes of iodine dissoved in 2033'68 of our units of volume, which gives '014055 gramme of iodine in one of our units of volumes. The solution was kept in the tightly stoppered 2 -litre flask.

Potassium Hydrate. — Potassium hydrogen carbonate was twice re-crystallised from water. The crystals were heated in a platinum crucible, fitted with a platinum hood, to a dull red heat for six hours. The potassium carbonate so obtained was dissolved in water, and silver carbonate added, and the mixture thoroughly agitated for three hours. The precipitate, composed chiefly of silver carbonate but probably containing traces of silver chloride and other substances, was allowed to settle and the supernatant liquid filtered into a silver dish through a filter filled with clean pieces of broken marble.

The solution in the silver dish contained one part of potassium carbonate in twelve of water. It was heated to the boiling-point, and two parts of lime (prepared by heating calcium carbonate to bright redness in a platinum crucible, and previously slaked in ten parts of water) were added by degrees, the liquid being boiled for a few minutes after each addition of lime to ensure its complete conversion into calcium carbonate. The addition of lime completed, the solution was boiled for half-an-hour and allowed to clarify by standing. The clarified solution was then filtered through another marble filter into a silver dish and boiled down until the hydrate commenced to evaporate. The semi-solid mass was then poured into a silver dish and allowed to cool in vaauo over calcium chloride. It was then divided into four portions, the first was broken into small fragments and introduced as rapidly as possible into the potash drying tubes ; the second was broken into larger pieces with which the chlorine absorption tube (fig. 3) was filled ; the third was dissolved in pure distilled water and the solution employed in the preparation of potassium iodide, whilst the remainder was used in the purification of water.

Pure Water. — The water used in these experiments was prepared by rectifying hot distilled water from the laboratory still. This was distilled over potassium hydrate (purified as described) and potassium' permanganate, twice re-crystallised from water. The retort employed was made of hard Bohemian glass, the condensing tube and receiver of Gerate glass. Immediately before use these were cleaned and steamed. 100 cub. centims. of this water, when slowly evaporated in a small platinum retort, gave no solid residue.

Phosphorus Pentoxide. — KAHLBAUM'S purest pentoxide, contained in Jena hard glass tubes, was distilled, at a bright red heat, in a current of pure dry oxygen through spongy platinum, kept in position by two platinised asbestos plugs. The distilled oxide condensed as a fine white crystalline powder in the cooler part of the Jena-glass tubes. It was kept in a tightly stoppered bottle until its introduction into the drying tubes. It answered all the tests recommended by SHENSTONE and

PROFESSOR H. B. DIXON AND MR. E. C. EDGAR

BECK for the identification of pure phosphorus pentoxide : (l) it did not reduce a 10-per cent, solution of silver nitrate; (2) it did not reduce mercuric chloride when boiled with it ; and (3) on evaporating an aqueous solution of it to dryness and igniting moderately, no odour of phosphine was detected.

Palladium Foil. — The palladium, which was used in the form of thin foil cut into very small pieces, was heated to dull redness in a current of pure dry air for twenty- four hours, in order to eliminate any grease which might have been acquired during rolling. It was then heated in glazed porcelain tubes to a bright heat, in vacua, for six hours.

Sodium Thiosulphate. — The sodium thiosulphate used was re-crystallised from water four times and was dried, in vacuo, over calcium chloride ; it was pure white in colour and its solution was neutral to litmus.

Sodium Hydrogen Carbonate. — The sodium hydrogen carbonate used for the preparation of carbonic acid, in an atmosphere of which the titration of the iodine contained in the combustion bulb was carried out, was purified by exposing the solid, at 70° C., to the action of a slow stream of carbonic acid gas passing through it. The carbonic acid was prepared by the action of hydrochloric acid on marble, and, before reaching the carbonate, was washed thoroughly with water. When the current of gas had been passing for three hours, the carbonate was allowed to cool in it until the ordinary temperature had been reached. Sodium hydrogen carbonate so prepared had no effect in impairing the accuracy of titrations of thiosulphate by means of the standard solution of iodine in potassium iodide. The gas obtained on heating the acid carbonate was completely absorbed by potassium hydrate.

Starch Solution. — The solution of starch, used as an indicator, was prepared by adding soluble starch, in very small quantities at a time, to boiling water which had been purified. When the solution commenced to assume a faint opalescent blue, the addition was discontinued. The solution, on cooling, was preserved in a tightly stoppered bottle, and to prevent any fermentation, a little mercuric iodide was added and dispersed through the solution by vigorous shaking.

Platinised Pumice. — Pumice stone was ground into small fragments and sifted through two sieves — the first of 2 sq. millims. mesh, the second 1 sq. millim. ; the part remaining on the second was transferred to a porcelain basin and washed thoroughly with aqua regia. After decanting the supernatant acid, the mass was washed with water until the washings were no longer acid. It was then dried in a porcelain crucible contained in an air-bath at 120° C. The dried product was saturated with a concentrated solution of platinic chloride, excess of ammonium hydrate added, and the mass stirred until the yellow colour of the platinic chloride had disappeared from the supernatant liquid, which was then decanted and the platinised pumice carefully dried. It was then heated in a deep porcelain crucible until fumes were no longer evolved. A lid was placed on the crucible and the whole heated to a dull red heat for twelve hours. On cooling, the platinised pumice was

ON THE ATOMIC WEIGHT OF CHLORINE. 185

packed into the small U-tube B (fig. 1), which was then fused to the apparatus for the preparation of hydrogen.

Purification of the Mercury used in the Pumps. — The mercury was frequently cleaned as follows : — It was placed in a suction flask, and on to its surface was poured a weak solution of nitric acid. The side tube of the flask was attached to the water pump, which drew air through the mercury by means of a glass tube held in position by a cork in the neck of the flask.

This stream of air, coupled with the intimate mixing of the mercury and the nitric acid, resulted in the rapid oxidation and solution of all metallic impurities contained in the metal. When this had been accomplished, the mercury was thoroughly washed with water, dried with filter paper, and filtered, by means of very fine holes, through clean white paper.

Cleaning of Glass Apparatus. — Before use, all glass apparatus was filled with a hot mixture of potassium dichromate solution and concentrated sulphuric acid and allowed to stand for six hours. It was then washed out with boiling distilled water, and filled with hot concentrated nitric acid and allowed to stand overnight. The next morning the vessel was emptied, thoroughly washed out with hot distilled water, and steamed for three hours. Finally, a current of hot air, filtered through cotton-wool and dried through sulphuric acid, was passed through it until it was completely dried.

6. Weighing the Bulbs,

The balance, made specially for atomic weight determinations, was placed on a stone pedestal in a cellar, situated in the basement of the chemical laboratories. Observations with a maximum and minimum thermometer showed that the tempera- ture in this cellar varied but little. Three filter funnels filled with calcium chloride were kept inside the balance case ; the air in it was assumed to be half dried. The doors of the balance case were closed and half-an-hour allowed to elapse before a weighing was made.

The vibrations of the pointer over the scale were viewed through a mirror by means of a telescope. Assuming the number of divisions on the scale to be 1000, and the average zero at no load 500, then the range of the zero variations, during our experiments, was 9 divisions, between 49G to 505.

The sensibility of the balance, during the weighings of the chlorine bulb, was approximately 206 divisions for 1 milligramme, with a range of variation of 8 divisions. During the weighings of the hydrogen bulb, the sensibility was approximately 198 divisions for 1 milligramme, with a range of variation ecpual to 10 divisions. The method of weighing adopted was GAUSS' method of reversals. Generally, five weighings were taken on one side and four on the other. The concordance of the individual weighings showed that their mean could be relied on to 4 divisions or •00002 gramme.

VOL. CCV. — A. 2 B

186

PROFESSOR H. B. DIXON AND MR. E. C. EDGAR

The weights employed were a brass hectogramme and its subdivisions to a gramme, and, for the submultiples of a gramme, small platinum weights. The hectogramme was taken as the unit and the separate weights were carefully compared with it. Since all our measurements of mass were relative and not absolute, it was not necessary to determine the absolute mass of our unit. In comparing the gramme of platinum with the brass gramme marked Z. a correction was applied for the different weights of air displaced by them. The values of all the weights are given below :—

VALUES of the Brass Weights.

Nominal value.	Value found.
100 grammes (unit) 50	100-00000 grammes 50-00005
20		19-99973
10	(A)	10-00002
10	(B)	9-99986
5		4-99991
2		1-99994
1	(I)	1-00025
1		•99998
1	(H)	•99993

VALUES of the Platinum Weights.

Nominal value.	Value found.
• 5 gramme •2	•49998 gramme •19994
•1 •1 •05	(1) (2)	•09996 •09992 •04995
•02		•02001
•01 •01 Ptri	(1) :ler	•01001 •01000 •01007

The palladium bulb, when charged with hydrogen and sealed off, varied in weight from about 419 grammes to 425 grammes. It was counterpoised by a vessel made of the same glass and of approximately the same volume, weighing 400 '00097 grammes. The brass and platinum standardised weights were used to complete the equilibrium. The only vacuum corrections necessary to apply to the weighings were — (i.) that for the difference in volume between the small weights used before and after the com- bustion, i.e., the volume occupied by (approximately) 1 gramme of brass, and (ii.) for possible changes in the buoyancy of the bulb.

The glass counterpoise was made the same volume as the bulb when first used in Experiment I. It was not considered necessary to alter it so as to make it exactly

ON THE ATOMIC WEIGHT OF CHLOEINE.

187

the same volume as the bulb in the subsequent experiments, since the maximum variation in the displacement of the bulb did not exceed 1'3 cub. centim. This variation in volume, caused by differences in sealing off the thick-walled capillary tube, may be assumed to be due to the solid glass drop at the sealed end. When the density of the air altered between the first and second weighings of the bulb, a difference between the displacement of the bulb and the counterpoise might affect the apparent weight of the bulb, but in only one experiment (No. 8) was a correction necessary, and that only a unit in the fifth place of decimals.

The weighings of the chlorine bulb were carried out in the same manner, with a similar glass counterpoise. It was not, of course, necessary to obtain the same degree of accuracy in weighing the chlorine as in weighing the hydrogen, since a unit in the fourth place of decimals is insignificant. Variations in the displacement of the chlorine bulb, caused by sealing-off, though considerably larger than those of the hydrogen bulb, did not affect the determination of the " chlorine taken."

In illustration of the method of weighing we may refer to Experiment V. The palladium bulb (charged with hydrogen) required the following weights to be added to the opposite pan : —

Weights used.

Value.

Brass 20 grammes

Ft '5

•02 „

•01 „ (1)

•01 „ (2)

Pt rider on 2nd division

19-99973 grammes •49998 •02001 •01001 •01000 •00201

20-54174 grammes

Five weighings with the weights in the right-hand pan gave a mean zero of 398 '2 divisions on the sale. Four weighings with the weights reversed gave a mean zero of 597. With no load the mean zero was 497. The two differences are :

Eight 98-8. Left lOO'O. Mean 99'4.

The sensibility under this load was found to be 202 divisions of the scale for a difference of 1 milligramme. The mean displacement of the zero was, therefore, equal to a weight '00049 gramme to be subtracted.

Adding these weights together

Counterpoise + 400 • 00097 grammes Weights + rider + 20-54174 „ •00049

420-54222

2 B 2

188 PROFESSOR H. B. DIXON AND MR. E. C. EDGAR

In this experiment the palladium bulb has a volume below that of the counterpoise by rather less than '5 cub. centim. The mean barometric pressure at the first weighing was 7G6'1 millims., and the mean temperature was 14°'5 C. At the second weighing, after the combustion, the mean barometric pressure was 761'2 millims., and the tempei'ature was 120-1 C. The difference in weight of -5 cub. centim. of air measured under these conditions is only -001 milligramme, and is therefore negligible.

Subjoined are the details of the weighings of the palladium bulb in Experiment V

EXPERIMENT V. Before Combustion.

Temperature of balance (at start of weighing) 14° -5 C.

(at end „ )14°-GC.

Barometric height (at start of weighing) 766' 7 millims.

„ (at end „ 765-5

Weights used were: — 20, -5, '02, -01 and -01. Rider on 2nd division on beam. Zero at no loud .... 498.

,, (weights in right pan) 393. Mean zero at no load 497.

,, ( ,, ,, left ,, ) 009. Zero (weights in right pan, mean of 5) 398 -2.

„ ( „ „ right „ ) 390.

„ ( „ ,, left „ ) 591. „ ( „ „ left „ „ 4) 597.

„ at no load .... 49G.

„ (weights in right pan) 404. Sensibility 202.

„ ( „ „ left „ ) 599.

„ ( „ „ right „ ) 408.

„ ( „ „ left „ ) 589.

„ ( „ „ right „ ) 396.

„ at no load .... 498.

Weight of bulb (before experiment) 420-54222 grammes.

EXPERIMENT V. After Combustion.

Temperature of balance (at start of weighing) 11° -8 C.

„ (at end „ )12°-4C.

Barometric height (at start of weighing) 760-5 millims.

„ (at end „ )761'9 „

Weights used were:— 10 (A), 5, 2, 1 (I), 1 (Z), -5, -02, -01 (1), and rider on 5th division. Zero at no load .... 503.

„ (weights in right pan) 480. Mean zero at no load 501. .. ( „ „ left „ ) 514.

» ( » » rignt » ) 488. Zero (weights in right pan, mean of 5) 489 • 4. ,, at no load .... 501.

„ (weights in left pan) 513. „ ( „ „ left „ „ 4)512-8.

,1 ( „ „ right „ ) 496.

» ( » .. left „ ) 509. Sensibility 201.

» ( ., ,i right „ ) 492.

» ( ,, ,, left „ ) 515.

„ ( „ „ right „ ) 491.

,, at no load .... 499.

Weight of bulb after experiment 419-53605 grammes.

ON THE ATOMIC WEIGHT OF CHLORINE. 189

7. Method of Carrying Chit the Combustion.

The weighings of the palladium bulb and the chlorine condensation bulb completed, the next step was to set up the combustion apparatus (fig. 5). This consisted of a stout glass globe A, the " combustion globe " made of Jena glass. Its capacity was about 750 cub. centims., and it was provided with three ground-glass tubulures. In order to ignite the hydrogen at the jet, two platinum-iridium wires* (totally enclosed, save for their extreme tips, in glass covers) were fused into the combustion globe on each side of the hydrogen tubulure. By the passage of electric sparks between their tips, the jet of hydrogen was easily ignited.

Into the combustion globe was run sufficient water to absorb all the hydrochloric acid gas formed during the combustion, and to leave dilute acid of a not greater strength than one-seventh concentrated. Then two very thin glass bulbs (capacity of each about 6 cub. centims.), which had been previously filled with a hot, concentrated solution of potassium iodide and sealed, were cautiously slid into this water through one of the tubulures. The palladium bulb B, the chlorine condensation bulb 0, and the three-way tap D were then, respectively, fitted to the tubulures E, F, and G, care being taken that none of the lubricant (phosphoric acid) was squeezed into the combustion globe through -the interstices of the ground-glass joints. To one limb of the three-way tap D, a generator of carbonic acid in an atmosphere of which the subsequent titration of residual iodine was carried out, was attached by a short length of thick-walled indiarubber tubing ; to the third limb was fused the apparatus H, through which any residual gases from the combustion were drawn. It consisted of a wash-bottle which could be taken to pieces by means of the ground-glass joint J. The tap K controlled the passage of the gases through the liquid, an alkaline solution of sodium thiosulphate, contained in the wash bottle.

The tube L was attached to the mercury pump by a short piece of thick-walled indiarubber tubing.

These two short lengths of indiarubber tubing were employed so as to enable us to give a jerking motion to the combustion globe and the bulbs when fitted together : (i.) to break the drawn-out cul-de-sac of the chlorine bulb, and (ii.) to break the potassium iodide bulbs after the combustion. The only danger arose from a possible in-leakage of air through the tube connecting the wash-bottle with the pump, by which the residual oxygen, nitrogen, and hydrogen were withdrawn from the globe. This tube was wired on to the glass when hot, and was well " drowned" before being used to evacuate the globe. We found that no readable volume of air had leaked through into the highest vacuum attainable during three days.

The different parts of the combustion apparatus having been fitted together, the strength of a neutral solution of sodium thiosulphate was determined by titrating a

* The position of these wires is shown by the dotted lines P and PI ; they lie in a plane at right angles to the vertical section shown in fig. 5.

190

PROFESSOR H. B. DIXON AND MR. E, C. EDGAR

I

I o

60

ON THE ATOMIC WEIGHT OF CHLORINE. 191

measured volume against the standard solution of iodine in potassium iodide. A measured amount (about 6 to 7 cub. oentims.) of the sodium thiosulphate solution was run into the wash-bottle and made alkaline by the addition of sodium hydrogen carbonate. The taps D and K were now opened, and the combustion apparatus evacuated (in a stream of water-vapour) as far as possible by the pump. A rapid stream of water- vapour was produced by immersing a large condenser, fused to the pump, in a freezing mixture of ice and salt, and by gently warming the lower part of the combustion globe with warm water. This was done to facilitate the removal of traces of air and nitrogen, and that this was accomplished we concluded from the small amount of nitrogen discovered in the subsequent gas analysis. During the last period of the exhaustion, the calcium chloride and ice freezing mixture, in which the bulb containing liquid chlorine was immersed during the combustion, was prepared, placed in a wide-necked, unsilvered Dewar tube, and packed well round the liquid chlorine bulb.

The evacuation completed, the taps D and H were closed, and the glass cul-de-sac M broken by jerking the glass rod N against it.

The heating of the palladium bulb, enclosed in a stout copper box covered witli asbestos sheet, was next started, the temperature being noted by means of a mercury- nitrogen thermometer.

The temperature of the liquid chlorine was now between —25° C. and —30° C., and the pressure on the special tap Q was therefore not greatly above atmospheric. Q was slightly turned so as to admit chlorine slowly into the combustion globe. When the pressure of gas in the globe had become nearly atmospheric, the tap Q was closed. This point was determined by the change in the faint hissing noise which attended the entry of chlorine into the vacuum. When the palladium bulb had reached a suitable temperature, all lights were turned out.

Next came the ignition of the jet of hydrogen. Whilst a rapid succession of sparks was passed between the platinum-iridium tips, the tap N was very cautiously opened so as to admit the hydrogen slowly into the combustion globe. The moment the jet of hydrogen had ignited the sparks were discontinued, and all attention was centred on the flame. To cool the globe during the combustion, ice was packed round the lower portion, while that part which was immediately above the flame was cooled by a stream of cold water.

To avoid, as far as possible, any diffusion of hydrogen through the flame, the combustion was carried out at a pressure only slightly below atmospheric. The atmosphere of chlorine was constantly replenished through the tap Q, whilst the tap N regulated the admission of hydrogen to the flame.

The combustion of hydrogen in chlorine at a glass jet is an interesting phenomenon. The flame can be divided into two zones — an inner zone of a light apple-green colour, with an outer zone of less pronounced hue. We learnt by experience that three points in connection with the flame were important for our purpose. Firstly, the

!92 PEOFESSOE H. B. DIXON AND ME. E. C. EDGAE

gradual elongation of the outer zone, together with a lessening of the luminosity ol the inner zone, indicated that the atmosphere of chlorine was riot being renewed quickly enough. Secondly, when the flame became smaller and more luminous, we knew that the pressure of chlorine was in excess, and that the gas was being admitted into the globe too quickly. Lastly, a gradual shrinking in the size of the flame, unattended by any change of luminosity, indicated that the supply of hydrogen was failing. This was, of course, remedied by raising the temperature of the palladium bulb.

When the combustion had been carried to such a point that only a drop of liquid chlorine was left in the condensation bulb, the tap Q was finally closed and the flame made very small. As the atmosphere became rarefied, the outer zone of the flame became elongated and less luminous ; the inner zone changed also, but to a less extent. In one experiment (IV.), the flow of hydrogen not being reduced as the chlorine- atmosphere became rarefied, a flame passed through the whole globe. Just before the point of extinction the tap N was closed and the combustion was ended. The duration of the combustion was about three hours, during which constant watching was necessary. The palladium bulb was now allowed to cool to the ordinary temperature.

The two small bulbs, containing concentrated solution of potassium iodide, were then 'broken by dashing them against the interior of the combustion globe, when the residual chlorine was absorbed with precipitation of iodine. The precipitated iodine, however, soon dissolved in the excess of potassium iodide. The tap D was opened and the residual gases were sucked out of the combustion globe in a current of water- vapour through the alkaline solution of sodium thiosulphate contained in the wash- bottle H, in which the vaporised iodine was absorbed. The residual gases were collected in the gas analysis apparatus.

During this exhaustion the long glass tube R connected with the three-way tap D, and containing NaHCO:i, had been heated. The evacuation completed, D was turned and COa admitted until the combustion globe was full. This was indicated by the escape of gas through the manometer. The tubulure G was now opened, cleaned from adhering phosphoric acid, and the residual iodine titrated in the atmosphere of carbonic acid by means of the sodium thiosulphate solution of known strength contained in a calibrated burette.* As sufficient potassium iodide was originally contained in the thin glass bulbs to dissolve easily the precipitated iodine, the titration was quickly and accurately carried out, five drops of starch solution being added towards the end of the titration. f One drop of the standard solution of iodine restored the blue starch-iodide colour to the decolourised liquid in the combustion

h The two burettes employed were carefully calibrated by means of an Ostwald calibrator of 2 cub. centims. volume. The mean results of two calibrations were tabulated and used in determining the volumes.

t Owing to the action of hydrochloric acid on a solution of sodium thiosulphate, we were unable to add excess of the sodium thiosulphate solution and titrate back with the standard solution of iodine.

ON THE ATOMIC WEIGHT OF CHLORINE.

193

B-

Fig. G. Gas-analysis apparatus.

globe, so it was evident that the error in our volumetric determinations of iodine must have been small. The alkaline sodium thiosulphate solution in the wash-bottle was exactly neutralised with very dilute hydrochloric acid, and the residual thiosulphate estimated with the standard iodine solution.

The determination of the residual gases was effected as follows : — Fig. 6 is a sketch of the gas-analysis apparatus employed.* It consisted of a graduated pipette A attached to a bent capillary tube, with stopcocks C and D and a graduated tube B. The weight of the apparatus, filled with mercury from ' the tap C to the end of the capillary tube E, having been determined, the whole was filled with mercury and placed in the trough. The gases, sucked out by the pump from the combustion globe, were collected in B and passed into A. It was assumed that the gases consisted of hydrogen, oxygen, and nitrogen.

When the residual gases had been collected, the whole was transferred to the balance room and allowed to reach the temperature of the room. During this time the taps C and D (fig. 6) were, of course, left open ; they were then closed. In order to maintain the gases during analysis at not only constant temperature but constant pressure, it was necessary that the height of the mercury in the limb A of the gas-analysis apparatus above the surface of the mercury in the trough should be kept constant. With the aid of the two etched scales on A and B, the divisions of which were 1 millim. apart, this constant pressure could be easily attained by raising or depressing the apparatus in the trough until the mercury in the limb A stood the same height as before above the level of the mercury in the trough.

When the gases had reached the temperature of the balance room, and the difference in level of the mercury in the limb A and of the mercury in the trough had been noted, the taps C and D were closed, B was emptied and the apparatus was then ready for weighing. The difference in the weights of the apparatus (i.) full of mercury, and (ii.) containing the residual gases of the combustion (corrected for the weight of these residual gases) gave, by an obvious process, their volume.

After being weighed, the gas apparatus was transferred to the mercury trough, the platinum spiral F was then cautiously heated by an electric current so as to bring about the combination of all the hydrogen with the oxygen. Sometimes the oxidation was attended by an explosion ; this, of course, occurred when the percentage of hydrogen was relatively great.

As can be seen from Table II., the oxygen was always in excess of that required * This form of gas-analysis apparatus was first used by D. L. CHAPMAN and E. HOPKINSON.

VOL. CCV. — A. 2 C

194" PROFESSOR H. B. DIXON AND MR. E. 0. EDGAR

for the complete combustion of the hydrogen. After B had been re-filled with mercury and the apparatus inverted in the trough, the taps C and D were opened, and consequent upon the contraction in volume of the gases, mercury rose in the two limbs A and B. When the whole had cooled to the temperature of the room, and the pressure had been equalised, C and D were closed and the apparatus again weighed as before. The difference between the second weighing and this last one gave the volume of contraction, i.e., gave the volumes of hydrogen and oxygen which had combined.

The apparatus was again transferred to the mercury trough, C and D opened, and hydrogen admitted to the apparatus, sufficient to burn up the residual oxygen. The mercury levels were again adjusted, C and D were closed, and the apparatus again weighed. From weighings three and four the volume of the added gas was easily calculated. The gases were fired by heating the platinum spiral, and, on cooling, the apparatus was weighed as before. The final weighing, coupled with weighing four, gave the volume of contraction, i.e., gave the volume of residual oxygen. From the data thus obtained the composition of the residual gases of the combustion, assuming them to have been hydrogen, oxygen, and nitrogen, was easily calculated.*

The analysis of the residual gases from Experiment 2 is given below in illustration :—

Mean temperature of balance room ............. 13°-8C.

,, barometric height ................. 768 '5 millims.

Difference in level between mercury in limb A and mercury in trough . 198 '5 „

Weight of gas apparatus full of mercury ........... 489-029 grammes.

,, „ „ and residual gases .......... 413'756 „

,, ,, ,, af tor combustion ........... 440 '846 „

,, ,, addition of EL .......... 309-846

» „ „ „ combustion ........... 450-838 „

From these weights the composition of the residual gases was calculated to be as follows : —

4-12 cub. centims. of oxygen ~\ ftt 13°.8 c an(1 768.5 _ 198.5 millims. pressure. 1'33 „ „ hydrogen >

•09 „ „ nitron-en I Gases saturated with aqueous vapour.

These volumes gave, on reduction to N.T.P.,

2-87 cub. centims. of oxygen from steam-) at 0« c alld 760 millimS-

"93 ,, „ hydrogen S.

•08 „ „ air J Gases dry.

Now, in accordance with the equation

2-87 cub. centims. of oxygen were produced by the action on aqueous vapour of 5 -74 cub. centims. of chlorine.

Weight of 5-74 cub. centims. of chlorine is 5-74 x -00317 = -0182 gramme.

Weight of • 93 cub. centim. of hydrogen = • 00008 gramme.

* In preliminary experiments, carried out in the same way, we failed to detect any trace of C02 in the products of combustion.

ON THE ATOMIC WEIGHT OF CHLORINE. 195

8. Results of the Experiments.

In the following tables we have put together the results obtained in the nine experiments. Table II. contains the volumes of the several residual gases, reduced to normal temperature and pressure, as determined by the gas-analysis.

We have assumed that the nitrogen found at the end of the experiments is due to residual air left in the evacuation of the large combustion globe. It is conceivable that a trace of this nitrogen came from the palladium bulb and was weighed as hydrogen. If that were so, the atomic weight of chlorine we have found would be too low. The total volume of nitrogen found in the nine experiments was '8 cub. centim. To make an extreme supposition — if all this nitrogen had been introduced from the palladium bulb, and weighed as hydrogen, and therefore all the oxygen had come from the steam, the atomic weight of chlorine found by us would be '005 too low.

In Table III. we have put together the several portions of residual chlorine not combining with the weighed hydrogen : — (i.) that calculated from the iodine found in the globe ; (ii.) that calculated from the iodine vapour drawn over with the residual gases and caught in the wash-bottle; and (iii.) that calculated from the oxygen found in the residual gases less the oxygen assumed (from the nitrogen) to be present as air.

In Table IV. the weights of the bulbs before and after the combustion are given, with the corrections for buoyancy and for the unburnt gases. We set out the hydrogen weighings to five places of decimals, although it is not, of course, suggested that the absolute weight of the palladium bulb can be determined to this degree of accuracy. The fifth figure does not affect the mean atomic weight deduced from the experiments.

2 C 2

196

PKOFESSOR H. B. DIXON AND ME. E. C. EDGAR

TABLE II. — Determination of Volumes of Residual Gases, cub. centims. at N.T.P.

Experiment	I.	II.	III,	IV.	V. VI.	VII.	VIII.	IX.
Volume of oxygen liberated from					1
	3-39	2-87	5-07	2-91	4-30 3-41	3-65	3-11	4-80
Volume of hydrogen unburnt. . .	1-51	•93	2-02	•08	•40 -74	1-04	3-61	2-45
Volume of residual air	•11	•08	•06	•11	•1C -06	•18	•13	•12

TABLE III. —Determination of the Weight of Chlorine uncombined with the Weighed Hydrogen (in grammes).

Experiment.	I.	II.	III	IV	V	VI	VII	VIII	IX
Unburnt chlorine calculated from iodine in globe	•6941	•6603	•7865	•6238	•6767	•5981	•6838	• 7999	•7260
Unburnt chlorine calculated from iodine vapour Chlorine corresponding with oxygen liberated	•0005 •0215	•0006 •0182	•0004 •0321	•0003 •0185	•0007 •0273	•0005 •0°16	•0005 •0231	•0006 •0197	•0005 •0304
Total excess of chlorine . .	•7161	•6791	•8190	•6426	•7047	• 6202	•7074	•7425	•7569

ON THE ATOMIC WEIGHT OF CHLORINE.

197

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PROFESSOE H. B. DIXON AND MR. E. C. EDGAR

APPENDIX.

1. The Action of Chlorine on Glass.

The following experiments were made to determine the action, if any, of pure, dry chlorine on soft glass. Two glass bulbs of approximately equal volume and weight were made. To one of these, A, was fused the inner portion of a ground glass joint ; it was then cleaned and dried. The two bulbs were then suspended from different pans of the balance and small weights added to one pan to bring them to equilibrium. A was then fitted to the apparatus for generating chlorine, and the whole was evacuated and filled with pure dry chlorine. The bulb was separated from the rest of the apparatus by fusion beyond the ground-glass joint and was then kept for one week. At the end of that time the ground-glass joint was taken to pieces, and the chlorine sucked out and replaced by dried air. It was then weighed, the other bulb acting as a counterpoise.

Weight to counterpoise the bulb (before exposure to chlorine) . . 1 -32468 grammes. „ „ (after „ „ ) . . 1 • 32464 „

The experiment was repeated with two similar bulbs, but the chlorine was left in contact with the glass for a fortnight.

Weight to counterpoise the bulb (before exposure to chlorine) . . 2-67931 grammes.

(after „ ) . . 2-67925

Two more bulbs were subjected to similar treatment, the time of contact, in this case, being a month.

Weight to counterpoise the bulb (before exposure to chlorine) . . 1 • 12884 grammes.

(after „ ) . . 1-12879

These weighings show that on allowing chlorine to remain in contact with soft glass for a considerable period of time, the latter loses weight very slightly.

The bulb used in Experiment 1 was again filled with chlorine, which was allowed to remain in contact with the glass for a week.

Weight of bulb before exposure 1 • 32464 grammes.

„ after „ 1-32465 „

An exposure for a further period of two weeks gave : —

Weight of bulb before exposure 1 • 32465 grammes.

„ after „ 1-32463 „

If any action of chlorine on the soft glass bulb may be assumed to have taken place, it must have occurred during the first week, as further exposure to chlorine gave a constant result.

ON THE ATOMIC WEIGHT OF CHLORINE. 199

The solvent action of liquid chlorine, if any, on soft glass was also examined. The same bulb was employed. After its weight had been determined as above, 10 cub. centims. of liquid chlorine were condensed in it by means of a freezing mixture of solid carbonic acid and ether. The bulb was then separated from the chlorine apparatus by fusion between the ground-glass joint and the bulb, and was laid aside for a week. The ground-glass portion was cleaned and dried, a mark was cut with a clean glass cutter in the glass capillary tubing attached to the bulb, and a clean fracture effected. When the chlorine in the bulb had been totally replaced by air, the three parts of the original apparatus, i.e., the bulb, the piece of glass broken off from it, and the inner portion of the ground-glass joint were weighed, the companion bulb acting as a counterpoise.

Weight of bulb before exposure 1 • 32464 grammes.

„ after „ 1-32466

A similar apparatus was constructed, and, after being subjected to the action of pure dry gaseous chlorine for a week, the last experiment was repeated, the time of exposure being a month.

Weight of bulb before exposure 3 • 49842 grammes.

„ „ after „ 3-49839

There seemed to be no appreciable action of liquid chlorine on soft glass.

Though the combined effect of gaseous and liquid chlorine on soft glass was so exceedingly small, the bulb of the chlorine condensation bulb was subjected, before use, to the action of pure dry gaseous chlorine for a fortnight.

2. The Reaction between Iodine and Sodium Thiosulphate in Presence of Carbonic Acid and of Hydrochloric Acid.

Titrations of sodium thiosulphate by iodine in potassium iodide, carried out in an atmosphere of carbonic acid, showed that the gas had no influence on the accuracy of the residual iodine determinations. A known volume of sodium thiosulphate solution was run into a small Erlenmeyer flask and titrated with the standard solution of iodine in potassium iodide. An equal volume was run into another flask and pure carbonic acid (from sodium hydrogen carbonate) was passed through the solution for ten minutes, it was then titrated as usual. No difference in the volumes of iodine in potassium iodide solution required to combine with the thio in the two flasks could be detected. Several repetitions gave similar results.

S. U. PICKERING* has shown that iodine in potassium iodide solution can be correctly titrated by thiosulphate in presence of hydrochloric acid, if allowance is made for the slow oxidation of the liberated hydrogen iodide by the oxygen from the

* ' Jouru. Chem. Soc.,' 1880, p. 134.

200 ON THE ATOMIC WEIGHT OF CHLOEINE.

air. We have confirmed these experiments with different strengths of hydrochloric acid and found that practically no iodine was liberated in the oxygen free solutions employed.

Approximately equal volumes of iodine in potassium iodide solution were run into small Erlenmeyer flasks X and Y from the calibrated burette B. The iodine in X was then titrated by means of thiosulphate solution from burette A ; hydrochloric acid of known strength was then added to Y and the titration immediately completed.

The experiments were repeated several times with the addition of hydrochloric acid of j concentration :

EXPERIMENT I.

120 cub. centims. of -^ concentrated HC1 were added to the solution in flask Y.

Volumes of iodine in K 1 taken. Burette B. Flask X, 25 • 04 cub. centims. Flask Y, 25 • 04 cub. centims.

Volumes of thio required by above Burette A. Flask X, 25-13 cub. centims. Flask Y, 25 • 16 cub. centims.

EXPERIMENT II.

120 cub. centims. of i concentrated HG1 were added to the solution in flask Y.

Volumes of iodine in K 1 taken. Burette B. Flask X, 25-08 cub. centims. Flask Y, 25-10 cub. centims.

Volumes of thio required by above Burette A. Flask X, 25-21 cub. centims. Flask Y, 25-20 cub. centims.

EXPERIMENT III.

120 cub. centims. of i concentrated HC1 were added to the solution in flask Y.

Volumes of iodine in K 1 taken. Burette B. Flask X, 25 • 23 cub. centims. Flask Y, 25 • 19 cub. centims.

Volumes of thio required by above Burette A. Flask X, 25 • 31 cub. centims. Flask Y, 25 • 29 cub. centims.

EXPERIMENT IV.

120 cub. centims. of i concentrated HC1 were added to the solution in flask Y.

Volumes of iodine in K 1 taken. Burette B. Flask X, 25-05 cub. centims. Flask Y, 25-06 cub. centims.

Volumes of thio required by above Burette A. Flask X, 25-14 cub. centims. Flask Y, 25 • 1 7 cub. centims.

Since hydrochloric acid of ^ concentration has then no influence on the titration of iodine in potassium iodide solution by sodium thiosulphate solution, we felt justified in using, in our experiments, such volumes of water as never permitted of the acid solution attaining a greater strength than j concentrated.

[ 201 ]

VI. Researches on Explosives. — Part III.

By Sir ANDREW NOBLE, Bart., K.C.B., F.R.S., F.R.A.S.

Received June 8, — Read June 8, 1905.

[PLATES 1-13.]

THE Researches which I venture to communicate to the Royal Society are, for the new explosives cordite, modified cordite, and nitro-cellulose, a continuation of the same modes of research, adopted in the experiments I made many years ago upon fired gunpowder with regard to the pressure and other phenomena attending its decomposition, and which appeared in the ' Philosophical Transactions.' * In the present investigations the same general methods have been followed, but with apparatus greatly improved and of much greater delicacy.

The Academy of Sciences of France did Sir F. ABEL and myself the great honour to appoint MM. le General MORIN and BERTHELOT to report on our paper, and after giving an extended analysis of the results of our experiments the reporters con- cluded t : " Par cette analyse trop succincte de rimportant travail que MM. NOBLE et ABEL ont soumis au jugement de 1' Academic, on pent voir que malgre" certaines critiques auxquelles nul travail humain ne saurait echapper, 1'ensemble de leurs recherches n'en constitue pas moins une oeuvre capitale, propre a, jeter un grand jour sur toutes les questions qui se rattachent aux effets des poudres."

A paper by M. BERTHELOT in the same No. of the ' Comptes Rendus ' draws attention to the chief point upon which that eminent chemist differed from ourselves.

A study of the variations in the products when the decomposition of gunpowder was conducted under pressures widely different, varying in fact between 1 ton per sq. inch and 35 to 40 tons per sq. inch, led my lamented friend Sir F. ABEL and myself to state that, according to our view, "any attempt to express even in a complicated chemical equation the nature of the metamorphosis which a gunpowder of average composition may be considered to undergo, would only be calculated to convey an erroneous impression as to the simplicity or definite nature of the chemical results, and their uniformity under different conditions, while possessing no important bearing upon an elucidation of the theory of the explosion of gunpowder. "

* NOBLE and ABEL, 'Fired Gunpowder,' Part I., 1875. t ' Comptes Rendus,' vol. 82, p. 492. VOL. CCV.— A 392. 2 D 23.9.05

202 SIE ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

M. BERTHELOT, in the memoir to which I have referred, considers that the view which we took was contrary to all that was known in chemistry.

It is no light thing to differ from so great an authority as M. BERTHELOT ; but the innumerable experiments I have since made with various modern explosives, in which the decomposition is of a simpler nature than that of fired gunpowder, have only confirmed me in the opinion that Sir F. ABEL and I then expressed.

Thus in a paper published in the ' Proceedings of the Royal Society,'* I pointed out that when gun-cotton was fired under a great variation of pressure, the variations in the proportions of the resulting gases were both great and regular. In passing, for instance, from explosions under a pressure varying from 1'5 ton per sq. inch (2287 atmospheres) to 50 tons per sq. inch (76217 atmospheres) the volume of carbonic anhydride increased from 26'49 per cent, to 36'18 per cent., while the carbon monoxide decreased from 36'G6 per cent, to 27'57 per cent.

There were also other differences, though not quite so marked, such as the steady decrease of free hydrogen and the large and steady increase of marsh gas.

In the researches on gun-cotton to which I have alluded, certain data, such as the units of heat and the quantity of water formed by the explosion, although deter- mined, were not determined under the varying conditions with regard to pressure and the quantity and nature of the gases generated, under which the explosion took place.

In the researches I am about to refer to, all the data connected with the explosion have been carefully determined, and I preface an account of the experiments themselves by a description of the varied apparatus adopted, or specially designed, for determining the tension of the gases generated by the explosion, the volume of the permanent gases and their nature, the quantity of water formed, the units ot heat generated, the time taken to complete the explosion under different pressures and different dimensions of the cords, tubes, or ribbons, these being the forms under which the explosives are generally made up.

I have made experiments also to determine the time in which the exploded gases part with their heat to the walls of the vessel in which they are confined.

These investigations have opened out many suggestive points, but in the present paper I propose to confine myself to a description of the apparatus used and the results obtained, giving also a resume of the calculations made to test the accuracy of the observations.

Commencing with the apparatus for firing the explosives experimented with at different densities, obtaining the gases for analysis, and measuring their total volume, the vessel A., in Plate 1, is one of the explosion .cylinders used for these experiments ; B is the plug closing the vessel, on which also is shown the arrangement by which, when desired, the gas is allowed to pass at a small pressure through the tubes, either

* ' Roy. Soc. Proc.,' vol. 56, p. 209.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES. 203

to the gasometer C or at pleasure into the gas tubes D, which, before the experi- ment, are filled with mercury, the stop-cocks above and below being closed ; E is a thermometer for determining the temperature of the gas when its volume is measured.

Immediately after the explosion, if the vessel be quite tight, the valve at B is very slightly opened and the gas allowed to pass slowly through the tube F, containing pumice-stone and concentrated sulphuric acid, into the gasometer.

When it is quite certain that all air is removed from the conducting tubes, the gas is allowed to flow into one of the gas tubes D, and shortly afterwards or at fixed intervals of time into the other two tubes, the quantity of gas in the tubes being added to that measured in the gasometer, the height of the barometer and the temperature of the gas at the moment of measurement being also determined.

When the whole of the gas has been transferred to the gasometer, and the temperature and barometric pressure taken, the cylinder is opened. A considerable quantity of water is always found ; as much as possible of this water is collected by means of a weighed sponge placed in a weighed vessel, and closed by a ground glass plate. The amount of the water so collected is determined by weighing in the usual manner.

After all the water that it is possible to remove with the sponge is collected, a weighed vessel of calcium chloride is placed in the cylinder, which is then closed, and left for one or two days, when the same procedure is followed with a second calcium chloride vessel, after which the cylinder is generally found to be perfectly dry.

The next point to be determined is the amount of heat generated by the explosion.

For this purpose a strong steel vessel, the section of which is shown in Plate 2, and of which the heat capacity is carefully determined, is employed. The calori- meter used is practically of the same construction as that described by OSTWALD in his ' Manual of Physico-Chemical Measurements.'

A section of this calorimeter is also shown in Plate 2, the corresponding inner and outer surfaces of the several vessels being nickel plated. For some hours before the experiment the calorimeter is kept in a room maintained at as even a temperature as possible, the explosion vessel itself with the charge to be exploded being kept in the water as shown, so that the whole system may assume practically the same temperature.

The rise of temperature due to the explosion being approximately known from previous experiments, the water in the outer cylinder before firing is kept at a temperature about half way between the initial and final temperatures of the inner vessel.

The thermometers employed for these determinations are calorimetric, specially made for calorimetric experiments, and are only used for observing changes of temperature, and not for determining absolute values. The range of measurement in the thermometers I used was about 8° C., but by a special contrivance these 8°

2 D 2

204 SIE ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

can be brought to any point of the thermometric scale that may be desired. The temperature can be read approximately to O'OOl0 C.

Full illustrations of a few of the calorimetric observations will be given with the corresponding calculations, and a resume of the results of the experiments at the end of the paper.

The analysis of the gaseous products of explosion was carried out by means of SODEAU'S gas analysis apparatus,* the principal features of which are shown in Plate 3.

Mr. SODEAU'S apparatus is admitted to be the most convenient that has been yet devised. I am indebted to him for the description of his apparatus and the mode of analysis followed.

The tubes used for measuring and correcting for variations of temperature and pressure are placed in a cylindrical water-jacket. The measuring tube M is of 50 cub. centims. capacity, and is graduated in -^-cub. centirn. divisions. Its upper end terminates in a capillary three-way stop-cock N, arranged so that the capillary K may be placed in communication either with the interior of the measuring tube or with the bent tube U containing water. The zero point of the graduation is at the outer side of the plug of the stop-cock N. The level tube L communicates with the measuring tube by means of a side branch, bent so as to prevent any entangled air bubbles from reaching the measuring tube. The lower end of the level tube is connected to a T piece, one end of which is provided with a stop-cock and leads to the mercury reservoir, whilst the other is prolonged across the table to a point near the reading telescope, where it terminates in a piece of thick-walled rubber tubing, the compression of which by a broad screw clip affords a means of accurately adjust- ing the level of the mercury without taking one's eye from the reading telescope.

In order to render the apparatus more compact, the reading telescope is placed on the gas analysis table instead of on a separate support, and all graduations are consequently on the side opposite to that from which the stop-cocks are manipulated. An illuminating arrangement slides on the rod P.

The corrections for variations of temperature and pressure are found by means of the "Kew Principle" correction tube C, which is so called because, as in the " Kew " barometer, the disturbance of the level of the liquid is allowed for in the graduation of the instrument, instead of being adjusted before each reading is taken. It consists of a cylindrical bulb having a stop-cock at its upper end, and attached below to a U tube, which is graduated on one limb and filled with water up to the zero mark whilst the stop-cock is open. The volume of air contained in the bulb is such that the water is displaced to the extent of one small division by a change of temperature and atmospheric pressure, which will cause a gas to experience an alteration of volume amounting to O'l per cent. These small divisions are further

* 'Journal of the Society of Chemical Industry,' Feb. 28, 1903, page 187.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES. 205

subdivided into tenths by eye estimation. Errors of parallax are avoided by the use of a mounted lens sliding on the rod E, and the corrections are thus read directly in percentages as easily as the temperatures would be read by means of a thermometer.

Absorptions are carried out in separate pipettes, one of which is shown in position. About 20 cub. centims. of the absorbent is usually confined over mercury in the bulb E, which is slightly inclined in order to facilitate the return of the unabsorbed gas. The horizontal bulb D receives the mercury displaced by the gas. The bulb F contains clean mercury, and, like the bulb E, can be placed in communication with the capillary G by means of the three-way stop-cock H.

The explosion pipette resembles that of DITTMAR, but has a three-way stop-cock and mercury bulb arranged as in the absorption pipettes.

In conducting an analysis, the sample tube is connected to the measuring tube by means of a capillary tube previously filled with mercury, and the gas drawn in by lowering the reservoir. After the mercury has been roughly levelled, the stop- cock N is turned so as to connect the capillary K with the tube U, and an absorption pipette, containing caustic potash solution, connected to the measuring tube by means of thick- walled rubber tubing, the ends of the capillaries being made to meet. A little water is then sucked through the capillaries into the bulb F, and mercury allowed to run back and fill the capillaries. The stop-cock leading to the large mercury reservoir having been closed, and the level tube being open to the atmo- sphere, the mercury is accurately levelled, as already described, and the volume ol the gas read by means of the reading telescope. A reading of the correction tube is also taken.

In order to determine the amount of carbon dioxide present, the gas is driven over into the absorption pipette, followed by sufficient mercury to clear the capillaries, and the pipette well shaken in order to make the absorption complete. A little more mercury is then run over in order to clear away the potash from the bottom of the capillary attached to the absorption bulb, and the stop-cock N reversed so that the mercury in the capillaries runs into the tube U. The stop-cock N is then again turned and the gas slowly passes into the measuring tube, the rate being controlled by the stop-cock H, which is reversed as soon as the absorbent reaches it, so that the gas may be swept out of the capillaries by means of clean mercury from the bulb F. The stop-cock N is closed as soon as it is reached by the mercury. The gas is again carefully measured and the decrease of volume (after the correction for alteration of temperature and pressure has been applied) is equal to the amount of carbon dioxide originally present. The residue is then treated with alkaline pyrogallol, in order to ascertain whether any trace of air has been left in the connecting tubes during the collection of the sample and has so contaminated the gas. (This is more likely to occur when the explosion has taken place under feeble pressures and but little gas been produced.) If any oxygen is absorbed by the pyrogallol, its volume is multiplied by 4 '8 and the product (representing the volume of air present) deducted from the

206 SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

volume of gas taken for analysis, in order to obtain the volume of uncontaminated gas in the sample, and hence the correct percentages of the various constituents.

Carbon monoxide is next removed by prolonged treatment with two successive portions of acid cuprous chloride solution. After absorption in the first cuprous chloride pipette the gas is directly transferred to a second pipette containing a solution which has not previously absorbed more than a trace of carbon monoxide, this transference being accomplished in practically the same manner as the return of the gas to the measuring tube, which takes place after transference to a pipette containing a little water, which removes the traces of hydrochloric acid derived from the cuprous chloride solution. An excess of oxygen* is then added, and, after measuring, the mixture is transferred to the explosion pipette, where it is exploded by means of an electric spark after expanding to such a volume as to prevent any marked oxidation of the nitrogen, whilst ensuring the complete combustion of the methane and hydrogen. The residue is next measured in order to ascertain the reduction of volume resulting from the explosion, and the carbon dioxide, produced by the combustion of the methane, is determined by absorption with potash. The volume of the carbon dioxide produced is equal to that of the methane originally present. The contraction due to the combustion of the methane, or in other words, twice the volume of the carbon dioxide, is deducted from the total contraction resulting from the explosion, and two-thirds of the corrected contraction so obtained is equal to the volume of hydrogen.

Finally, the excess of oxygen remaining after explosion is determined by means of alkaline pyrogallol as a check upon the amounts of hydrogen and methane calculated as above. The nitrogen is estimated by difference.

The above represents the routine determination of carbon dioxide, carbon monoxide, hydrogen, methane and nitrogen, as usually carried out, but additional tests have also been employed in order to ascertain whether certain other bodies were present in measurable quantities, but with negative results. Thus some of the gas samples were examined for unsaturated hydro-carbons (ethylene, &c.) immediately after the removal of the carbon dioxide, by shaking the gas with fuming sulphuric acid,t and removing acid fumes in the potash pipette before again measuring. No change of volume was ordinarily observed, and in no case did the change exceed O'l per cent., hence the samples did not contain any appreciable quantity of unsaturated hydrocarbons.

The ordinary determinations of contraction resulting from explosion, carbon dioxide

* The oxygen is prepared by the electrolysis of dilute sulphuric acid in a Hof mann voltameter and freed from traces of hydrogen by treatment in a Winkler combustion pipette. A supply is stored over mercury in one of the ordinary absorption pipettes ready for use.

t Fuming sulphuric acid was used in one of the ordinary absorption pipettes, provided with a guard tube containing sulphuric acid, in order to prevent moisture from gaining access to the upper bulb D. Of course no mercury was employed in the absorption bulb, and that in the capillaries was driven into the bulb F when sending the gas into the pipette.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES. 207

produced, and oxygen consumed, do not afford a means of distinguishing methane from its homologues in presence of an excess of hydrogen ; thus ethane, together with its own volume of hydrogen, would give the same numerical results as two volumes of methane. A process of fractional combustion was therefore applied to some of the samples obtained from high density charges, as these contained large proportions ot saturated hydrocarbons. After removing carbon dioxide, carbon monoxide and unsaturated hydrocarbons, an excess of oxygen was added and the hydrogen was removed by repeatedly passing over gently heated palladinized asbestos contained in a capillary tube attached to a pipette containing water, as in the ordinary Orsat- Lunge apparatus, until no further decrease of volume occurred. The residual mixture was then examined by explosion, &c., in the usual manner. In each case the volume of carbon dioxide produced almost exactly half that of the decrease, resulting from the explosion, which latter was equal to the volume of oxygen consumed. These ratios agree with those required by the equation

CH4+2Oi = CO,+2H,0,

but differ markedly from those which would result with the homologues of methane, thus even with ethane the proportions are 4:5:7 instead of 1 : 2 : 2. It therefore follows that the saturated hydrocarbons should be calculated as methane, none of the other members of the series being present in appreciable quantities. Examination of the water condensed in the closed vessel showed that the gas could not contain either ammonia or cyanogen in marked quantities, as the distribution under high pressure would so greatly favour the water. The presence of oxides of nitrogen is, of course, incompatible with that of a large proportion of hydrogen, as the gases have slowly cooled from a very high temperature. A trace of sulphuretted hydrogen, sufficient to markedly discolour mercury, exists in the gas when black powder is used as a lighter, but for all practical purposes the gaseous products of explosion may be regarded as consisting entirely of carbon dioxide, carbon monoxide, hydrogen, methane and nitrogen.

One other arrangement of apparatus remains to be described, and that apparatus is used both for determining the time that explosives of various forms and natures require for their transformation, and for determining the rate at which they communicate the heat accompanying the explosion to the walls of the vessel in which the explosion takes place.

The apparatus (see Plate 4) consists of an explosion vessel of the usual form, the explosion vessel being closed at its two ends by gas-tight plugs, through one of which pass the firing wires, while to the plug at the other end is fitted a pressure indicator.

The pressure indicator is provided with a steel plunger of small area, which is exposed to the gas pressure.

An enlarged continuation of this plunger engages the end of a spiral spring a, the

208 SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

resistance of which has been carefully determined. Attached to this plunger at I is a lever, the fulcrum of which, c, is fixed to the stationary bracket of the indicator, so that, when the spring is compressed, motion is given to the ends of the lever.

Fixed to the lever are two electric magnets d, the one to record seconds, the other to perfect the firing circuit. A rocking bar e is coupled up over the seconds magnet, which is again coupled at the other end by a link /, thus conveying the seconds beats of the chronometer to the pen tracing its path on the revolving drum.

The revolving drum itself is of light wood ; fixed to the frame are two rods gg, upon which slides the carriage for carrying the recording pen. The pen is held up by a detent, which is liberated by the firing current passing through the electro- magnet to which the detent is attached. There are two speeds given to the drum, the first a high speed (about 40 inches per second), the second very slow, about one inch per second. The drum is revolved by means of cord bands, which lead from the speed gear of the motor.

Before firing, the fast-speed cord is made to drive the drum, the slow-speed cord running free ; about one or two seconds after the explosion the change speed lever is raised, thereby releasing the fast cord and tightening the slow cord. The fast speed is obtained approximately by watching the tachometer, but the actual speed is determined by measuring the length of the second on the recording diagram. The diagram is traced on a sheet of tin foil backed by paper. This is placed on the drum as shown on Plate 4, the edges being joined with gum, the surface being smoked black by camphor.

The chronometer is of the ordinary marine type, but is furnished with a seconds make-and-break arrangement ; this being coupled up through a relay to the pressure lever, causes the recording pen to beat seconds till the desired curve is complete.

The action of the apparatus during an experiment is as follows : —

All connections being made, the chronometer is coupled up, the pen carriage beating seconds, but no mark is yet made on the recording tin foil, the pen being held by the detent. The drum is started, and when it has reached the desired speed, as shown by the tachometer, the button of the firing battery is pressed and the circuit is completed at the beat of the next second.

The current simultaneously releases the pen and fires the charge. As quickly as possible the speed is reduced by raising the speed lever and at the same time reducing the speed of the motor. The chronometer continues to beat seconds, thereby giving the relation between time and pressure until the experiment is concluded. The diagram is then removed from the drum by cutting through the point where the pen dropped, this being the beat of the second firing the charge. The sheet is then laid on a tray face up, flooded with thin varnish, and hung up to dry.

For the purposes of these reseaches, which are specially directed to ascertain the differences in the phenomena attending the transformation of explosives fired under diiferent pressures, I have employed three explosives, viz., the cordite known as

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

209

Mark I (for which the country has been indebted to the labours of Sir F. ABEL and Sir J. DEWAR), the modified cordite known as M.D., and a tubular nitrocellulose known as R. R. Rottweil.

The general results, which I need not say have necessitated much calculation, are given in tables, but I think it necessary to give the results of a few experiments worked out in full, these being a fair sample of the whole series. In each case I give the reconciliation between the elements determined in the explosive and the same elements found in the gases after explosion.

Taking into account the fact that the explosives themselves are not always of precisely the same composition, and also the nature of the experiments, the recon- ciliation to which I have referred is a very great deal closer than I expected.

It has been suggested to me more than once that the mixture of the gases might not be homogeneous, that is, that tubes taken at different times from the explosion vessel might not give the same analysis. I have not found this to be the case. Thus, in an experiment where a charge of Rottweil R. R. was fired under a pressure of 20'5 tons per sq. inch (3125 atmospheres), and a tube of the resultant gases was taken so soon as it was certain that all the air contained in the conducting tubes, &c., was displaced, a second tube being taken 6 or 7 minutes later, the analysis gave for the two tubes of permanent gases the following percentages : —

1st tube.

2nd tube.

C02

CO

H

CH4

N

28 '06 percentage volumes. 34-02 17-16 7-41 13-35

28-02 percentage volumes. 33-92 17-00 7-40 13-66

Taking now the data given by the explosion of a charge of 3 2 '6 8 grammes of M.D., which was fired at a density of O'l under a pressure of G-9 tons per sq. inch (1051 '8 atmospheres), the resultant quantity of gas was

27,486 cub. centims. at 160-6 C., and under bar. pressure of 75T33 millims. = 25,916 cub. centims. at 0° C. and 751 "33 millims. = 25,621 cub. centims. at 0° C. and 760 millims.

The quantity of water collected was 4136 grammes, equivalent to 5145'! cub. centims. aqueous vapour at 0° C. and 760 millims.

The percentage results of the analysis of the permanent gases in volumes are given in Column I., the total volumes in Column II, the percentage volumes, including aqueous vapour, are shown in Column III., and the percentage weights of the total gases in Column IV.

VOL. ccv. — A. 2 E

210

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

	I.	II.	III.	IV.
	Percentage volumes,	Total, permanent gases,	Percentage volumes, total	Percentage weights, total
	permanent gases.	cub. centims.	gases.	gases.
C02.	20-10	5,149-8	16-74	30-82
CO ....	40-70	10,427-7	33-90	39-71
H . . . .	23-10	5,918-4	19-24	1-61
CH4 . . .	1-00	256-3	0-83	0-55
N . . . .	15-10	3,868-8	12-57	14-74
H20 ...			16-72	12-57

The reconciliation between the amounts of C, O, H and N, contained originally in the explosive, and found in the products of explosion, were obtained as follows : —

	C,.	02.	H2.	N2.
CO.,* CO ... H . . . CH4* . . N . . .	0-2010 0-4070 0-2310 o-oioo 0-1510	0-1005 0-2035 0-0050	0-2010 0-2035	0-2310 0-0200	0-1510
Total	s . . . .	0-3090	0-4045	0-2510	0-1510

Multiplying the carbon, oxygen, and nitrogen by 12, 16, and 14 respectively, we obtain : -

	C2. grammes. 3-708	0,. grammes. ' 6-472	H2. grammes. 0-2510	N2. grammes. 2-114
And again multiplying by 2 • 295	25,621-0 ,
11,160-7
Add the H20 In M.D. cordite Difference	C2. grammes. 8-510	02. grammes. 14-85 3-67	H2. grammes. 0-57 0-46	N2. grammes. 4-850
8-51 9-11	18-53 18-66	1-03 1-030	4-85 4-49t
-0-60	- 0-13	o-oo	+ 0-36

In this experiment the quantity of gas and water measured shows that 1 gramme of M.D. under the pressure named above gave rise to 788 '4 cub. centims. of permanent gases, or to 946 '4 cub. centims. including aqueous vapour.

@ A

* Using HOFMANN'S notation : t Including N in cylinder.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

211

Again, with a charge of 99 '65 grammes of M.D. cordite and a density of 0'3, the gaseous pressure being 27'62 tons per sq. inch (4210'3 atmospheres), the quantity of gas measured, after being reduced to 0° C. and 760 millims. pressure, was 72,768 cub. centims., while the quantity of water collected was 11 '162 grammes, equals 13,885'5 cub. centims. aqueous vapour at 0° C. and 760 millims. pressure.

	I.	II.	III.	IV.
Percentage volumes, permanent gases.	Total, permanent gases, cub. centims.	Percentage volumes, total gases.	Percentage weights, total gases.
CO, . . . CO ... H . . . . CH4 . . . N . . . . H20 . . .	29-40 31-10 17-75 6-55 15-20	21,393-7 22,630-9 12,916-4 4,766-3 11,060-7	24-69 26-12 14-91 5-50 12-76 16-02	42-07 28-32 1-16 3-41 13-91 11-13

From the above data it appears that the explosion gave rise to 735 cub. centims. of permanent gas and 87 5 '3 cub. centims. total gas when reduced to 0° C. and 760 millims. pressure.

RECONCILIATION.

	C2.	02.	H,.	N2.
CO,. . . 0-294 CO ... 0-311 H . . . ; 0-1775 CH4. . . 0-0655 N . . . 0-152	0-147 0-1555 0-03275	0-294 0-1555	0-1775 0-1310	0-152
1-000 0-3352 0-4495 0-3085 0-152
x!2 x!6 xl4
= 4-022 7-192 0-3085 2-128

72 768'0 Multiplying again by — — = 6 '520, we have as the weights found in grammes :—

Adding for H20

Totals In cordite before explosion

Difference

C2. 26-22	02. 46-89 9-92	H2. 2-01 1-24	N2. 13-88
26-22 27-05	56-81 56-26	3-25 3-33	13-88 13-36
-0-83	+ 0-55 2 E 2	-0-08	+ 0-52

212

SIR ANDREW NOBLE: EESEARCHES ON EXPLOSIVES.

Again, taking from Experiment 1416 an example of transformation at a high pressure, a charge of 81 grammes (including lighter) of Mark I cordite fired under a pressure of 22'5 tons per sq. inch (3429'8 atmospheres), the quantity of gas generated after being reduced to 0° C. and 760 millims. pressure was 54, 961 '5 cub. centims. As before, the percentage in volumes of the permanent gases is shown in Column L, of the total gas in Column III., and the respective weights of the total gases in Column IV.

The quantity of H2O collected was 11 '96 grammes = 14,878'2 cub. centims. aqueous vapour at 0° C. and 760 millims.

	I.	II.	III.	IV.
	Percentage volumes,	Total, permanent gases.	Percentage volumes, total	•Percentage weights, total
	permanent gases.	cub. centims.	gases.	gases.
CO, . . .	31-30	17,203-0	24-63	41-95
'CO ...	29-50	16,213-7	23-22	25-15
H . . . .	18-50	10,167-8	14-56	1-13
CH4 . . .	1-95	1,071-7	1-53	0-95
N . . . .	18-75	10,305-3	14-76	16-02
H,0 . . .	—	—	21-30	14-80

These data give the quantity of permanent gases generated at 686 "4 cub. centims. and the total gases at 869 -7 cub. centims. per gramme.

Proceeding to compare as before the elements in the cordite and in the exploded gases, we have : —

	C*	02.	H2.	N2.
C02. . . CO ... H . . . CH4. . . N . . .	0-3130 0-2950 0-1850 0-0195 0-1875	0-1565 0-1475 0-0098	0-3130 0-1475	0-1850 0-0390	0-1875
0-3138 0-4605 0-224 0-1875
x!2 x 16 x!4
=3-766 7-368 0-224 2-625

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

213

	' 11,1607		we nave lur >	veigrits in
Add for H20 Totals The cordite	C2. 18-55	02. 36-28 10-63	H2. 1-10 1-33	N2. 12-93
18-55 18-47	46-91 46-94	2-43 2-43	12-93 12-46

Difference

+ 0-08

-0-03

0-00

+ 0-47

Experiment 1401.— At a density of 0'4 a charge of 128'5 grammes of Rottweil nitrocellulose E.R., giving a pressure of 34 -9 tons per sq. inch (5320'0 atmospheres), generated 88,689 cub. centims. at 0° C. and 760 millims. pressure, also 15 '2 3 grammes of water = 18,946'! cub. centim. of aqueous vapour at 0° C. and 760 millims.

GAS ANALYSIS.

	I.	II.	III.	IV.
	Percentage	Total,	Percentage	Purcenta
	volumes,	permanent gases,	volumes, total	weights, tc
	permanent gases.	cub. centims.	gases.	gases.
C02. . . .	33-70	29,888-2	27-44	45-80
CO ....	28-90	25,631-1	23-53	24-99
H . . . .	14-20	12,593-8	11-56	0-88
CH4 ...	9-85	8,735-9	8-02	4-87
N . . . .	13-35	11,840-0	10-87	11-57
H20 . . .		—	18-58	11-89

Hence we have 690 '1 cub. centims. of permanent gases, or 8 46 "8 cub. centims. including aqueous vapour per gramme of explosive. Proceeding to reconcile the elements, we have : —

	C2.	02.	Ha.	N2.
C02. . . CO ... H . . . CH4. . . N . . .	0-3370 0-2890 0-1420 0-0985 0-1335	0-1680 0-1450 0-0495	0-3370 0-1340	0-1420 0-1970	0-1335
1-0000 0-3625 0-4810 0-3390 0-1335
x!2 x!6 x!4
= 4-350 7-700 0-339 1'870

214

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

Multiplying again by

Add H20

Totals In the R.R. nitrocellulose

Difference

= 7'947, we have the weights :-

C2.	02.	H2.	N»
grammes.	grammes.	grammes.	grammes.
34-57	61-08	2-69	14-83
—	13-54	1-69	—
34-57	74-62	4-38	14-83
35-85	73-33	4-06	15-16

-1-28

+ 1-29

+ 0-32

-0-33

In the reduction of the experiments hitherto considered, it has not been necessary to make any correction to the quantity of gas, as the weight of the gases sufficiently accurately represents the weight of the explosive experimented on, but it occasionally happens, especially at high pressures, that, at the moment of firing, a puff of gas escapes, the leak, however, being generally only momentary, the explosion vessel becoming later perfectly tight. In these cases of course the weight would be in defect, but in a few cases the weight of the gases was in excess, and I proceed to show how these experiments were dealt with.

Experiment 1417.— At a density of 0'45 a charge of 143'91 grammes of M.D. cordite, giving a pressure of 43'22 tons per square inch (6588'2 atmospheres), generated 98,231-9 cub. centims. at 0° C. and 760 millims. pressure; the water collected was 15'59 grammes = 19,384'0 cub. centims. The analysis of the perma- nent gases in volumes gave : —

	I.	II.	III.	IV.
Percentage volumes, permanent gases.	Total, permanent gases.	Percentage volumes, total gases.	Percentage weights, total gases.
C02 . . CO .... H . . . . CH, . . . N . . . . H.O . . .	36-6 24-8 11-9 10-7 16-0	cub. centims. grammes. 35,952-9 = 70-86 24',361-5 = 30-55 11,689-6 = 1-05 10,510-8 = 7-54 15,717-1 = 19-76	30-56 20-71 9-94 8-94 13-36 16-49	48-75 21-02 0-72 5-19 13-59 10-73

Now if to the weights given in Column III. we add the weights of water, it will be found that the total weight is 1'44 grammes greater than the charge actually employed. The volume of the gases has therefore been reduced to 97,589'5 cub. centims., thus giving 676'3 cub. centims. of permanent gases or 810'6 cub. centims. total gas for each gramme exploded.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES. RECONCILIATION.

215

	C2.	02.	H2.	N2.
C02. . . CO ... H . . . CH4 . . N . . .	0-366 0-248 0-119 0-107 0-160	0-183 0-124 0-054	0-366 0-124	0-119 0-214	0-160
1-000 0-361 0-490 0-333 0-160
x!2 x!6 x!4
= 4-332 7-840 0-333 2-240

Multiplying again by

Add H2O

Totals In M.D.

Difference

Q7

'

"* 11,160	•7 "
C,.	0,.
grammes.	grammes.
37-88	68-56
—	13-86
37-88	82-42
39-06	81-24

= 8744:-

-1-18

+ 1-18

H2.

grammes.

2-91 1-73

4-64 4-80

-0-16

N2. grammes.

19-59

19-59 19-29

+ 0-30

Experiment 1496. — At a density of 0'5 a charge of 155 "84 grammes of cordite were fired under a pressure of 52 '84 tons per sq. inch (80547 atmospheres). On firing, a slight escape of gas passed the firing plug, which, however, became immediately tight. The quantity of gas measured was 93,1 99 -8 cub. centims., when reduced to 0° C. and 760 millims. pressure. 21 '135 grammes of water were collected, representing 26,291 cub. centims. aqueous vapour. At the standard temperature and pressure, the gas analysis was as follows : —

	I.	II.
	Percentage		Total, grammes.
	volumes,	Total, permanent gases.
	permanent gases.
		cub. centims. grammes.
C02 . .	41-95	39,097-3 = 77-06
CO ...	19-10	17,801-2 = 22-32
H . . . . CH4 . . .	12-05 7-05	11,230-6 = 1-01 6,570-6 = 4-71	-149-49
N . . . .	19-85	18,500-2 = 23-25
H20 . . .		21-14	-

216

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

If the column of weights be added up, it will be found that there is a deficiency of 6 '3 5 grammes. The quantity of gas measured must therefore be increased to 97,158-9 cub. centims., and the corrected calculation will stand thus: —

	I.	II.	III.	IV.
	Percentage volumes,	Total, permanent gases.	Percentage volumes,	Percentage weights, total
	permanent gases.		total gases.	gases.
	cub. centims. grammes.
CO, . . .	41-95	40,758-2 = 80-38	33-02	51-84
CO ...	19-10	18,557-3 = 23-29	15-03	15-03
H . . .	12-05	11,707-7 = 1-04	9-48 0-67
CH4 . . .	7-05	6,849-7 = 4-93	5-55 3-18
N . . .	19-85 19,286-0 = 24-26	15-62	15-65
H,0 . . .	—	27,408-4 = 21-94	21-30 13-63
Total gases ....	124,567-3 =155-84	—	—

RECONCILIATION.

	C,.	02.	Hg.	N»
C02 . . . CO ... H . . . CH4. . . N . . .	0 4195 0-1910 0-1205 0-0705 0-1985	0-2098 0-0955 0-0353	0-4195 0-0955	0-1205 0-1410	0-1985
0-3406 0-5150 0-2615 0-1985
x!2 x!6 x!4
=4-087 8-240 0-2615 2-779

Multiplying again by

H20

Totals Originally in cordite

J 11,1607
grammes. 35-58	O2. grammes. 71-73 19-50	H2. grammes. 2-28 2-44	N2. grammes. 24-19
35-58 35-65	91-23 90-69	4-72 4-68	24-19 24-05

-0-07

+ 0-54

+ 0-04

+ 0-14

The heat units evolved by the explosion were, as has been already mentioned, determined in a calorimeter of the type of that described by Professor OSTWALD in his " Physico-Chemical Measurements."

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

217

The heat capacities of the explosion vessels were carefully determined, as were those of the calorimeters and their equipage, including the stirrers and the mercurial thermometers. The latter were of the differential type described by Professor OSTWALD, those I used having a range of about 8° C. Two observations for each density were sufficient if the observations were accordant. If not accordant, three were generally taken.

Not unnaturally, the observations at the higher densities were considerably more accordant than those at the lower.

Commencing with the Chilworth R. R. nitrocellulose (tubular) in Experiment 1344, 9'17 grammes were fired, the explosion vessel, when fired, being suspended in 4000 grammes of distilled water in the calorimeter, the water equivalent of the explosion vessel and the calorimeter being 680 cub. centims. It was then found that immediately before explosion the calorimeter differential thermometer showed 1°'1G1 (equivalent to 19° '9 C.).

	Degrees Cent.	Difference.
Temperature before explosion	1-161
„ 2 minutes after explosion . . .	2-600	—
4 „ „ „ ...	3-043
6 „ „ ...	3-055
8	3-057
		- -002
» 10 ,,	3-055
		- -003
» 12 „ ,, „	3-052
		- -001
i) 14 >, ,, >,	3-051
		- -003
16 „ „ ...	3-048
		- -004
18 „ „ ...	3-044
		- -002
20 „ „ ...	3-042

C.

It will be observed that the maximum temperature reached was. . 3° '057 Subtracting temperature before explosion 1°'161

we have 1°'896

Adding correction for lost heat during rise 0°'010

•1°-906 Hence ^— = 9727 heat units.

VOL. CCV. — A.

2 F

218

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

The temperature of the water in the outer vessel did not move during the experiment, being 21°'0 C. before and after.

A second determination of the heat, developed under the same conditions, was made in Experiment 1345, which was simply a repetition of Experiment 1344. Here, immediately before explosion, the differential thermometer gave a temperature of 0°722 (equivalent to 19°'4 C.).

	Degrees Cent.	Difference.
Temperature at moment of explosion ....	0-722
„ 2 minutes after explosion . . .	2-050
4. 11 11 » 11	2-575
6 „ „ ...	2-604
,. 11 11 it ...	2-618
		-•001
„ iv -. ,, ,* ...	2-617
		- -003
19 11 *•« 11 11 11 ...	2-614
		-•002
11 11 11 ...	2-612
		-•003
-| r> 11 11 11 11 ...	2-609

Here the maximum temperature reached in 8 minutes was Subtracting temperature before explosion

Correction for loss of heat

C.

2°-618 0°722

l°-896 0°'009

1°-905

Hence units of heat developed = 4680x1'905 = 972 '2 units.

9'17

The previous experiment having given 9727 heat units, the mean may be taken as 972'5. It is unnecessary to say that this degree of accuracy is exceptional, but still, considering the nature of the experiments, the accuracy, even at moderate densities, cannot be considered unsatisfactory.

Thus in Experiment 1392, at a density of 0'25, 7737 grammes of M.D. were fired, the differential thermometer being at 2°'012 (equivalent to 18°'3 C.). Hence

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

219

	Degrees Cent.	Difference.
Temperature at moment of explosion ....	2-012
93	2 minutes after explosion . . .	3-510
)»	4 j» » »» ...	3-707
»	6 „ ...	3-721
»»	8 „ „ „ ...	3-721
			- -002
>i	10 „ „ ...	3-719
			- -003
>)	12 „ „ ...	3-716
			- -002
n	14 „	3-714
			- -003
i)	16 „	3-711

C. . . 3°721

. . 2°'012

1°-709 Correction 0°'011

Hence maximum temperature reached . Less

1°-720

TT ., f , 4680x1-720 ,nono Hence units ot heat = = 1039 '2 units.

7737

The repeat Experiment 1393 gave

TT -4. c v, 4. 4680x1-720 iriOAO •,

Units of heat = = 1030 '2 units.

7'737

Again, in Experiment 1390, at the same density, 0'25, the same number of grammes were fired, the differential thermometer immediately before the explosion being at 0°-581 (equivalent to 18° "6 C.). Hence :—

	Degrees Cent.	Difference.
Temperature at moment of explosion ....	0-581
n	2 minutes after explosion . . .	2-362
11	4 „ ...	2-599
ii	6 „ ...	2-617
ii	8 „ „ „ ...	2-618
			- -004
n	10 „ „ ...	2-614
			- -003
ii	12 „,, „ ...	2-611
			- -004
ii	14 „ „ ...	2-607
			- -004
n	16 „ „ ...	2-603

220 SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

C. Hence maximum temperature reached . . . . 2° '6 18

Less 0°-581

2°'037 Add correction . 0°'012

2°'049

TT ., ,., 4680x2-049 10QQ K

Hence units of heat = = 1239 '5.

7737

The repeat Experiment No. 1391 gave

TT ., „, 4680x2-060

Units of heat = ^^ = 1246'2.

To illustrate the remark I made as to increase of accuracy when taking the calorimetric observations at the higher densities, I give the whole of the observations on this point with Mark I cordite. Thus, at the density of 0'05 the three observations were, respectively, 1265'!, 1303'0, 1248'8, or a mean of 1272'3 units. With density O'l the three observations were 1275"8, 1240'5, 12357, or a mean of 12507 units. Density 0-15 gave 12597, 1247'2, 12427, or a mean of 1249'9. With density 0'20, 1245'2, 1246-5, 1241-0, or a mean of 1244'2. Density 0'25, 1246'2, 1239'5, and 1241'3, giving a mean of 1242'3. 0'3 density gave 1276*9, 1280'0, 1264'0, mean 1273'6 : for 0'4, 1305-0 and 1294'3 or 12997 mean, and for 0'45, 1326'3 and 1320'0 or mean 1323'2.

We are now in a position to give in a tabular form the result of the series of experiments on the three explosives fired under a variety of densities and pressures, and with regard to which the essential constants have been determined.

These tables give : —

(1.) The densities under which the various charges were fired.

(2.) The volumes of permanent gases generated at 0° C. and 760 millims. of barometric pressure per gramme of explosive.

(3.) The total volume of gas per gramme, aqueous vapour being included.

(4.) The percentage volumes of permanent gases.

(5.) The percentage volumes of the total gases.

(6.) The percentage weights of the total gases.

(7.) The pressures at each density in tons per sq. inch.

(8.) The same pressures in atmospheres.

(9.) The units of heat determined, the water being fluid. (10.) The imits of heat, water being gaseous.

(11.) The specific heat of the products of explosion for each density. (12.) The comparative temperatures of explosion determined by dividing the

units of heat (water gaseous) by the specific heats in (11).

(13.) The comparative potential energy, the highest energy determined being taken as unity.

SIE ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

221

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222

SIE ANDEEW NOBLE : EESEAECHES ON EXPLOSIVES.

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SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

223

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224 SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

If the figures given in these tables be carefully examined, it will be observed that in the three explosives the transformation on firing appears in all to follow the same general laws.

Thus in all three there is, with increase of pressure, at first a slight increase, afterwards a steady decrease in the volume of permanent gases produced.

This increase in the total gases is much less marked with cordite, and in the case of M.D. and nitrocellulose there is practically a steady decrease in the volume of the total gases.

In all three explosives there is, with increased pressure, a large increase in the volume of carbonic anhydride and a large decrease in the volume of carbonic monoxide.

In the case of hydrogen, this decrease of volume with increase of pressure is very great, while with methane, the percentage which with low pressures is quite insignificant, very rapidly increases and at the highest density is from twenty to thirty times greater than the lowest density.

There are some variations in the percentages of nitrogen and H2O, but on the whole these constituents may be considered to be nearly constant.

The units of heat with a slight decline at first afterwards increase and somewhat rapidly at the highest pressures.

But the changes which take place under different pressures are more readily appreciated if the observations are graphically recorded by means of curves.

Accordingly in Plate 5 I have given for three explosives the pressures in tons per sq. inch and in atmospheres, deduced from the experiments under consideration, and which pressures vary from about 3 tons per sq. inch (457 atmospheres) to (in the case of Mark I cordite) 53 tons per sq. inch (8078 atmospheres).

It will be observed also that from densities of about 0'25 upwards the curve expressing the relation of pressure to density, both in the Mark I cordite and in the M.D., differs inappreciably from a straight line. This remark also appears to be, in some degree, corroborated by an experiment I once made at a density of unity, and which gave a pressure of about 112* tons per sq. inch (17,070 atmospheres).

With nitrocellulose there appears, at high densities, to be a tendency to detonate, from which tendency Mark I cordite appears to be free. By way of showing the enormous superiority of the new explosives as regards potential energy, I have added to Plate 5 the curve showing the relation of pressure to density of fired gunpowder.

In Plate 6 there are three sets of curves : — (l) The changes in the volumes of the permanent gases due to increase of density ; (2) The changes in the volumes of the total gases which do not differ very greatly from those of the permanent gases ; and (3) The changes in the units of heat at different densities (water fluid).

It may be noted that, while at pressures under 20 tons per sq. inch the heat

* On the occasion referred to, I was not sure that the pressure might not be higher, as there was considerable friction between the piston and the cylinder, due to compression of the gauge.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES. 225

developed does not vary greatly ; at higher pressures the heat increases considerably, thus compensating for the loss of potential energy due to the decrement in the volume of gas generated.

Plates 7, 8 and 9 show graphically the great changes that take place in the decomposition of the gases in passing from densities of 0'05 to 0*45.

In all, carbon monoxide and dioxide change places, the two gases having equal volumes in the case of cordite 24'2 per cent, at a density of 0'19, in the case of M.D. 25 '5 per cent, at a density of 0'32, and with nitrocellulose 26 per cent, at a density of 0-36.

The changes with hydrogen and methane are equally striking, the hydrogen in cordite falling from a maximum of nearly 16 per cent, in volumes to about 9 '5, while the methane increases from about 0'2 per cent, to about 5 '5 per cent. In M.D. the volumes of hydrogen fall from about 19 per cent, to about 10 '4 per cent., while the volume of methane increases from about 0'3 per cent, to nearly 9 per cent., and in nitrocellulose the volume of hydrogen falls from 207 per cent, to about 11 per cent., the methane increasing from 0'5 per cent, to a little over 9 per cent.

In the tables I have submitted it will be observed that the specific heats and the temperatures of explosion have been given, but in regard to temperatures so far above those in regard to which accurate observations have been made the figures I give can only be taken as provisional. The specific heats of the various gases have been taken at the values usually assigned to them. Of course, it cannot be assumed that these specific heats remain unchanged over the wide range of temperature necessary, although I believe it has been found that the specific heats of some permanent gases such as nitrogen and oxygen are but slightly altered up to 800° C.

The temperatures of explosion which, as I have said, can only be taken as provisional, have been obtained by dividing the units of heat (water gaseous) by the specific heats, and, although provisional, can safely be used in comparing the temperatures of explosion of the three explosives. The temperatures of explosion, for example, of cordite and nitrocellulose at the density of 0'20 may tolerably safely be taken to be in the ratio of 51 to 36.

I am, from other considerations, inclined to believe that the temperatures I have obtained and given in the tables are not very far removed from the truth. I tried with cordite to confirm the results by using the equation of dilatability of gases. At the high pressures the results were satisfactory, but quite the reverse at the lower densities.

The comparative approximate potential energies are obtained by multiplying the volume of gas produced by the temperature of explosion. The means for the three explosives are respectively: cordite, 0'9762 ; M.D., 0'8387 ; nitrocellulose, 07464. The highest potential energy (taken as unity), it will be noted, was obtained from cordite at a density of 0'5.

I submit that the wide differences in the transformation of the three explosives

VOL. OCV. — A. 2 G

226 SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

with which I have experimented justify the conclusion at which Sir F. ABEL and I arrived with respect to gunpowder, viz., that any attempt to define by a chemical equation the nature of the metamorphosis which one explosive may be considered to undergo would only be calculated to convey an erroneous impression as to the definite nature of the chemical results and their uniformity under different conditions.

The apparatus shown in Plate 4 was employed for two purposes, (l) to determine the time for the complete ignition of various explosives, or for various forms and thicknesses of the same explosives ; (2) to determine the rate at which the exploded gases part with their heat to the walls of the vessel in which they are enclosed.

The high and low speeds that can be given to the drum permit these two observations to be made by a single experiment. Thus, in Plate 10, I show the commencement and part of the curves of two experiments, the one (fig. 1) fired at a pressure of a little over 12 tons per sq. inch (1829 atmospheres), the other (fig. 2) nearly 18 tons (2744 atmospheres). At the point "A" the charge is fired, and it will be noted that the circumference of the drum is travelling at about 40 inches per second. From fig. 2 it will be seen that at 2 seconds after firing the speed has, in this experiment, been reduced to about an inch per second.

The times required for the completion of ignition are given in Plate 11, and are obtained from the curves shown on Plate 10 and from two similar curves. The vertical scale in Plate 11 for the three last densities is doubled to make them accord with density O'l, the spring employed in that experiment being half the strength of that used for the last three.

I may point out that when fired in close vessels the rate of combustion of the explosives, even in the cord form, appears to be very constant, the increase of pressure apparently nearly compensates for the reduction of surface, the differences in time of burning being due to want of uniformity in the lighting, which in many cases is very variable. This is illustrated by comparing the times of ignition of densities O'l 5 and 0'2 in Plate 11, where the total time from firing to complete ignition is less for density O'l 5 than for density 0'2. In reality, however, after complete lighting the latter is burning quicker, as may be seen by comparing the angles made by the curves with the axis of abscissae.

But this question is too large to enter into fully in the present paper. I therefore only give the times of approximate complete combustion of cordite and M.D. cordite of different diameters when exploded at a pressure of about 9 tons on the square inch.

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

227

Diameter of cordite.	Time of burning, seconds.	Diameter of M.D.	Time of burning, seconds.
0-033	0-0163
0-045	0-0172	—	—
0-091	0-0207	—	—
0-181	0-0274	0-192	0-0377
0-266	0-0337	0--235	0-0480
0-343	0-0395	0-263	0-0547
0-482	0-0499	0-318	0-0679
0-577	0-0570	—	'

Comparing the times of burning of 0'2 cordite and rifle cordite, the times are approximately as follows :—

0-2 cordite.	Rifle cordite.
Tons per sq. inch.	Seconds.	Tons per sq. inch.	Seconds.
3-5 10-0 11-5	0-03844 0-01896 0-01700	3-5 10-0 11-5	0-00972 0-00553 0-00498

The rates of cooling of cordite (charges and densities being as stated) are shown in Plate 12, the interior surface of the explosive vessel being 54'9 sq. inches (354'3 sq. centims.). The communication of heat to the vessel is extraordinarily rapid. The pressure and approximately the temperature of the exploded gases is in the case of

Density O'l (32 grammes) reduced to one half in 0'87 second, and to one quarter in 270 seconds.

Density O'l 5 (48 grammes) reduced to one half in 0'93 second, and to one quarter in 2 '8 2 seconds.

Density 0'20 (64 grammes) reduced to one half in 1'54 seconds, and to one quarter in 3 '8 3 seconds.

Density 0'25 (80 grammes) reduced to one half in 2'40 seconds, and to one quarter in 6 '04 seconds.

1020'6 grammes of the same cordite fired at a density of O'l in a vessel whose interior surface was 3271 sq. centims. reduced its pressure to one half in 3'1 seconds, to one fourth in 10 seconds.

1247'4 grammes fired at a density of 0'12 in the same vessel had the pressure reduced to one half in 4'2 seconds, and to one quarter in 13'8 seconds.

1360-8 grammes fired at a density of O'l 31 in the same vessel recorded a pressure of one half in 6 '3 seconds, and of one quarter in 31 seconds.

2 G 2

228

SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

I venture to allude to two other points of interest. I have always thought it probable that the dissociation, for example, of carbonic dioxide into carbonic monoxide and oxygen might be very greatly modified or extinguished by the extremely high pressure at which my experiments have been made ; and I thought it possible that, if dissociation did take place, some indication of the re-formation of carbonic dioxide would appear in the cooling curves, which have been obtained under a variety of conditions and pressures. These curves, however, are singularly free from any indication of disturbance, so that, if any recombination does take place, it has no effect on the extremely regular coolings to which I have alluded, and would seem to prove that the re-formation of CO2 and H2O must take place gradually and in no case per saltum. I have found also, and this point is of some interest, that gases I have taken from the chamber of a 9 '2-inch gun immediately after firing have, when corrected for the air with which they are mixed, the same composition as those which have been fired under similar densities in a close vessel.

The experiments I have made on erosion with the three explosives referred to in this paper, and on some others, have satisfied me that the amount of absolute erosion is governed practically entirely by the heat developed by the explosion. I had thought that increase of pressure would considerably increase the amount of erosion, but in experiments carried on with cordite and nitrocellulose under pressures varying from 5 tons to 32 tons per sq. inch the erosion was practically entirely independent of the pressure both for the cordite and the nitrocellulose. The results of these experiments are given in Plate 13.

APPENDIX.

Abstract of Experiments Referred to in Paper.

CORDITE MARK I.

Experiment 1380.— Fired in explosion vessel Q, 16 '75 grammes of Mark I cordite. Density of charge 0-05.

Pressure 2-9 tons per sq. inch (442 -1 atmospheres).

Permanent gases 11,186-7 cub. centims. at 0° C. and 760 millims.

Aqueous vapour 3296 • 6 cub. centims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in cordite . . .	grammes 3-72 3-68	grammes 9-47 9-35	grammes 0-47 0-48	grammes 2-91* 2-48
Differences ....	+ 0-04	+ 0-12	- o-oi	+ 0-43

* The N and 0 contained in air in cylinder not taken into account.

SIE ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

229

Experiment 1383.— Fired in explosion vessel Q, 32-73 grammes of Mark I cordite. Density of charge 0-10.

Pressure 7 -8 tons per sq. inch (1189-0 atmospheres). Permanent gases 23,124-7 cub. centims. at 0° C. and 760 millims. Aqueous vapour 6277 -2 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.
Found by analysis .... Originally in cordite . . . Differences ....	grammes 7-66 7-53	grammes 19-26 19-13
+ 0-13	+ 0-13

H.

grammes

0-98 0-98

0-00

N.

gram tnes

5-56

5-08

+ 0-48

Experiment 1386. — Fired in explosion vessel Q, 47 -77 grammes of Mark I cordite. Density of charge 0-15.

Pressure 11-49 tons per sq. inch (1751-5 atmospheres). Permanent gases 33,646-2 cub. centims. at 0° C. and 760 millims. Aqueous vapour 9104-7 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in cordite . . .	grammes 11-14 11-20	grammes 28-26 28-48	grammes 1-44 1'47	grammes 7-98 7-56
Differences ....	- 0-06	- 0-22	- 0-03	+ 0-42

Experiment 1371. — Fired in explosion vessel Q, 63 -96 grammes of Mark I cordite. Density of charge 0 • 20.

Pressure 17 -2 tons per sq. inch (2621-9 atmospheres). Permanent gases 46,440-3 cub. centims. at 0J C. and 760 millims. Aqueous vapour 11,594-1 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in cordite . . .	grammes 15-45 15-02	grammes 38-59 38-18	grammes 1-98 1-98	grammes 10-86 10-14
Differences ....	+ 0-43	+ 0-41	o-oo .	+ 0-72

230

SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

Experiment 1389.— Fired in explosion vessel Q, 80-3 grammes of Mark I cordite. Density of charge 0 • 25.

Pressure 21-08 tons per sq. inch (3213-3 atmospheres). Permanent gases 55,834 '4 cub. centims. at 0° C. and 760 millims. Aqueous vapour 14,480 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in cordite . .	grammes 18-69 18-76	grammes 47-33 47-70	grammes 2-40 2-46	grammes 13-17 12-67
Differences ....	- 0-07	- 0-37	- 0-06	+ 0-50

Experiment 1375. — Fired in explosion vessel Q, 95-94 grammes of Mark I cordite. Density of charge 0 • 30.

Pressure 30'5 tons per sq. inch (4649 -3 atmospheres). Permanent gases 64,453- 7 cub. centims. at 0° C. and 760 millims. Aqueous vapour 16,653-4 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in cordite	grammes 22-11 22-31	grammes 56-35 56-71	grammes 2-88 2-93	grammes 15-12 15-05
Differences ....	- 0-20	- 0-36	- 0-05	+ 0-07

Experiment 1497. — Fired in explosion vessel Q, 124-67 grammes of Mark I cordite. Density of charge 0'40.

Pressure 41 -4 tons per sq. inch (6310-8 atmospheres). Permanent gases 80,403-1 cub. centims. at 0° C. and 760 millims. Aqueous vapour 21,832 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	O.	H.	N.
Found by analysis .... Originally in cordite . . . Differences ....	grammes 28-54 28-67	grammes 72-79 72-89	grammes 3-74 3-76	grammes 19-36 19-35
- 0-13	- 0-10	- 0-02	+ o-oi

SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

231

Experiment 1496. — Fired in explosion vessel Q, 155-84 grammes of Mark I cordite. Density of charge 0-5.

Pressure 52-84 tons per sq. inch (8063 '8 atmospheres). Permanent gases 97, 158- 9 cub. centims. at 0° C. and 760 millims. Aqueous vapour 26,291 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.
Found by analysis .... Originally in cordite . . . Differences ....	grammes 35-58 35-65	1
- 0-07	-

0.	H.	N.
;rammes 90-52 90-69	grammes 4-63 4-68	grammes 24-19 24-05
0-17	- 0-05	+ 0-14

Experiment 1387. — Fired in explosion vessel Q, 16 grammes M.D. Density of charge 0-05.

Pressure 2 -7 tons per sq. inch (411-6 atmospheres).

Permanent gases 12,899-8 cub. ceYitims. at 0° C. and 760 millims.

Aqueous vapour 2861 -2 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	1
Found by analysis .... Originally in M.D	grammes 4-24 4-66	grammes 9-34 9-53	grammes 0-53 0-53	gnu 2 1
Differences ....	- 0-42	- 0-19	o-oo	+ 0

N.

2-55* •97

* Chiefly due to air in explosion vessel.

Experiment 1388. — Fired in explosion vessel Q, 31-98 grammes M.D. Density of charge 0-10. Pressure 6 -9 tons per sq. inch (1051-8 atmospheres). Permanent gases 25,621-0 cub. centims. at 0° C. and 760 millims. Aqueous vapour 5145-1 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in M.D	grammes 8-51 9-12	grammes 18-53 18-66	grammes 1-04 1-03	grammes 4-85* 4-49
Differences ....	- 0-61	- 0-13	+ 0-01	+ 0-36

Including N in air.

232

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

Experiment 1357. — Fired in explosion vessel Q, 47 '97 grammes M.D. Density of charge 0-15. Pressure 10-2 tons per sq. inch (1554-8 atmospheres). Permanent gases 38,458-3 cub. centims. at 0° C. and 760 millims. Aqueous vapour 7600-8 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C. 0.	H.	N.
Found by analysis .... Originally in M.D	grammes grammes 13-58 28-00 13-53 28-13	grammes 1-59 1-66	grammes 7-24 6-69
Differences ....	+ 0-05 - 0-13	- 0-07	+ 0-55

Experiment 1356. — Fired in explosion vessel Q, 63 -96 grammes M.D. Density of charge 0-20. Pressure 15-2 tons per sq. inch (2317'0 atmospheres). Permanent gases 50,229 • 8 cub. centims. at 0° C. and 760 millims. Aqueous vapour 9,558-7 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

Found by analysis Originally in M.D.

Differences

C.

grammes

17-17 17-66

- 0-49

0.	H.	N.
grammes 37-00 36-74	grammes 2-14 2-17	grammes 9-32 8-74
+ 0-26	-0-03	+ 0-58

Experiment 1370.— Fired in explosion vessel Q, 79 -95 grammes of M.D. Density of charge 0-25. Pressure 20'74 tons per sq. inch (3155-4 atmospheres). Permanent gases 60,611 -2 cub. centims. at Oc C. and 760 millims. Aqueous vapour 11,631-4 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis . . -. . Originally in M.D. ,	grammes 21-21 21-99	grammes 46-17' 45-74	• grammes • 2-62 2-71	grammes 11-52 10-8fi
Differences ....	- 0-78	+ 0-43	-0-09	+ 0-66

SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

233

1354. — Fired in explosion vessel Q, 98-94 grammes of M.D. cordite. Density of charge 0'3. Pressure 27 '62 tons per sq. inch (4210 • 3 atmospheres). Permanent gases 72,768-0 cub. centims. at 0° C. and 760 millims. Aqueous vapour 13,885-5 cub. centims. at 0° C. and 760 millims.

RECONCILIATION'.

	C.	O.	H.	N.
Found by analysis .... Originally in M D	grammes 26-22 27-05	grammes 56-81 56-26	grammes 3-25 3-33	grammes 13-88 1 V 36
Differences ....	- 0-83	+ 0-55	-0-08	+ 0-52

Experiment 1405. — Fired in explosion vessel R, 128-48 grammes of M.D. cordite. Density of charge 0-40.

Pressure 38'1 tons per sq. inch (5807-8 atmospheres). Permanent gases 89,410-2 cub. centims. at 0° C. and 760 millims. Aqueous vapour 17, 887 '2 cub. centims. at 0' C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in M.D	grammes 33 • 63 35 • 87	grammes 73-06 73-37	grammes 4-25 4-06	grammes 17-50 15-18
Differences ....	- 2-24	- 0-31	+ 0-19	+ 2-32

Experiment 1417. — Fired in explosion vessel Q, 143-91 grammes of M.D. cordite. Density of charge 0 • 45.

Pressure 43 • 22 tons per sq. inch (6587 • 3 atmospheres). Permanent gases 97,589-5 cub. centims. at 0° C. and 760 millims. Aqueous vapour 19,394-0 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

Differences

Found by analysis .... Originally in M.D. . .

C.

grammes

37-88 39-06

- 1-18

0.	H.	N.
grammes 82-42 81-24	grammes 4-64 4-80	grammes 19-59 19-29
+ 1-18	- 0-16	+ 0-30

VOL. CCY. A

2 H

234 SIR ANDEEW NOBLE: RESEARCHES ON EXPLOSIVES.

Experiment 1339. — Fired in explosion vessel L2, 41-5 grammes of Rottweil R. R. Density of

chcirge 0-05.

Pressure 3-35 tons per sq. inch (510-7 atmospheres). Permanent gases 33,811-8 cub. centims. at 0° C. and 760 millims. Aqueous vapour 7402-8 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in R. R	grammes 11-26 11-59	grammes 24-49 23-71	grammes 1-44 1-32	grammes 5-79 4-91
Differences ....	- 0-33	+ 0-7S	+ 0-12	+ 0-88

Experiment 1340. — Fired in explosion vessel L>, 83 grammes of Rottweil R. R. Density of charge 0-10.

Pressure 6 -26 tons per sq. inch (954-2 atmospheres).

Permanent gases 66,802-6 cub. centims. at 0' C. and 760 millims.

Aqueous vapour 13,646-7 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.
Found by analysis .... Originally in R R . .	grammes 22 • 38 23-17
Differences ....	- 0-79

0.	H.	N.
grammes 48-60 47-40	grammes 2-81 2-62	gramn 10-8 9-8
+ 1-20	+ 0-19	+ i-c

* Partly due to air in explosion vessel.

Experiment 1341. — Fired in explosion vessel Q, 47 • 97 grammes of Rottweil R. R. Density of charge 0 • 15. Pressure 10'4 tons per sq. inch (1585-3 atmospheres). Permanent gases 38,585-8 cub. centims. at 0° C. and 760 millims. Aqueous vapour 7949 -2 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Originally in R. R Differences ....	grammes 13-15 13-39	grammes 28-45 27-40	grammes 1-66 1-52	grammes 6-17 5-38
- 0-24	+ 1-05	+ 0-14	+ 0-79

SIR ANDREW NOBLE: RESEARCHES ON EXPLOSIVES.

235-

Experiment 1342.— Fired in explosion vessel L.>, 166 grammes of Rottweil R. R, Density of charge 0 • 20. Pressure 14-41 tons per sq. inch (2196-6 atmospheres). Permanent gases 127,643-1 cub. centims. at 0° C. and 760 millims. Aqueous vapour 26,721 -6 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	II.
Found by analysis .... Originally in R. R. .	grammes 44-7 46-6	grammes 96-5 95-6	grammes 5-7 5-4	gr< •
Differences ....	1-9	* + 0-9 ..	+ 0-3	+

X.

grammes

20-9 19-7

1-2

Experiment 1338. — Fired in explosion vessel Q, 70-99 grammes of Rottweil R. R. Density of charge 0-222. Pressure 16-47 tons per sq. inch (2510-6 atmospheres). Permanent gases 53,898-2 cub. centims. at 0° C. and 760 millims. Aqueous vapour 11,576-7 cub. centims. at 0' C. and 760 millims.

RECONCILIATION.

	C.	0.	II.	X.
Found by analysis .... Originally in R. R	grammes 19-22 19-82	grammes 41-76 40 • 55	grammes 2-46 o . 05	grammes 9-06 x • .in
Differences ....	- 0-GO	+ 1-21	+ 0-21	+ 0-66

Experiment 1337. — Fired in explosion vessel Q, 92-74 grammes of Rottweil R. R. Density of charge 0-29. Pressure 21-5 tons per sq. inch (327 7 -4 atmospheres). Permanent gases 68,427 -3 cub. centims. at 0° C. and 760 millims. Aqueous vapour 13,972'6 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	o.	H.	X.
Found by analysis .... Originally in R. R Differences ....	grammes 25-55 26-24	grammes 54-12 53-68	grammes 3-19 2-97	grammes 11-54 11-10
- 0-69	+ 0-44	+ 0-22	+ 0-44

2 H 2

236

SIR ANDREW NOBLE: EESEARCHES ON EXPLOSIVES.

Experiment 1346.— Fired in explosion vessel Q, 95'94 grammes of Rottweil R. R. Density of charge 0'30.

Pressure 20-54 tons per sq. inch (3131'0 atmospheres). Permanent gases 70.802'3 cub. centims. at 0° C. and 760 millims. Aqueous vapour 13,834-3 cub. centims. at 0° C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
Found by analysis .... Ori^inallv in R R .	grammes 26-43 27-16	grammes 56-14 56-16	grammes 3-32 3-07	grammes 12-00 11-50
Differences ....	- 0-73	- 0-02	+ 0-25	+ 0-50

Esperime-nt 1401.— Fired in explosion vessel Q, 127-92 grammes of Rottweil R. R. Density of charge 0-40.

Pressure 34'9 tons per sq. inch (5320-0 atmospheres).

Permanent gases 88,689'0 cub. centims. at 0" C. and 760 millims.

Aqueous vapour 18,946'! cub. centims. at 0" C. and 760 millims.

RECONCILIATION.

	C.	0.	H.	N.
	grammes	grammes	grammes	grammes
Found by analysis .... Originally in R. R	34-57 35-85	74-62 73-33	4-38 4-06	14-83 15-16
Differences ....	1-2S	+ 1-29	4-0-32	- 0-33

Experiment 1402. — Fired in explosion vessel R, 144-54 grammes of Rottweil R. R. Density of charge 0'45.

Pressure 40-5 tons per sq. inch (6173-6 atmospheres). Permanent gases 98,819'4 cub. eentims. at 0° C. and 760 millims. Aqueous vapour 19,568-1 cub. centims. at 0' C. and 760 millims.

RECOXCILATIOX.

	C.	0.	H.	N.
Found by analysis .... Originally in R. R	grammes 39-29 40-27	grammes 83-47 82-38	grammes 4-86 4-56	grammes 16-61 17-03
Differences ....	- 0-98	4- 1-09	+ 0-30	- 0-42

[ 237 ]

VII. Colours in Metal Glasses, in Metallic Films, and in Metallic

Solutions. — //.

By J. C. MAXWELL GARNETT. Communicated by Professor L ARMOR, Sec.R.S.

Received May 15, — Read June 8, 1905. CONTENTS.

Pages

1. Introduction 237-239

2. Expressions for the optical constants of media containing metal in amorphous or

granular forms 239-242

3. Formulae applicable only when the volume proportion of metal is small 242-243-

4. Calculated numerical value of the optical constants of metal glasses, &c 243-248

5. Diffusions of gold. The nature and form of the suspended particles 248-255

6. Diffusions of silver. The nature and form of the suspended particles 255-259

7. Blue reflection from the stained face of silver glass 259-263

8. Diffusions of copper. The nature and form of the suspended particles 263-265

9. Colouring effects of the radiation from radium, cathode rays, £c 265-2G7

10. Numerical values of the optical constants of media containing large volume proportions

of certain metals 267-276

11. Colour changes caused by heating metal films 277-282

12. The exceptional case of beaten metal leaf 282-283

13. CAREY LEA'S "allotropic" silver 283-285

14. HERMANN VOGEL'S silver 285-286

15. Allotropic forms of metal 286-288

1. Introduction.

THIS paper is an extension of a previous memoir on the " Colours in Metal Glasses and in Metallic Films "* ; it is concerned with the application of mathematical analysis, akin to that already there developed, to the explanation and coordination of the colours which certain metals are, under a great variety of circumstances, capable of causing.

* 'Phil. Trans.,' A, 1904, vol. 203, pp. 385-420. VOL. CCV. — A 393. 11.10.05

238 ME. J. C. MAXWELL GARNETT

From observations on gold and copper ruby glasses, it has been shown* that the first stage in the formation of a crystal of those metals is the small sphere ; and from observations on the growth of sulphur crystals in CS2, VoGELSANGt arrived at the conclusion that the small sphere is always the first stage in the formation of a crystal. He remarked, however, that it is by no means necessary that each of the small spheres, formed as crystallisation commences, should give rise to a separate crystal : the small spheres tend to coagulate; forming first rows and then groups of other and more complicated shapes, until the crystal is ultimately formed. To the intermediate bodies he gives the name of crystallites.

That the spherical form of the nascent crystal is governed by surface tension, was suggested in the former paper.;}; If this suggestion is correct, we should expect that when the conditions are not the same in all directions, the spherical form of the nascent crystal will be replaced by an ellipsoidal form. In particular, when a very thin film of amorphous metal is heated until the molecules are sufficiently free to allow crystallisation to commence, the nascent crystals may be expected to be spheroids of the planetary type, having their axes normal to the film. Mr. G. T. BKILBY§ has observed such spheroids in thin films of gold and silver.

Now it will appear below that metals are not only dichroic, exhibiting one colour by reflected light and, in thin films, another by transmitted light ; but that one and the same metal may, as its physical condition is altered, show a great variety of colours by reflected light, and a corresponding other series of tints by transmitted light. The ultimate cause of all these colours is to be found in the structure of the molecule itself. Juxtaposition, however, causes one molecule to afl'ect the vibrations of another. Thus consider a substance composed of molecules of a given metal separated from each other by the oether or by any other non-absorbing medium :|| the " effective free period " of the molecule of such a substance is dependent on the geometrical arrange- ment and density of distribution of the molecules in question. The optical properties of the substance will therefore depend on its microstructure. The object of this paper is to obtain information concerning the ultramicroscopic structure of various metal glasses, colloidal solutions, and metallic films, by calculating optical properties corresponding to certain assumed microstructures, and by comparing the calculated properties with those observed.

* Lot:, cit. (pp. 388-392). When writing the former paper here cited I was unaware of VOGELSANG'S work.

t H. VOGELSANG, "Sur les Cristallites," 'Archives Neerlandaises,' V. (1870), p, 156; VI. (1871), p. 223; VII. (1872), pp. 38-385.

J Loc. cit., p. 392. The further suggestion there made that iu the colourless gold glass there are the molecules of gold present is, as will appear below, p. 251, erroneous. It is almost certain that in the colourless glass a gold salt is in solution, so that the heating has first to reduce the gold and then to allow the isolated molecules to run together into spheres.

§ 'Hurter Memorial Lecture,' Glasgow, 1903, p. 46.

il As, for example, glass in a metal glass, or water in a colloidal solution.

ON COLOURS IN METAL GLASSES, ETC. 239

The microstructures to be assumed are suggested by the preceding remarks cm crystallisation. Calculations will be made for three types of microstructure, namely,

(1) amorphous — that in which the metal molecules are distributed at random ;

(2) granular — that in which the metal molecules are arranged in spherical groups ;

(3) spicular* — that in which these small spheres are replaced by oblate spheroids. It will subsequently appear that when the surrounding non-absorbing medium is of refractive index unity, an amorphous and a granular microstructure produce the same colours.

In order to calculate the optical constants — the refractive index and the coefficient of absorption — which correspond to any given microstructure, it is necessary to know the values of the constants for some standard amorphous state of the metal. Now BEILBY t has shown that the process of polishing a metal surface causes the surface layer to " flow " as a liquid, and thus the polished surface is that of the metal in the amorphous state. It follows that the optical constants which we are to use as data for our calculations should, so far as possible, be those which have been determined by means of reflection from the polished surface of the metal in its normal state according to DRUDE'SJ method, rather than those obtained by means of the light transmitted through thin prisms of the metal, after the method adopted by KUNDT.

2. Expressions for Optical Constants of Media containing Metal in Amorphous or

Granular Forms.

The optical properties of a homogeneous isotropic medium are determined when the values of the refractive index n and the absorption coefficient x, which correspond to light of every frequency, are known. We proceed to obtain the values of n and HK for a substance composed of molecules of metal embedded in an isotropic non-absorbing medium, the microstructure being amorphous.

Consider then a medium consisting of one substance A, in solution in another C, so that the molecules of each substance are distributed at random. Let the number of molecules present per unit volume in the standard amorphous forms of A and C be respectively 9tfA and 9RC, and let the number of molecules present per unit volume of the composite medium be MA^A and /AC^C respectively. We shall assume /AA and p.c to be constant throughout the medium ; or, more precisely, we assume that a length •?•„, very small compared with a wave-length of light, can be found such that, for all values of r greater than r0, the number of molecules of A contained by a spherical surface situated wholly within the medium having a radius r and its centre being

* The calculations for a spicular microstructure are reserved for subsequent publication, see note p. 241. t Loc. cit., Lord KAYLEIGH (Royal Institution Lecture on Polish, March, 1901) also holds the view that the process of polishing is a molecular one. J 'Ann. der Phys.,' XXXIX. (1889).

240 MR. J. C. MAXWELL GAENETT

situated at any point, is independent of the position of that point; thus f w/i^S^r*' depends only on r : and similarly for C.

Suppose that when electromagnetic waves traverse this medium, the moments of" the average molecule of A and C in the vicinity of the point (x, y, z) are

fA (*) = (/*„ A, A) and fc(t) = (fCl,fc,fc).

Then fA and fc are both proportional to E', the electric force exciting the average

molecule* ; thus

f* = 0,vE', fc = #cE'.

The polarisation f ' (t) of the compound medium is given by

Writing now E for E0 in the general equation t

E' = E0+tnf',

we obtain

E' = so that

ft (f\ _

' But MAXWELL'S equations for the composite medium are

. df'ft) clE , „ cZH , ,,

4?r — ji-i + — r = c curl H, - = — c curl E, dt dt dt

where c is the velocity of light in racuo. These may be written

N'2 — j- = c curl H, ^-=- = — c curl E, dt dt

when we put

If now we write

N7 = «'fl iu-'\ (9\

r\ — /t \L iK ) \^j-

then n' and /c' are the refractive index and absorption coefficient of the composite medium.

But the same analysis will show that if NA = nA(l — IKA) and Nc = nc(l-iKc);, where WA> KA and nc, KC are the optical constants of A and of C, then

h and

* Of. 'Phil. Trans.,' A, vol. 203, pp. 392, 393. t Loe. cit., equation (9), p. 393.

ON COLOURS IN METAL GLASSES, ETC. 241

Substituting these expressions in equation (I) we obtain

N/2-l NA2-1 Nca-l N'2+2 ~ MANA2+2 +McNc2+2 '

If, now, we suppose that C is a transparent isotropic substance of refractive index v, and that A is a metal, we have, by omitting the suffix A and putting /u,c = 1— ft,

N/2-l N2-l , .v2-! N'2+2 /AN2+2 +( MV+2' or,

22 ,

N'3+2 When fj, is very small this equation becomes

......... (4).

These equations give the optical constants of the metalliferous medium in the amorphous state. When the microstructure is granular, these equations (3) and (4) are, as has been already shown,* replaced by

N'2-V = _N2-v2 /r,

N'2+2v2 I so that when p. is small,

Comparison of equations (3) and (5) shows that the optical properties of a metalliferous medium, containing a given volume proportion ju, of the metal, vary according as the metal is in small spheres or in a state of molecular subdivision, except when //,= !. Thus when metal is in solution in water or glass the colour of the compound medium will change as crystallisation commences. When, however, v = 1, the equations (3) and (5) both reduce to

N'2-l N2-l m

N/2+2 ^N2+2 '

It follows that the optical properties of a metal in a state in which its specific

* Loc. tit., equations (11) and (12), p. 394. The mathematical treatment of the optical properties of media containing minute metallic ellipsoids, instead of the spheres which give the granular microstructure, is under consideration, but, with the exception of the case wherein the volume proportion, p., of metal is small, it is not yet complete.

[Note added 1st August, 1905. — The investigation of the general case (any value of /*) has now been completed. The results for the case when p. is small, which, when this memoir was communicated to the Royal Society, were given in § 12, have therefore been reserved for subsequent publication in a more complete form.]

VOL. COV. — A. 2 I

242 ME. J. C. MAXWELL GARNETT

gravity has any known value, are unaltered by a change in the raicrostructure from amorphous to granular.* Or, again, Professor R. W. WOOD'S clouds of sodium vapour, t for which v = I nearly, do not change colour as condensation commences.

3. Formula? Applicable only when the Volume Proportion of Metal is Small.

The volume proportion of metal present in all the coloured glasses and colloidal solutions which we shall discuss below is small. We proceed to obtain, from equations (4) and (6) above, expressions for the optical constants of media, such as glass or water, holding in suspension metal in the amorphous and granular states.

Let N" EE n" (I — LK") denote the optical constants of the compound medium when the metal is in the amorphous state — in true solution. Then, replacing N' by N" in equation (4), we have

2 say.

Equating real and imaginary parts, we find that

, ^(l-*"2)-^ K(K2+l)}2+ft2(*2-l)(v2-2)-2v2

2--*

(2 + j/V " K(/c2-l)-2}2+4nV J

From (8) we have

n"2 ( 1 - K"2) = S + (2 + v2) pa', n"*K" = (2 + v2) /*£', so that, neglecting ju,2, we obtain

wV = (2 + !/»)/?. ^ n" = v{l + (2 + v2)/2v2.ljM'} .... (9).

The corresponding values of W = n'(l — iKf), the optical constants when the metal is in the granular state, have already been obtained.! They are reproduced here for convenience of reference : thus

n'K' = 3fjLV/3, w' = v(l+|./4«) ....... (10),

where

a =

, ,

* For example, the tables given in the previous paper ('Phil. Trans.,' A, 1904, p. 406), and the curves shown (loc. cit., pp. 411-414), as well as the tables and curves given in § 10 of the present communication, represent optical properties of the media as /* diminishes from unity to zero, whether that diminution is associated with the formation of small spheres or whether the metal retains its amorphous state throughout the change in /«.

t Brit. Assoc., Cambridge, 1904.

t Loc. cit., § 5, pp. 394, 395.

ON COLOURS IN METAL GLASSES, ETC. 243

Thus, when light of wave-length X traverses a thickness d of a metalliferous

n"n" n'n'

medium, the intensity of the light is reduced in the proportion* e~M A or e~M * according as the metal is in true solution or in spherical aggregates.

Suppose now that two kinds of monochromatic light, of wave-lengths Xj and X2, are, by traversing a distance d in an absorbing medium, reduced in intensity by e~K<d and e-:M respectively. Then the absorbing medium reduces the proportion of the inten- sities of the two kinds of light in the ratio e~(Kl~v>d, which is a function of d. Thus the tint of a coloured medium, viewed by transmitted light, depends on its thickness.! We shall, however, speak of two absorbing media as possessing the same colour when, whatever be the values of Xx and X2, the ratio KI : K2 is the same for either medium ; for, if suitable thicknesses of such media be chosen, the light transmitted by them will be of precisely the same tint.

Since, therefore, it appears from equations (8) and (9) above that the ratio

is independent of v, it follows that a niolecularly subdivided metal produces the same coloration (by transmitted light) in all non-dispersive transparent isotropic " solvents," irrespective of their refractive indices. \ Thus, neglecting the small dispersion, a borax bead and a glass bead, each containing a metal in solution, will be of the same colour ; but so soon as crystallisation of the metal begins, so that part of the metal is in small spheres, the beads will cease to be of the same colour, since the ratio

is not independent of v.

4. Numerical Values of Optical Constants of Metal Glasses, &c.

Consider any transparent isotropic non-dispersive medium of refractive index v, containing either molecules or small spheres of a metal, the optical constants of which, for light of wave-length X, are n and UK, the particles of metal being so distributed that there are many of them to a wave-length of light. The "absorptions," nV/X and w'V/X, of the compound medium can be easily determined by means of equations (9) and (10), when the values of a, ft, a!, /3', are known for light of wave-length X. These values can be calculated by means of equations (11)

* Of. 'Phil. Trans.,' A, 1904, p. 395.

t Thus, for example, a thin sheet of gold ruby glass will appear pink, a considerable amount of blue light being transmitted, whereas a thick sheet of the same glass will appear deep red, almost like a copper ruby. Again, by increasing the depth of a silver stain on glass, we get all gradations in colour from canary yellow through amber to red.

| This, then, must be the colour of the vapour of the metal provided that the molecules are monatomic, or, at least, do not dissociate when the metal is vapourised. We shall term it the " vapour colour."

2 I 2

244 MR. J. C. MAXWELL GARNETT

and (8), when the quantities n and UK have been determined for the light in question by direct experiment on the metal in the standard amorphous state. R. S. MINOR* has made such experimental determinations, for various kinds of monochromatic light, from the polished surfaces of silver and copper. His values of n and n/c for silver and copper, together with the numerical values of a, ft, a!, ft', and of certain other functions as calculated for various values of v, are shown in Tables II. and III. The values of n and n/c, for X = '630 and X = '589 only, have also been determined by DRUDEf from the polished surface of gold ; but values of n and HK for other values of X have been obtained by HAGEN and RUBENS J from gold prisms deposited on glass. Since however the state of the metal in the prisms is not known, these latter values cannot be depended upon for our purpose ;§ but as a rough estimate of the values of a, ft, a', ft', &c., may be formed by their means, the numerical values of these quantities have been calculated ; the results are shown, together with all the observed values of n and UK, in Table I. The wave-length X of the light will throughout be measured in thousandths of a millimetre.

All the calculated numbers given in Tables I., II., and III. have been carefully checked with a " Brunsviga " machine. I believe that in no case does an error amounting to 1 per cent, survive in these Tables, which must accordingly supersede those -given in the former communication. ||

* R. S. MINOR, 'Ann. der Phys.,' vol. X., 1903.

t 'P. DRUDE, 'Physik. Zeitschr.,' January, 1900.

\ RUBENS, ' Wied. Ann.,' vol. XXXVIL, 1889.

§ It will appear in the sequel that on this account the optical properties of silver glasses and of colloidal solutions of silver are much more accurately represented by our calculations than is the case with gold ruby glasses and colloidal solutions of gold.

|| Lot. tit., p. 396.

ON COLOURS TN METAL GLASSES, ETC.

245

TABLE I.— Gold.

X		. {	•6562	•6300	•5892	•5269	•4584 /F + QX
		I	(C).		(D).	(E).	( 2 /
UK			2-91*	3-15t	2-82t	1-86*	1-52*
• n			•38*	•31f	•37f	•53*	•79*
		a.	2-50	2-27	2-65	•83	•46
		ft	•479	•251	•584	1-068	•552
		a? + ±p				C ft . 1 Q	00 . J 0
Glass \ v = 1-56 / '		A4	38 Go	o4 44	oy o<
		ri'ic" (v* + 2)P
		H*A. vX	47	oU	Oo
		M'K' 3^ TX x~	3-42	1-86	4-64	9-49	5-63
Glass \ v = 1-5J- '		«V 3vj8 7U ~ "T	2-33	1-30	3-12	8-97	5-17
Water at 19° C.\ v= 1-3333 J		UK 3vft ~jl\ T	1-293	•766	1-687	9-906	6-034
Vacuum v = 1 • 0 "1 (Vapour colour) J		TI'K' 3/2 TA ~: T	•338	•214	•417	3-191	4-019

RUBENS.

t DRUDE.

246

MR. J. C. MAXWELL QARNETT

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ON COLOURS IN METAL GLASSES, ETC.

247

TABLE III. — Copper.

X	. . .	•3467	•395	•450	• 500	•535	•550	•575	•5892	•630
UK	. . .	1-466	1-763	2-149	2-341	2-276	2-233	2-428	2-630	3-012
n		1-190	1-173	1-131	1-098	1-004	•892	•651	•617	•562
	ft	•435	•560	•683	•701	•781	•891	1-114	•890	•465
Glass ~1	H"K"	4. £ AO	3. OOQ	O • M*7	2. AKA	*) . 1 AO	o . f)f\Q	1 • f>71	1 • 1 1 f\	. KQQ
i/ = 1-56 / ' '	/.A
	mV TX	5-875	6-661	7-107	6-559	6-829	7-579	9-279	7-065	3-453
Glass \ v = 1-5J	wV TX	5-797	6-420	6 • 568	5-897	6-180	6-890	7 • 634	5-699	2-759
Vacuum v = 1 "1 (Vapour colour) J	T>'K' 7IT	1-104	•921	•642	•492	• 504	• 530	•404	•268	•142

We have now to compare the observed optical properties of various media coloured by gold, silver, and copper with those corresponding properties which, according to the Tables I., II., and III., would be exhibited by the media if the colouring agent were the metal itself, in either the molecularly subdivided or the granular state.

The simplest of the optical properties to observe and to measure is the absorption of light by the medium. Although the absorptions of colloidal solutions of various metals, and even of suspensions of metals in gelatine, have already been measured for several values of the wave-length, X, of light, no such measurements of the absorption of glasses coloured by metals appear to have yet been made.* Owing, however, to their permanence, such media seemed likely to yield the surest information as to the chemical and physical nature of their colouring agents. The absorptions of a series of glasses coloured with gold, silver, and copper have therefore been measured for me at the National Physical Laboratory, under the supervision of Mr. F. J. SELBY. The silver glasses consisted of a silver stain on one side of a colourless glass, the refractive index of which was, for sodium light, equal to v = 1'579. The gold ruby glasses were flashed on to colourless glass. Both the silver and gold glasses were, to ensure purity of materials, specially prepared at the Whitefriars Glass Works by

* Except two gold glasses, the absorption curves for which are given by ZSIGMONDY ('LiEB. Ann.,' vol. 301, pp. 46-48).

248

MR. J. C. MAXWELL GARNETT

Mr. H. J. POWELL, to whom I am much indebted for the trouble he has taken on my account. The copper ruby glass used was the ordinary commercial flashed glass. The absorptions of the various glasses are indicated by the curves marked Au (A) and Au (B) in fig. 2, Ag (B) in fig. 4, and Cu (X) in fig. 7 ; the ordinates representing the quantity K, where e~K represents the proportion of light of wave-length X, transmitted by the glass after allowance has been made for reflections, and the abscissae representing the corresponding values of X. The scales on which Au (A) and Au (B) are represented are such that K has the same value for both at the D line

(X = -589).

The following are the values of K measured at the National Physical Laboratory

for the respective glasses :—

TABLE IV.

Glass	Au(A).	Au (B).	Ag(B).	Cu(X).
X.	K.	K.	X.	K.	X.	K.
•698		•080	•696	0	•698	•300
•696	•113		•664	0	•664	•378
•664	•154	•140	•606	•063	•634	•563
•606	•324	•406	•562	•112	•606	1-038
•562	•878	1-532	•528	•210	•584	2-496
•553	1-400	2-175	•500	•384	•573	3-309
•544	1-990 3-0065	•478	•809	•567	3-484
•537	2-487	3-7485	•458	1-363	•562	3-484
•528	2-551	4-050	•442	2-514	•544	3-091
•514	2-122	3-224	•436	2-894	•528	2-952
•500	1-495	2-344	•429	3-003	•514	2-985
•478	1-030	1-628	•422	2-225	•500	3-023
•458	•954	1-456	•406	1-520	•478	3-270
•442	•981			—	•458	3-484
					•441	3-789
—	—		—	— •	•436	3-807

5. Diffusions of Gold. TJie Nature and Form of the Suspended Particles.

The present section will be concerned with the colours produced by diffused particles of gold. The values of the expression nV//A for v = 1'5, v = 1'3333, and v = 1 given in Table I. are plotted in fig. 1. The curves shown in that figure have been drawn to pass through the plotted points, the coordinates of the maximum for each curve being determined by assuming n and n/c to vary continuously for values of X intermediate between the abscissae of the plotted points on either side of those maxima. According to the remarks in the preceding section, these curves must not be regarded as accurately representing the absorption of gold spheres in glass, of gold spheres in water, and of gold vapour (or gold spheres in vacuo) respectively ; they should, however, enable us to form a fair estimate of the absorption in question.*

* See footnote §, p. 244.

ON COLOURS IN METAL GLASSES, ETC.

249

Fig. 1. GOLD— calculated values of -^- .

A

Fig. 2. GOLD. (1) K, observed fov gold ruby glass Au(A) :

(1) Spheres (or molecules) in vae.uo, v= 1 • 0 : (3) K, observed for gold ruby glass Au(B): -

(2) Spheres in water at 19° C., v= 1-3333: - Observed values : 0 .

(3) Spheres in glass, v = 1 • 5 :

Calculated values are shown thus : 0.

(3) _, calculated for gold spheres in

A

glass, v — 1 • 5 :

Calculated values : x .

VOL. CCV. — A.

2 K

250 MR. J. C. MAXWELL GARNETT

In fig. 2 the graph of nY/X for glass (v = I'd), shown by a broken line in fig. 1, is again represented ; but here on such a scale as to have the same ordinate at X = '589 as that of the graphs of the observed absorptions of Au (A) and Au (B). The calculated and observed curves resemble one another in having a minimum in the red and a maximum in the green, although the calculated maximum occurs at about X = '550, while the observed maximum falls at X "= '533. Also both calculated and observed absorptions fall from green to blue, while the dotted curve in fig. 1 shows that the absorptions produced by molecularly divided gold will increase from green to blue, having a maximum at about X = '475. These results then, so far as they go, are in accordance with the suggestion, put forward in the former memoir, that the colouring agent of gold ruby glass consists primarily of diffused spheres of gold, although some discrete molecules may also be present. The following is the evidence which has accumulated to show that a gold ruby glass contains minute spheres ot gold, many to a wave-length of light, and that it is to these small gold spheres that the pink colour of the glass is primarily due : — -

(1) There are particles, presumably of gold, visible in all specimens of gold ruby glass in which the

colour has been developed.*

(2) Whenever these particles are of diameter less than 10~5 centim. they are spherical in shape.t

(3) SIEDENTOPF and ZSIGMONDY statej : " It is only in the case of ruby glasses that the particles are

so dense that they cannot be fully separated under the microscope." In other words, whenever there are many small spheres to a wave-length of light, the glass is ruby.

(4) We have just seen that, within the limits of experimental error, this ruby colour is that which

would be produced by small spheres (but not by molecules) of gold, many to a wave-length, embedded in the glass.

(5) The polarisation of the cone of light emitted by the particles in the path of a beam of white light

traversing any of the three ruby glasses examined by SIEDENTOPF and ZsiGMONDY§ is that which would be possessed by the cone of light emitted by small spheres of metal embedded in the glass. Further, the colour of the cone of light in the case of these three glasses was green, while it has been shown in the former paper|| that the intensity of light of wave-length X emitted by small spheres of gold embedded in the glass is proportional to (a2 + 4y8a)/X4, and, according to Table I. above, this expression for gold spheres in a glass of refractive index v = 1 -56 has a maximum in the neighbourhood of X = -560, i.e., in the yellowish-green.

We conclude then that the colouring agent of gold ruby glass is metallic gold,! the major portion of which is in the form of small spheres.

The irregular blue and purple which often appear, instead of the ruby at which the glass manufacturer aims, can be explained as indicated in the appendix to the former

* SIEDENTOPF and ZSIGMONDY, ' Ann. der Phys.,' January, 1903. t Cf. former paper, 'Phil. Trans.,' A, 1904, p. 391. I Loc. tit., p. 27.

§ See their table reproduced at p. 397 of the former paper and discussion following it on pp. 398-401 ('Phil. Trans.,' A, 1904). || Loc. cit., p. 400. U Not aurous oxide, as stated in the text-books on glass making.

ON COLOURS IN METAL GLASSES, ETC. 251

paper. We have already* seen that if gold glass when first made in the furnace be rapidly cooled, the glass remains colourless. In order to obtain ruby glass, the molten glass must be left in the annealing oven and maintained at a high temperature for about three days. If the glass is too violently heated or is kept too long at a high temperature, it becomes turbid, reflects brown light, and develops first an amethystine and then a blue tint by transmitted light. But it now appears that the gold cannot, as previouslyt suggested, be in solution in the colourless glass when first heated ; for if metallic gold were in true solution in the glass it would have the vapour colour indicated by the dotted curve in fig. 1. The gold must therefore be gradually reduced during the annealing process. So long as the glass remains hot enough to admit of molecular movement, the molecules of gold go together to form spheres, and these small spheres tend to coagulate into crystallites.} If the glass cools before the coagulation of the small spheres, a gold ruby glass is obtained. If, however, some of the small spheres have coagulated into crystallites, the density of which exceeds '6 of that of normal gold,§ these crystallites will reflect light which is predominantly yellow or red.|| The glass will thus reflect brownish light; and since the more refrangible rays are less reflected than those of longer wave-length, the red end of the absorption curve will, owing to the crystallites, be raised relative to the blue. The glass will thus appear bluer than when no coagulation has occurred. Further, as these crystallites may be of dimensions comparable with a wave-length of light, they destroy the optical continuity of the medium and produce turbidity. Now the blue colour of a gold glass is always associated with turbidity and a brown appearance by reflected light, so that the formation of crystallites of gold in the glass accounts for the irregular blue and purple colours which gold glass sometimes exhibits. 1

Diffusions of gold particles in water — the so-called "colloidal solutions" of gold- have been prepared by FARADAY**, ZsiGMONDY,tt and STOEKL and VANINO^, who

* Loc. cit., p. 392.

t Loc. cit., p. 392.

J In the case of copper ruby glasses the process continues until actual crystals of copper are formed, but I have not seen gold crystals in a glass, although it is probable that they are occasionally formed.

§ KIRCHNER and ZSIGMONDY ('Ann. der Phys.,' XIII., 1904, p. 591) estimate that a clump of gold particles in a blue gold-gelatine preparation contains at least 50 per cent, of gold. See below, p. 254.

|| See fig. 12, below.

U The blue and violet [purple] colours of the glasses D and E in SIEDENTOPF and ZSIGMONDY'S table (see ' Phil. Trans.,' A, 1904, p. 397), as well as the red, yellow and brown colours of the cone of light emitted by them, are thus explained. STOKES (Royal Institution Lecture, 1864, 'Collected Works,' vol. IV., p. 244), without entering into the question why gold glass ordinarily transmits pink light, says that, it being the property of gold to transmit bluish light, the metallic gold in suspension causes the blue appearance.

** FARADAY, Bakerian Lecture for 1857, printed in 'Phil. Trans.' for 1857, and reprinted in his 'Researches in Chemistry and Physics.' References will be made to the pages of the reprint.

ft ZSIGMONDY, 'LiEB. Ann.,' vol. 301 (1898), p. 29, and 'Zeitschr. f. Electrochem.,' vol. IV., p. 546.

II STOEKL and VANIXO, 'Zeitschr. f. Phys. Chem.,' XXX. (1899), p. 98.

2 K 2

252 ME. J. C. MAXWELL GAENETT

precipitated the gold from its chloride by means of various reducing agents ; and by BREUIG* and later by EHRENHAFT, t who used a gold terminal for an electric arc which was caused to spark under water.

All these preparations exhibited a gradual change in colour from red through purple to blue ; this change was greatly accelerated by the introduction of a trace of salt into the water. ZSIGMONDY^ gives the absorption curves of a number of "solutions" of gold. STOEKL and VANINO§ measured the absorptions of a red suspension con- taining a known volume proportion of gold. Lastly, EHUENHAFT|| has made careful measurements of the absorptions of "colloidal" gold. The curves plotted from his measurements of the red "solutions" resemble the continuous curve shown in fig. 1. Again, EHRENHAFT statesll that the absorption band of a gold suspension which possessed a beautiful red colour began at X = "560 and attained a maximum at X = -520, while the solution was almost transparent in the ultra-violet. Now the maximum of the calculated absorption curve for spheres of gold in water (v = 1/3333) occurs at X = '533.** Again, the dotted curve in fig. 1, which will represent the absorption produced by a true solution of gold, does not sufficiently agree with the measured absorptions to admit of the gold being in true solution in the water. These results suggest that the coloration is due to diffused spheres ft of gold, although some discrete gold molecules may also be present.

* BHKDIG, ' Zeitschr. f. Phys. Ghcm.,' XXXIL, p. 127.

t F. EHREXHAFT, 'Ann. der Phys.,' XI. (1903), p. 489.

{ ZSIGMONDY, 'LiKP,. Ann.,' vol. 301 (1898), pp. 46-48.

§ Loc,. cit., p. 108. For a discussion of their rusults see below (footnote, p. 253).

|[ Loc. fit., pp. 505, 506.

H Of., table given, lor. fit., p. 507.

** Thus the differences in wave-length between the observed maximum absorption of gold ruby glass and of the calculated maximum for gold spheres in glass (;' = 1'5), and between the observed maximum for colloidal gold and the calculated maximum for gold spheres in water, are respectively '017 and "013, and these differences are of the same si/.e.

ft EHREXHAFT also supposed that the gold was present in the form of small spheres ; but he proceeded to define the average size of these spheres (and also of those of Ag, Pt, &c., in the " colloidal" solutions of these metals) by means of J. J. THOMSON'S equation connecting the radius of a conducting sphere with the wave-length corresponding to the free periods of its vibration, this wave-length being assumed to be that of the absorption maximum. KIRCHXER and ZSIGMOXDY (' Ann. der Phys.,' 1904, p. 575), however, point out that there is no connection between size of particles and the absorption of light produced by them, and this we have seen to be the case, provided there are many particles to a wave-length ; also the very small size (if spherical, their average diameters would be 7///J.) of the particles of gold, the gold content of which ZsiGMONDY measured would require the absorption maximum to be in the ultra-violet. KlRCHXER and ZsiGMOXDY add that it would only be possible to get a large enough linear dimension to give a free period if the particles were not iso-dimensional, and they conclude therefore that the gold particles must be in the form of leaves or of rods ; but they do not reconcile such a form with the polarisation and green colour of the cone of light emitted by the smaller particles. Since, however, we find that the small-sphere hypothesis accounts for the observed phenomena, we must agree with EHREXHAFT that the particles are spherical, although we cannot admit that the average diameter of the spheres is correlated to the wave-length of the light most absorbed.

ON COLOURS IN METAL GLASSES, ETC. 253

This view enables us to explain the change of colour from red to blue, by the coagulation of the small spheres, just as in the case of the glasses coloured by gold ; the simultaneous development of a brown reflection and a turbid appearance is at the same time explained. The following quotations must suffice to describe the phenomena in question.

FABADAY observes that

"A gradual change goes on amongst the particles diffused through these fluids, especially in the cases where the gold is apparently abundant. It appears to consist of an aggregation. Fluids at first clear, or almost clear, to ordinary observation, become turbid; being left to stand for a few days, a deposit falls."*

When common salt, or any other substance which dissociates in water, was added to the fluid

"... The salt diffused gradually through the whole, first turning the gold it came in contact with blue, and then causing its precipitation.!

" Such results would seem to show that this blue gold is aggregated gold, is., gold in larger particles than before."}

Again

" The supernatant fluid in specimens that had stood long and deposited was always ruby . . . there was every reason to believe that the gold was there in separate particles, and that such specimens afforded cases of extreme division."§

Observations made by subsequent physicists agree with those of FARADAY. Thus ZSIGMONDY writes

" In every case the bright red colour [of suspensions of gold in water] changed to blue on the addition of salt; and decoloration of the upper part of the liquid showed that precipitation has then begun."]]

Again, STOEKL and VANINO, who examined a large number of suspensions of gold in water prepared by many different methods, state that

" When the particles [of gold] are very small ... the fluid appears red-yellow, ruby-red. When, however, the particles increase in size, the red and yellow rays are quite cut off and the transmitted light consists only of blue and violet rays, the fluid appearing blue-violet. "U

* ' Researches in Physics and Chemistry,' p. 414.

t We may suppose that by friction against the water the gold spheres obtain that negative change which ZSIGMONDY (' LIEB. Ann.,' vol. 301, p. 36) found that they possess. The mutual repulsion of these like charges prevents the spheres from coagulating and thus keeps the gold in suspension in the water. But when an electrolyte is introduced into the fluid, the positive ions discharge the gold spheres, so that coagulation and precipitation result.

\ FARADAY, loc.. cit., p. 420.

§ FARADAY, loc. cit., p. 418.

|| ZSIGMONDY, ' LIEB. Ann.,' vol. 301 (1898), p. 34.

II ' Zeitschr. f. Phys. Chem.,' XXX. (1899), p. 108.

As already stated, STOEKL and VANINO measured the absorption of light, for six different values of \ by a suspension containing a known volume proportion of gold. Using their value of /x(- 000003) to determine the scale of the continuous curve in fig. 1, and comparing the values of wV/A so obtained with

254 MR. J. C. MAXWELL GARNETT

Finally, KIRCHNER and ZSIGMONDY record that in a gold suspension in water

"... A given (generally large) number of particles which diffract green light [i.e., small spheres] were brought together by the addition of an electrolyte into a single particle which diffracted yellow or red light with much greater intensity* than its components. With this uniting of particles occurs the change in the colour of the fluid from red to blue."t

We have already shown that, theoretically, the coagulation of the small spheres of gold should produce a colour change in the fluid, from red through purple to blue ; and the above quotations have indicated that coagulation accompanies the change of colour. But that the coagulation takes place in the manner assumed for the purposes of the theory has been shown by KIRCHNER and ZSIGMONDY, who prepared suspensions of gold in gelatine, some of which preparations were red when wet, and changed to blue on being dried, at the same time developing a gold-bronze reflection. J

Now these dry blue membranes contained a number of clumps, each composed of hundreds of ultra-microscopic resonators^ (small spheres) ; and these clumps were comparable in size with a wave-length of light, being directly visible when examined with a numerical aperture of T4 : they would therefore be capable of reflecting light. Further, the change of colour to blue was most marked in those preparations in which the individual clumps were most dense, || and it appears from fig. 12 below that the selective absorption of red and yellow light by a gold crystallite is greater the greater its density. The theoretical explanation of the change to blue requires the rays of lower refrangibility to be stopped by reflections from crystallites,!! and this requirement is thus satisfied.

the absorption curve obtained from STOEKL and VANINO'K observations, we find that the observed curve lies below the calculated curve, except for red light. But SroEKr, and VANINO record that the observed fluid had a yellowish reflection, so that large particles (crystallites) must have been present in it ; and the presence of these crystallites requires the volume // of gold, which per unit volume of the liquid is in the form of small spheres, to be less than the total volume proportion /x This diminishes the absorptions throughout the spectrum. But the volume proportion /*-/*' of crystallites produces absorption which is much greater for the red and yellow than for the green and blue rays. The superposition of the absorptions produced by /<,' and by //. - // would thus produce an absorption curve in accordance with that observed.

^ The aggregate may be supposed to be comparable in size with a wave-length of light ; the intensity of the light reflected from it would thus be proportional to the square of its diameter, while the intensity of the light diffracted by the small spheres is proportional only to the sixth power of their diameters.

t KIRCHNER and ZSIGMONDY, he. tit., p. 592.

| Loc. cit., p. 589.

§ KIRCHNER and ZSIGMONDY, loc. cit., p. 576.

|| Loc. cit., p. 577.

U A similar explanation possibly applies to the fact that when light, transmitted through a stretched membrane containing gold in suspension, is polarised in the direction of stretching, the emergent light is red, but when the incident light is polarised in a perpendicular direction the colour is blue, the gold clumps being comparable with a wave-length in the direction of stretching, but not in a perpendicular direction. (Of. AMBRONN, 'Ber. d. math.-phys. Kl. d. k. Sachs. Gesellsch. d. Wissensch.,' December 7, 1896, and AMBRONN and ZSIGMONDY, do., July 31, 1899).

ON COLOUES IN METAL GLASSES, ETC. 255

In conclusion, we remark that most " colloidal solutions" of gold, even those which are of a ruby colour, contain crystallites in addition to the small spheres to which the colour is primarily due. Thus FAKADAY could detect the green " cone of light," which indicates the absence of large aggregations, only in those liquids which had been cleared by prolonged precipitation and frequent decantation ; and STOEKL and VANINO found that all the gold suspensions which they examined showed a yellowish reflection. A small number of the large aggregations may, however, cause the cone of light to appear yellow or red without appreciably altering the colour of the transmitted light. For, whereas the intensity of the (green) light emitted by a small sphere is proportional to the sixth power of the diameter, the intensity of the (brown) light reflected from a gold crystallite is proportional to the square of its linear dimensions. Gold solutions prepared chemically appear, however, to be freer from aggregated gold than are those prepared by BRE DIG'S method.*

6. Diffusions of Silver. TJie Nature and Form of the Suspended Particles.

We proceed to consider the absorption of light produced by diffused particles of silver. The values of nY//nX for v = T6, v = 1P5, v = 1'3333, and v = I'O given in Table II. are plotted in fig. 3, the positions of the maximum of each curve being determined as in the case of gold. Since (cf. above, § 4) the values of n and HK for silver were all determined from the polished surface of the metal, these curves should represent the absorption produced by diffused spheres of silver in glass, in water, and in vacua, with only a small error, t The dotted curve in fig. 1, which represents the absorptions of diffused molecules of silver in vacuo (and, on different scales, in other non-absorbing and non-dispersive media), shows that the silver molecule has a free period corresponding to X = "3GO, about. The existence of this free period is possibly responsible for the sensitiveness of silver salts to ultra-violet light.

In fig. 4 the graphs of wV/X for glasses of refractive indices v = 1'GO and v — 1'56 are shown on such a scale as to have the same ordinate at the D line as the graph of K for the measured glass Ag (B), of which the measured refractive index at the D line was T579. The measured curve resembles those calculated, following them very closely from X = '600 to X = "475, and having a maximum for a value of X intermediate between those values of X which correspond to the maxima of the two calculated curves.^ This close approximation of the observed absorptions to those

* Cf. ZSIGMONDY, ' Zeitschr. f. Electrochem.,' p. 547. BREDIG'S remark, that his gold solutions were blue red, points to the same conclusion.

t These curves show that in each case the absorption is less for red than for yellow. This is contrary to statements made in the previous paper (loc. tit., pp. 399 and 420) ; the errors therein made were due to miscalculation for silver (red) (loc. cit., Table I., p. 396).

J The cause of the depression of the observed maximum below those calculated is doubtless to be found in the fact, to which Lord RAY LEIGH has called attention in a recent lecture at the Royal Institution, that

256

MR. J. C. MAXWELL GARNETT

IIK

,„ „ Fig. 4. SILVER.

Fig. 3. SILVER— calculated values of —. ,,»«•! j * i A /D\

A. (1) K, observed for glass Ag (B) :

Observed values : x .

(1) Spheres (or molecules) in vacua, v=\-0:

(2) Spheres in water. v=l- 3333 :

(3) Spheres in glass, v = 1 • 5 :

(4) Spheres in glass, v = \ • 6 :

Calculated values shown thus: O-

(2) ^-, calculated for silver spheres in

A

glass, v = 1 • 56 :

(3) -— , calculated for silver spheres in

A

glass, v -• \ ' GO :

Calculated values : Q .

ON COLOURS IN METAL GLASSES, ETC. 257

calculated suggests that the colouring agent of the yellow silver glass consists primarily of diffused spheres of silver. Since discrete silver molecules would produce an absorption maximum at X = '360, not more than a comparatively small amount of silver can be present in the molecularly subdivided condition. The conclusion that silver glass owes its colour to diffused spheres of silver will be verified in the following section.

The absorption spectra of some colloidal solutions of silver, prepared by BREDIG'S method,* have been measured by EHRENHAFT. The continuous curve shown in fig. 3, representing the calculated absorptions of a diffusion of silver spheres in water, is of the same form as that which, according to EHRENHAFT'S measurements, represents the absorption of visible light by a colloidal solution of silver.! Using ultra-violet light, he further found that a brown colloidal solution of silver, examined before coagulation had seriously affected its colour, showed an absorption band which began a,t X = '503 and attained a maximum at X = '380, while the fluid was again quite transparent at X = '335. Except for the fact that the maximum ordinate of the calculated curve for silver spheres in water is at X = '389 instead of at X = '380, the above observations admirably describe the continuous curve shown in fig. 3. Since the dotted curve given in that figure shows a maximum at X = '360, and the absorp- tion band does not begin until X = '450, about, the colour of the " colloidal " solution is not that which would be exhibited by a suspension of discrete silver molecules, i.e., by a true solution. We conclude, therefore, that the silver in a " colloidal " solution is present in the form of small spheres ; discrete molecules may, however, also be present, and, as indicated above in the case of gold, prepared by BREDIG'S method, probably also crystallites, the number and size of which will increase with the age of the solution.

That the silver in a colloidal solution is in the form of small spheres is further shown by an experiment of BARUS and SCHNEIDER j who measured the refractive index of such a fluid. Their results are given in the following table, § in which n represents the measured refractive index :—

the spectrum formed by the light which has traversed the glass is not quite pure, so that that image of the slit which should be illuminated only by light of wave-length, say, A = -433, is also, owing to reflections from dust particles, &c., illuminated by light of other wave-lengths which has experienced a less absorption.

* BREDIG, ' Zeitschr. f. Electrochemie,' 4, pp. 514, 547.

t EHRENHAFT, loc. cit., p. 506.

J BARUS and SCHNEIDER on "The Nature of Colloidal Solutions," 'Zeitschr. f. Phys. Chem.,' VIII., p. 278.

§ Tabelle 5, loc. cit., p. 296.

VOL. CCV. — A. 2 L

258

MR. J. C. MAXWELL GARNETT

TABLE V.— Index of Refraction of Colloidal Solution of Silver for Sodium Light

(X = -589).

Solution.	Percentage of silver.	Percentage of foreign salts.	Temperature, °C.	n.
	0	0	18-0	1-3306
	0	0	18-2	1-3315
	1-16	0-18	18-6	1-3369
Silver Solution* . . . <	1-16	0-18	18-6	1-3363
	1-16	0-18	18-6	1-3369
	1-16	0-18	17-0	1-3363
Water J	0	0	18-7	1-3331
	0	0	19-0	1-3333

* The solutions were prepared by CAREY LEA'S method of precipitating silver nitrate with ferrous citrate — they were subsequently dialysed for 60 hours.

Thus the mean refractive index of silver in water at 18°'6 was n = T3367, while the refractive index of water at 18°7 was v = 1-3331. Taking the specific gravity of silver as 10, the volume proportion silver was /x = '00116.

The values of the functions « and a! for sodium light and water at v = T3331 are, according to Table II., a= 1-571 and a' = 1-333. Substituting these values of v, p., and a in equation (10), namely

we obtain

n' = *(l + f/ia), n' = ^(1-00273) = 1-33674. Similarly, from equation (9),

n" — v \ 1 H — ua'

2f we have

n" = v (1-002078) = 1-33587. Comparison of these values of n' and n" with the observed value, namely n = 1'3367,

ON COLOURS IN METAL GLASSES, ETC. 259

requires that practically the whole of the silver must have been in suspension in the form of small spheres.*

Once more, CAREY LEAt prepared suspensions of silver in water by precipitating the silver from the nitrate by means of a mixture of ferrous sulphate and sodic citrate. He describes how, after careful washing, the silver frequently " dissolved," forming a liquid which varied from red to yellow^ and was generally blood red ; he adds :—

" On one occasion the substance was obtained in a crystalline form. Some crude red solution had been set aside in a corked vial. Some weeks after the solution had become decoloured with crystalline deposit on the bottom, The bottle was carefully broken ; the deposit, examined by a lens, consisted of short black needles and thin prisms."

If, then, the diffused particles of silver when aggregated and precipitated had become crystalline, they must before have been in the form of nascent crystals, and for gold and for all the substances examined by VOGELSANG, § such nascent crystals were spherical.

7. Blue Reflection from the Stained Face of Silver Glass.

When clear glass is flashed with silver glass, or when a clear glass is so stained on one face with silver that the volume proportion /A of silver does not gradually diminish to zero as we proceed inwards from the stained face, but that the stained region ends abruptly, a blue reflection from the interface can be observed if the glass is held with the stained face away from the eye. No blue reflection can be seen from the air-glass interface when the stain is held towards the eye. STOKES observed this blue reflection, and stated that the interface presented the appearance of being coated with a fine blue powder. ||

We proceed to examine whether the presence of small spheres of silver, which has been shown to account for the colour of the light transmitted by silver glass, will also account for this blue reflection. Consider, then, plane polarised light travelling in a medium of refractive index v' and directly incident on the surface, z = 0, of an absorbing medium whose optical constants are n' and K', where N' = n' (I— IK'). Then we may take as the electric and magnetic vectors for

* BARUS and SCHNEIDER (foe. cit., p. 297) make the following comment on their experiment: — •

"KuNDT has found for normal metallic silver a refractive index of about 0'27. It would, therefore, be expected that the presence of the silver would diminish the refractive index of the water. It is by no means denied that it might be possible to explain the normal refractive indices of the above table in accordance with MAXWELL'S Theory of Light."

The investigation in the text attempts to give such an explanation.

t CAREY LEA, ' Amer. Journal of Science,' 1889, and 'Phil. Mag.,' 1891.

I Cf. above, p. 243, especially second footnote.

§ Cf. above, § 1.

|| STOKES, 'Collected Works,' vol. III., p. 316.

2 L 2

260 MR. J. C. MAXWELL GAKNETT

Incident light : —

X = exp {tp (t-i/z/c)}, Y = 0, Z = 0,

a = 0, /3 = v' exp {ip (t—v'z/c)}, y = 0.

Reflected light : —

X = B exp {ip (t+v'z/c)}, Y = 0, Z = 0,

a = 0, /3 = — z/B exp {ip (t + v'z/c)}, y = 0.

Light inside absorbing medium : —

X = C exp {ip (t-Wz/c)}, Y = 0, Z = 0,

« = 0, /3 = N'G exp {ip (t-Wz/c)}, y = 0.

Making X and /3 continuous at Z = 0, we have

C=1+B, N'C = »/(1-B). Hence

B = —

Taking the square of the modulus, we have, for the value II of the ratio of the intensity of the reflected light to that of the incident light,

R = (B)* = ="±! ........ (12).

' ''2IJ

If, now, the absorbing medium consist of minute spheres of metal embedded in a transparent medium of refractive index v, we have equations (10), namely,

n'K' = Sfjiv/3, n'= y(l+f/ta) ....... (10).

Substituting these values of nV and of n' in (12) we obtain

...l - - (13),

in which powers of /t higher than the second have been neglected.

Suppose first that v' = 1, so that we consider the reflection at the front face of the stained glass. Omitting powers of p. except the lowest which occur, we then have from (13)

It appears from equation (14) that light is reflected from the stained glass almost as if the stain did not exist, the effect of the stain being slightly to increase the reflection of those colours (in the blue) for which, according to Table II., a is greatest.

ON COLOURS IN METAL GLASSES, ETC.

261

Now, however, suppose that v' = v, so that the light is reflected at the interface between colourless glass and the same glass containing small spheres of metal. Neglecting p.3, equation (13) then reduces to

(15).

Since this expression for R contains no large constant term, the light from the interface will in this case be highly coloured in the case of those metals for which a3+4/82 varies greatly for different values of X.

If, however, the absorbing medium contain molecularly divided metal, equations (10) are replaced by

2 + "2 -A (11).

n" =

Replacing 3/*/3 and 3/j.a. in equations (14) and (15) by (2 + v2)/v2 . /3' and (2 + v2)/v2 . a' respectively, we obtain, as the intensities of the light reflect from the front face of the stain and from the interface respectively,

<>«>•

(17).

As before, it appears that when the stain is held towards the eye the reflection R/ is almost as if the stain were not there ; while when the stained face is away from the eye, the reflection is highly coloured.

Sir WILUAM ABNEY has kindly measured for me the intensities R0 of light reflected from the interface between the unstained and stained regions of one of STOKES' specimens of silver glass. The values of R0 are given in the following table : —

TABLE VI. — Blue Reflection from Silver Glass. Measured Value of

v at D Lines = T532.

A.	Ro.	A..	RO.	A.	Ro.
•4200	•25	•5000	•067	•5800	•014
•4300	•285	•5100	•050	•5900	•016
•4400	•290	•5200	•042	•6000	•018
•4500	•267	•5300	•032	•6100	•020
•4600	•237	•5400	•025	•6200	•021
•4700	•195	•5500	•020	•6300	•022
•4800	•146	•5600	•018	—	—
•4900	•095	•5700	•016	•6800	•022

262

ME. J. C. MAXWELL GAENETT

Fig. 5. (1) E

Blue reflection from silver glass. Calculated values : O- • spheres in glass, v = 1 • 56 :

(2) E' -r- molecules in glass, v = 1 • 56 :

Observed values : x .

(3) BO ~f~ observed :

•350 -400 -450 -500

•550 -600

•650

•700

Fig. 5.

The continuous curve shown in fig. 5 has been fitted to the plots of these values of E.0. In the same figure are also shown the calculated values of E and of E/, obtained from equations (15) and (17) by means of the values of a2+4/32 and of a'2 + 4/3/2 given for silver and glass of refractive index v = I' 56 in Table II. The scales on which E and E' are represented are so chosen that the ordinates corre- sponding to X = '589 shall be the same as that for the continuous curve.

ON COLOURS IN METAL GLASSES, ETC. 263

It appears that while the graphs of R' and R0 widely differ, the positions of the respective maxima falling near X = '360 and X = '436 respectively, the graph of R closely resembles that of R0)* the maxima of R and of R' occurring at almost the same value of X. We conclude that the presence of small spheres of silver throughout the stained region of the glass will account for the blue reflection ; and we thus confirm the view, to which absorption phenomena led us, that silver glass consists of a suspension of small spheres of silver in a colourless glass.

Before leaving the consideration of the blue reflection from silver glass, it may be noticed that the light is not reflected as from a plane interface between glass and silver glass. Thus when the source of light is an electric arc, the blue colour is clearly discernible by an observer whose eye is not in the straight line determined by the ordinary law of reflection. This effect is due to the irregularity of the interface, the silver not having penetrated the glass to a uniform depth. AH alternative explanation, however, suggests itself, — the blue colour might be due to independent radiation from discrete spheres (or molecules) of silver so far apart as not to form an optically homogeneous medium. The intensity of the emitted light would then be proportional to (a2+4/32)/X4 (in the case of spheres, or (a'- + 4/3'-)/Xl in the case of molecules). Further, the blue colour would be equally visible if the light illuminating the discrete spheres (or molecules) entered the silver glass from the air side or the clear glass side ; and this is not the case.

It is of interest to notice that while each individual sphere in glass radiates out light of an intensity proportional to (a*+ 4j62)/X4, a surface separating a glass, containing many of the spheres to a wave-length of light, from a region of the same glass in which no spheres are present, reflects light with an intensity proportional to or + 4/3". This is due to the fact that the number of spheres (on the reflecting surface), the phase of the forced vibrations of which lies at any instant between given limits, is proportional to X2 ; so that the intensity of the reflected light is proportional to X' times the intensity of the light emitted by a single sphere.

8. Diffusions of Copper. The Nature and Form of the Suspended Particle*.

We proceed to discuss the colours produced by diffused particles of copper in order to discover the cause of colour of copper ruby glass. The values of the expression nV//iX for v = 1-56, v = 1'5, and v = I'O given in Table III., are plotted in fig. 6, the maxima being determined as in the case of fig. 1 (cf. § 5 above). As in the case of silver, these curves should fairly accurately represent the absorptions produced by copper spheres in glass v = T56, in glass v = 1'5, and by copper spheres or molecules

* The fact that E0 increases from yellow to red, while E diminishes in the same range, would be accounted for if the black paper with which Sir WILLIAM ABNEY backed the stained face of the glass reflected 2 per cent, of the light incident on it. Further experiments are to be made on this.

Fig. 6. COPPER — calculated values of -^-.

Observed values : x .

264 MR. J. C. MAXWELL GARNETT

Fig. 7. COPPER. ( 1 ) K, observed for copper ruby glass Cu (X) :

(1) Spheres (or molecules) in vacuo, v = 1 • 0 :

(2) Spheres in glass, v = 1 • 5 :

(3) Spheres in glass, v = 1 • 56 :

Calculated values shown thus : Q. tv~

(3) n-!L! calculated for copper molecules in

glass :

Calculated values : O.

(2) ~, calculated for copper spheres in glass,

if

- . tf

.(

60

E

ON COLOUKS IN METAL GLASSES, ETC. 265

(copper vapour) in vacuo respectively. The absorption band in the yellow green shown by the top two curves in fig. 6 was observed by STOKES in the spectrum of a copper ruby glass.

In fig. 7 the graphs of nV/X for glass (v = 1'5) and of H"K"/\* are reproduced from fig. 6 on such a scale as to have the same ordinate at the D line as that possessed by the continuous curve which has been fitted to the plots of the measured absorption K of the glass Cu (X). The curves in fig. 7 all have a minimum in the red or infra red and a maximum in the yellow-green ; but while that (nV/X) which represents the absorptions of spheres in glass has a secondary maximum near X — '480, the dotted curve shows that the absorption of copper molecules in glass continues to increase till X<-350. Also the maximum in the yellow green for the "sphere" curve occurs for approximately the same value of X as corresponds to the maximum observed absorption ; while the value of X at the maximum of the dotted ("molecule") curve is about lO^i/A less, and the latter maximum is much less marked than are the former two. Finally, the last readings obtained for K in the violet indicate that the continuous (observed) curve rapidly approaches a maximum near X = '480.

We conclude that copper ruby glass is coloured by metallic copper,! and that the greater part of the copper is present in the form of small spheres, although some probably remains in the form of discrete molecules.|

The manufacture of copper ruby glass closely resembles that of gold ruby.§ Like gold ruby, the copper ruby glass becomes turbid if kept too long at a high temperature. This turbidity is also probably due to the formation of crystallites by the coagulation of small spheres, since, when the conditions necessary for the development of turbidity are long maintained, actual crystals, apparently of copper, are formed in the glass. ||

9. Colouring Effects of the Radiation from Radium, Cathode Rays, &c.

It has long been well known that cathode rays produce a blue-violet coloration in soda glass. Soda glass tubes, after containing the emanation from radium, show the

* The graph of »V/A for v = l can, by increasing all the ordinates in the proper constant proportion, be changed into the graph of w'V'/A for any value of A. Of. § 3 above.

t STOKES (' Math, and Phys. Papers,' vol. IV., p. 242) supposed that the colouring agent was suboxide of copper. The blue colour exhibited by overheated specimens of the glass (lo>;. cit., p. 243) is probably caused by the coagulation of the small spheres into crystallites and crystals which reflect out the red light.

| Measurements will have to be made with ultra-violet light in order to determine how much copper remains in the molecularly subdivided condition.

§ Of. above, §5, p. 251.

|| Of. 'Phil. Trans.,' A (1904), p. 392. Some of the crystalline glazes made by Mr. BURTON at PILKINGTON'S tile works exhibit the same effect. I have seen a pot with a copper glaze in parts of which the copper was apparently reduced, for in passing from the colourless glaze (where the copper was not reduced) into regions where the reduction had been effected, a deep red (copper ruby) was first reached ; that colour increased in intensity until, in the central portions of the region, crystals, apparently of copper, could be seen.

VOL. CCV. — A. 2 M

266 ME. J. C. MAXWELL GARNETT

same colour, and crystals of each salt acquire iinder cathode rays a beautiful violet tint.* Experiment has also shown that exposure to the emanation from radium gives to gold glass a ruby colour, to silver glass a yellow colour, and to potash glass a brown colour.

Now we have seent that a molecularly subdivided metal possesses the same colour by transmitted light whatever be the nature of the surrounding transparent medium, supposed non-dispersive and isotropic. This colour may be called the vapour- colour of the metal. It has further appeared that although the transmitted colour of a metal subdivided into small spheres, many to a wave-length of light, does depend on the refractive index v of the medium in which the small spheres are " embedded," yet this colour approaches to the vapour-colour as v approximates to unity. As is shown by the dotted curve in fig. 1, the vapour-colour of gold must be red.| The colour of glass containing molecularly distributed gold is thus red,;}; although when the gold is collected into spheres the glass is pink. Similarly, reference to the relative values of /3'/X in Table II. shows that the vapour-colour of silver is yellow. Glass coloured by small spheres of silver is also yellow. Again, Professor R. W. WOOD showed to the British Association^ in Cambridge that the vapour-colour of sodium is violet, this colour being due to the absorption at the D lines. This violet colour is also produced at the cathode in the electrolysis of sodium chloride, || the molecules of sodium formed at the cathode being distributed throughout the water in its neighbourhood and giving rise to the vapour-colour. 'I Analogy with the cases of gold and of silver indicates that small spheres of sodium would produce in glass a colour not greatly different from the vapour-colour produced by the molecularly subdivided metal.

Thus the colours developed in gold, silver, or soda glass by the radiation from the emanation from radium are approximately the same as the colours which would be given to the glass by the presence of the reduced metal, either molecularly divided or in small spheres (nascent crystals).

It is therefore very probable that the metal in the glass is reduced by the action of the radiation. This view finds considerable support in the discovery of VILLARU,** that cathode rays exert a reducing action, as well as from the fact, already cited, tf that ELSTER and GEITKL found the salts of the alkali metals, which had been coloured by exposure to cathode rays, to exhibit photo-electric effects as if they contained traces of the free metal.

* GOLDSTEIN, ' WIED. Ann.,' liv., p. 371, 1898.

t Fide ante, p. 243.

I Or yellow ; see the second footnote, p. 243.

§ August, 1904.

|| Cf. J. J. THOMSON, 'Conduction of Electricity through Gases,' pp. 495, 496.

IT BUNSEN found that common salt, after heating to about 900° C., exhibited a violet colour, due apparently to the reduced metal, although BUNSEN suggested a, subchloride, ** 'Journal de Phys.,' 3™' Series, VIII., p. 140, 1899. tt See 'Phil. Trans.,' A, 1904, p. 400.

ON COLOURS IN METAL GLASSES, ETC. 267

Sir WILLIAM RAMSAY, when I first called his attention to the explanation of the coloration of glass by radium which is afforded by supposing the radiation to reduce metal in the glass, suggested that the reduction might be effected by the discharge of free ions of the metal. Since that time the further evidence that has accumulated seems to 'favour the truth of this theory. Thus, as all the colour-changes from pink to blue exhibited by gold glass can be imitated with suspensions of gold in water, the glass appears to behave as a liquid, although a very viscous one ; and it seems, therefore, reasonable to suppose that the salt of a metal which will dissociate in water will dissociate also in glass. As an alternative hypothesis, we might suppose the compound molecules broken up by the rays. But, were this the case, the a-rays would be far more efficient than the /3 in producing the colour. And this is not true ; for the coloration produced in the splinters of gold and silver glass, as well as in soda and potash glasses, are not, apparently, stronger on the sides of the glass, but seem to be of uniform strength throughout. From this it appears that the /8-rays are alone capable of producing the colour. This is in accord with the former hypothesis. For the ions of the metal in the glass would be positively charged, and their discharge by the negatively-charged /3 particles (or cathode rays) would change them into molecules — just as the sodium ions in the electrolysis of common salt are discharged at the cathode, and thus are transformed into molecules of sodium, imparting a violet colour to the water and capable of forming caustic soda.

It appears, therefore, possible that all glasses contain free ions of metal, and that it is by the discharge of these ions, and consequent reduction of the metal, that cathode and Becquerel rays are able to produce coloration in them.

10. Numerical Values of the Optical Constants of Media containing Large Volume

Proportions of Certain Metals.

The preceding sections of this paper have treated only of the optical properties of those media for which the volume proportion, //., of metal is very small. The consideration of media in which p. may have any value up to unity will now, however, be resumed, in order to discover what may be the physical explanation of those colours and changes of colour which FARADAY,* BEILBY,! and others have found to be exhibited by thin metallic films. In § 11 of the former communication J the question whether films built up of small spheres of silver or of gold would, for any given volume proportion of metal, transmit red or yellow light more easily, was dis- cussed, and the conclusions reached were compared with the results of Mr. BEILBY'S experimentsf on the effect of heat on thin films of metal. The present section extends the scope of that enquiry.

* Bakerian Lecture for 1857, 'Phil. Trans.,' A, 1857. Reprinted in FARADAY'S 'Researches in Chemistry and Physics,' pp. 391 et seq. (Reference will be made to the pages of the reprint.) t 'Roy. Soc. Proc.,' vol. 72, 1903, p. 226.. t 'Phil. Trans.,' A, 1904, p. 415.

2 M 2

26 g ME. J. C. MAXWELL GARNETT

It has been shown that the optical properties of a metal, so diffused in vacuo (v = 1) that p. has some definite value, are the same whether the microstructure be amorphous (molecularly sub-divided) or consist of small spheres, these optical properties being in either case deducible from equation (7), p. 241.* If then, in accordance with the notation adopted in the former communication (pp. 403 et seq.), the accents' hitherto used to denote the optical constants, n and HK, when p. differs from unity, be now omitted, and the values of those constants corresponding to any particular value /*' of p be denoted by a suffix (e.g., HK^^), the values of n and HK are given by equation (17), p. 404, t namely,

a — ;

where, as in equation (13'), p. 403, t

- {n'(K'-l)-2}'+4nV > K(^-l)-2

By these equations the values of n and UK, determining optical properties of amorphous or " small-sphere " metallic media of any density, may be calculated for light of wave-length X, in terms of the values of n^=l and of nKIL=l for the same mono- chromatic light. The values of n and of UK for gold and for silver have already been calculated for all values of \L in the case of red light (X = '630) and in that of yellow light (X = '58 9). The results are given in Table IV. of the former com- munication. | But in order to obtain a true conception of the colours of such media, corresponding calculations must be effected for other colours also.

Now, in the case of silver, the numerical values of all those functions of HK^I and «M=1 which have hitherto been calculated for the case of v = 1— those functions, in fact, which relate to molecules or small spheres of silver in vacuo — vary continuously from red (X = -630) to blue (X = '450). If, therefore, the values of n and UK corresponding to X = '450 and X = '500 be now calculated for all values of p., the values of n and n/c for other colours may be obtained approximately by interpolation between X = '630 and X = '450.

The values of n and UK corresponding to X = '450 and X = '500 have therefore been calculated for certain values of /A, by means of equations (18) and (19). The values of ?i/c/1=1 and n^=l used for the calculations were carefully determined by B. S. MINOB.§ The results are tabulated below (Table VIII.).

In the case of gold it is not so easy to apply this method of interpolation. The values of n and HK corresponding both to blue (X = '458) and to green (X = '527)

* 'Phil. Trans.,' A, 1904, equation (16), p. 403.

t ' Phil. Trans.,' A, 1904.

I ' Phil. Trans.,' A, 1904, p. 406.

§ Loc. cit. (vide ante, Table II).

ON COLOUES IN METAL GLASSES, ETC.

269

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270 MR. J. C. MAXWELL GARNETT

have been calculated ; but the values of nK^=l and n^i used for the purpose are those given by HAGEN and RUBENS,* and are not by any means so accurate as those which MINOR has determined for silver.! The results are included in Table VII.

We return to the consideration of the transmission and reflection of light by a metallic film. We confine our present attention to films the microstructure of which is either amorphous or consists of small spheres of metal ; the films in question are thus optically isotropic. Suppose that, as explained in the preceding section, the optical constants of the film are n and K when its specific gravity is p. times that of the metal of which it is composed.

When light of wave-length \ in vacua is directly incident on such a film, ot thickness d, let Rft and T0 denote the ratios of the intensities of the reflected and of the transmitted light to that of the incident beam. Adopting the analysis already given in the previous paper, p. 409, we suppose the film to be bounded by z = 0 and z = d, and that

Incident wave is

E = 0, exp {ij> (t—z/<-)}, 0 ; H = -exp {ip (t-zjc}, 0, 0. Reflected wave is

E = 0, B exp { ip (I + z/c)} , 0 ; H = B exp { ip (t + z/c) } , 0, 0. Wave in film, i.e., hot ween z = 0 and z = d, is

E = 0, A' exp {q> (/-c/V)} t-B' exp {lp (t + z/V)}, 0,

H = -c/V[A' exp {tjp(«-*/V)}-F exp {q>(f + z/V)}], 0, 0.

Transmitted wave is

E = 0, Cexp{ip(*-z/c)},0; H= -(! exp [q, ((-z/c)}, 0, 0, where c/V = H (1 — IK).

We shall suppose:]: that TrdnK/\> 1, so that we shall be correct within 2 per cent. when we neglect B' in comparison with A'. The boundary conditions at 2 = 0, namely the continuity of the components of E and H which are parallel to the interface, then give

1 + B = A'; (l-B) = e/V. A' = »(!-«) A'. Eliminating A' we obtain, by taking the squares of moduli,

* Loc. cit.

t Cj. above, § 4.

| Of. 'Phil. Trans.,' A, 1904, p. 409.

ON COLOURS IN METAL GLASSES, ETC. 271

In equation (26), on p. 409,* we have already proved that

T = I 0 1 2 - 16^(1 + *") -^./A (9 - \

•»s

If we write

M 16n2(l + K2) ,

-{(l + n)2+nV}2 equation (21) may be written

T0 = Mo e-""'""/A ......... (23).

Equations (20) and (21) are thus correct within 2 per cent, for directly incident visible light, and for /A = 1 in the case of gold if c/>91ju//, or in the case of silver if (/> 60/J./A, where Ip-p EE 10~" millim.

For convenience of reference the corresponding results for obliquely incident light are given below. Let 0 be the angle of incidence. When the incident light is polarised in the plane of incidence, the ratios R, T of the intensities of the reflected and of the transmitted beams to that of the incident light are given by

'8ay ..... (2o)'

where u and v are defined by the equation

(u, v) cos (9 = [{(nV-1 + sin2 0}2 + 4nV}» + (n3/?^! + sin2 ^)]5 . . (20). v 2

When, however the incident light is polarised perpendicular to tlie plane of incidence, the corresponding ratios are given by

where

u'-<.v'={n(l-iK)YI(u-<.v) ........ (29).

Putting 9 = 0, we obtain

R = If = H,h T = T' = T0,

u = u' = n, v = v' = HK.

It appears from equation (23) that the colour of the light transmitted by a metallic film, although principally dependent on the values of nK/k for different values of X, is also affected by the corresponding values of M0. The thicker the film, however, the less important is M0 in determining the colour.

* 'Phil. Trans.,' A, 1904.

272

ME. J. C. MAXWELL GARNETT

The values of n/c/\, calculated from Table IV. of the former paper,* using however new and more accurate values in the case of silver with red light, and from Table VII. above, are shown in Table VIII. The corresponding values of M0 are given in Table IX., in which table the values of the reflecting power R0 have also been included.

In order to facilitate the consideration of the colours which should, according to the above analysis and calculations, be exhibited by gold and silver films when their specific gravities vary but their microstructures remain amorphous or granular (small spheres), graphs of n/c/\, of M0, and of R0 are given in the accompanying figures (figs. 8, 9, 10, 11, 12, and 13). In these figures the abscissse represent the volume proportion, /A, of metal, and the ordinates the value of the function. The curves have been fitted to the plots of the numerical values shown in Tables VIII. and IX. In each case the positions of the plots of calculated values have been indicated by small circles.

TABLE VIII. —Value of nK/K.

~a 43 y> M <5	o "3 • U	-4	i — i II a.	ci 11 a.	•* II 3,	id II a.	o II a.	i-^ II a.	oo" II a.	a> II a.	0 II a.	,-H 1 « II a.	1 a
	Blue .	•458	•420	•879	1-845	2-279	2-637	2-908	3-099	3-240	3-319	—	—
r— ' "3 i	Green .	•527	-.360	•861	2-432	3-149	3-526	3-646	3-642	3-592	3-529	3-524	•599
O	Yellow	•589	•046	—	—	•896	2-555	6-90	6-45	5-42	4-79	6-47	•685
	Red .	•630	•022	—	—	•381	•960	4-56	7-95	5-96	5-00	7-76	•734
	Blue .	•450	•064	—	•902	3-50	10-16	7-81	6-49	5-76	5-30	9-17	•561
|H to £ \	Green .	•500	•022	•053	•197	•431	1-401	12-557	8-901	6-850	5-882	11-585	•693
a	Yellow	•589	•008	—	—	•112	•238	•762	14-84	8-33	6-23	13-52	•791
1	Red .	•630	•006	—	•046	•082	•163	•444	4-885	9-005	6-286	13-599	•822

'Phil. Trans..' A, 1904, p. 406.

ON COLOUKS IN METAL GLASSES, ETC.

273

TABLE IX.

M

"

/c ~ _

a> ' u ~

Colour.

1 a

/»=•!

^=•4

P--6

/x=-6

/x=-7

.>=-8

^=•9

./i=l-0

/*--. a

Bed 1

•822^

' Mo

•985

•801

•695

•560

•380

•123

•430

•858

•087

A -630 j

I

. RO

•007

•105

•167

•252

•385

•669

•928

•953

•792

Yellow 1

•79lJ

; MO

•984

—

•675

•529

•331

•105

•563

•972

•100

Silver "{

A-589 j

I

RO

•008

—

•179

•274

•419

•800

•930

•951

•778

Green "1

•G93<J

Mo

•979

•725

•573

•365

•184

•634

1-056

1-384

•174

X-500 J

Ro

•010

•149

•245

•403

•736

•881

•916

•932

•709

Blue 1

•561<

M0

•969

•592

•367

•423

•894

1-29

1-60

1-79

•308

A. -450 j

I

Ro

•010

•238

•432

•716

•824

•867

•891

•907

•620

r

Red "I

•734^

Mo

•982

—

•622

•449

•247

•436

•848

1-190

1 •231

A -630 J

[

RO

•009

—

•215

• 338

•576

•798

•863

•895

•666

Yellow 1

•685^

Mo

•979

-

•578

•406

•374

•695

1-048

1-344

•354

Gold <j

X-589 J

I

Ro

•on

—

•252

•405

•626

•759

•820

•850

•595

Green "1

•599<|

Mo

•983

•915

•996

1-138

1-308

1-478

1-637

1-779 ,

—

A-527 J

RO

•0197

•239

•335

•421

•493

•550

•597

•634

•420

Blue 1

1-191 J

M0

1-006

1-078

1-129

1-196 j

L-276

1-363

1-454

1-544

—

(

A-458 J

Ro

•0114

•134

•188

•242

•294

•343

•387

•427

—

VOL. CCV. — A.

2 N

274

ME. J. C. MAXWELL GARNETT

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> •< GOLI 30): •589 527) 58):

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ON COLOURS IN METAL GLASSES, ETC.

275

Fig. 10. GOLD — M0.

Fig. 11. SILVER — M0.

Yellow (. Green (X Blue (X =

v •.

1 • 589): 27): 0' 1

itea (*.= Yellow (X Green (X = Blue(X = 5 M Q S ^

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: 89): 0): ) =

= -5 •451

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'276

MR. J. C. MAXWELL GAKNETT

Fig. 12. GOLD — RQ. Red (A =-630): Yellow (A =-589): -

Green (A =-527):

Blue (A= -458):

Fig. 13. SILVER — RO. Red (X= -630): Yellow (A =-589): —

Green (A =-500):

Blue (A. = "450):

I

ON COLOUES IN METAL GLASSES, ETC. 277

11. Colour Changes Caused by Heating Metal Films.

In the Bakerian Lecture* for 1857, FARADAY described a, number of experiments concerning the colours which gold and other metals were, in various conditions, capable of exhibiting.

Mr. G. T. BEILBY'S investigations on the colour and structure of films of metal are described in his paper on " The Surface Structure of Solids."t

The average thickness of the gold leaf which FARADAY used in his earlier expei'iments was about 90 /i/i.| Reference to p. 271 above will show that, with a probable error of 2 per cent., the optical properties of such a leaf will be subject to the analysis given in § 10 above.

Thus, for example, if we assume that, in a gold leaf as it leaves the beater, the gold is in an amorphous state, its colour by directly transmitted light is that for which T0, as given by equation (23), namely

T0 = Muexp {-4ml. riK/X} (23),

is a maximum. If, further, the metal has its normal specific gravity, so that /j. = 1, the values of nx/K and of M0 in this equation are those given for p, = 1 in Tables VIII. and IX., or figs. 8 and 10.

Now, when /JL = 1, the value of n/c/A. is much smaller for blue and green than for yellow and red, and is slightly smaller for blue than for green ; while the value of M0 is greater for green and blue than for yellow and red, and is considerably greater for green than for blue. Thus both the factors of T0 in equation (23) are greater for blue and green than for yellow and red. The former colours therefore predominate in the transmitted light. Further, in very thick films, for which n/c/A. is of supreme importance, blue should, in the transmitted beam, predominate over green ; while in thinner films, on account of the greater value of Mu for green than for blue, green light should be more intense.

FARADAY found that all his gold leaf appeared olive-green by transmitted light.

Again, Table IX., or fig. 12, shows that the reflecting power, R0, is, when p = 1, much greater for red and yellow than for green and blue ; and this result is again in accordance with the observed colour of gold leaf by reflected light.

FARADAY,§ however, states that gold leaf still appeared green by transmitted light when its thickness was reduced to only 10 JU./A or 5 pp,. Now equation (24) of the former communication, || namely,

T0 = l-47rd.n2K/\,

* Reprinted from the 'Phil. Trans.' in his 'Researches in Chemistry and Physics,' p. 391. References will be made to the pages of this reprint." t Glasgow, 1903. t Loc. tit., p. 394. § Loc. tit., p. 395. || 'Phil. Trans.,' A, 1904, p. 408.

278 MR. J. C. MAXWELL GAKNETT

shows that, for a film of such thinness, the intensity T0 of the transmitted light is greatest for red light.* This red colour has been seen both by FARADAY! and by BEILBY| in parts of their green films. FARADAY says the red colour was extremely faint but appeared to have an objective reality, while BEILBY describes the effect as that of " an irregular film of pink jelly."

It appears that extremely thin films of gold are, by surface tension, drawn up into green patches, leaving larger areas covered by an almost transparent, but faintly red, film. The effect on the unaided eye is that of a transparent green.

The silver leaf used by BEILBY was over 300 p,^ thick. It therefore comes well inside the range for which the analysis of § 10 applies. Now Table VIII. , or fig. 9, shows that, for amorphous silver of normal specific gravity (//. = 1), «/c/X is least for the more refrangible rays. Again, Table IX., or fig. 11, shows that, for p. = 1, M0 is greatest for the same rays. It follows therefore, from equation (23) above, that, on both these accounts, the light transmitted by silver leaf should be blue ; and, in fact, silver leaf transmits a deep blue light. The approximately equal values of the reflecting power, R0, shown in Table IX., or fig. 13, for /A = 1, correspond to the almost colourless reflection from polished normal silver.

Consider now the colour changes which, according to figs. 8 to 13, deduced from the calculations of § 10, should accompany a diminution in the density of gold and silver films from its normal value (/u. = 1) to zero (p. = 0). This diminution of density may be conceived either as an increase of the distance between adjacent molecules or as due to the aggregation of groups of neighbouring molecules into small spheres. For geometrical considerations show that so soon as two spheres form adjacent to one another in an otherwise amorphous mass of metal, the density of the mass must begin to diminish. And it has been shown that the calculations in question are applicable whether the metal is in small spheres or in an amorphous state, and thus when it is partly in the one condition and partly in the other.

Taking first the case of gold, it appears from figs. 8 and 10, in conjunction with equation (23), that, as /x begins to diminish from unity, the absorptions of red and yellow light increase rapidly, owing both to the increase of HK/\ and to the decrease in M0. Meanwhile, owing to the decrease of the ratio of M0 (green) to M0 (blue) and to the increase of (wic/X) (green) — (w/c/X) (blue), the relative intensity of green to blue in the transmitted beam diminishes. Thus the first effect is to make the transmitted light bluer, and this effect continues until //. = about 75. As p continues to diminish below this value, the absorption of red rapidly decreases until, at /u, = '68, in a very thick film,§ the absorption of red has become as small as that of blue. The film is

* Cf. Table IV., p. 406 of former paper, and Table VIII. above, t Loc. cit., p. 400. I Loc. cit., p. 40.

§ When exp ( - iird . WK/A) is the dominating factor in T0. The corresponding value of /* is less in thinner films for which M0 is very important.

ON COLOURS IN METAL GLASSES, ETC. 279

then purple. As p. still further diminishes, the relative absorption of red continues to become small, so that the film becomes pink. Finally, at p. = 0, the absorption of green is less than that of blue, and the colour has changed from pink to red.* It is further seen, from fig. 12, that the reflecting power E,0 has, as p, began to diminish, become more yellow. At p, = '60, when the colour of T0 is still purple or pink, the colour of R0 has become green ; and thenceforward R0 remains green as p. diminishes from '6 to zero.

Similar consideration of figs. 9 and 11 shows that in the case of silver, as p. begins to diminish from unity, the colour by transmitted light becomes at first bluer, then changes to purple in the neighbourhood of p, = '8, and thence, through pink, to red or " amber " as p, further diminishes to /A = 0. In fact, it appears from the four colours for which calculations have been made, that there is, for any value of p., a well-defined absorption band at some position in the (visible or invisible) spectrum, and that, as p. diminishes, the position of this absorption band moves from the infra-red through the visible spectrum towards the shorter wave-lengths, being at X = -630 for p, = about '83, at X = "589 for p, = about '80, at X = '500 for p, = about '69, and at X = '450 for p. = about "55. Fig. 13 shows that the colour of the reflected light becomes distinctly blue at about p = '75, and remains blue down to p. = 0.

With a view to determining what may be the explanation of the colours and changes of colour exhibited by gold and silver films, we have now to compare the latter colours with those which we have found above to be consequent upon a mere isotropic change in density.

BEiLBYf has prepared gold films by using paints in which " the metal had been brought into solution in an essential oil." Having smoothly coated a plate of glass or mica with the paint, he heated it to a temperature of about 400°, thereby driving off the oil and other volatile constituents. A film of pure gold with full metallic reflection, and transmitting green light, is left adhering to the glass.

When these films are kept at a high temperature for some time, they change colour. By transmitted light, the original olive-green colour becomes at first bluer, then changes to purple, in which, as the annealing process is still continued, the red predominates more and more over the blue, until finally the purple has given place to pink. The reflecting power of the film has, meanwhile, diminished. But the colour of the light reflected from the blue films remains yellow, while the pink films reflect a green colour. I have before me a gold film prepared in this way and subjected to lengthy annealing. By transmitted light it appears striated with pink and blue bands. By reflected light the blue striae become golden, but the pink striae green. Under the microscope the film appears continuous, and is quite thick.

These colour changes, both with transmitted and with reflected light, are just

* Or yellow, if the colour is faint. See the second footnote on p. 243 above. - t Loc, cit., p. 40.

2go MR. J. C. MAXWELL GARNETT

those which have been shown above to be consequent upon a continuous diminution in the density of a gold film, which throughout remains either amorphous or "granular" (i.e., possessing a microstructure of small spheres). The view that the film is initially amorphous or granular, and that heating diminishes its density, is supported, as has already been pointed out,* by the fact that the curves in figs. 8 and 10 show that the absorption of light increases rapidly as /* begins to diminish from unity, while BEILBY'S films exhibited just such an increase of absorptive power when first heated. This view is also in accordance with the loosening of structure which is suggested by the great decrease in electric conductivity which accompanies heating. But direct evidence of the correctness of the view that heating produces decrease in density is not wanting, for BEILBY! has estimated the thickness of a film which had been annealed to the purple stage. He found, by weighing the gold from a given area, that, had the density of the gold been then normal (p = 1), the film would have been lGO/x/.t thick, whereas, under the microscope, the thickness seemed to be much greater than this. The density of the gold in the purple film thus appeared to be less than in the normal green films.

We conclude, therefore, that (a) the films, as first prepared, are amorphous or granular in structure; and (/*) heating diminishes the density of the film, while pressure is able to increase the density ar/ain.^

Further, BEILBY found t that, when the heating of a film was continued after it had reached the purple stage, " the film assumes a frosted appearance by reflected light and becomes paler by transmitted light." The frosted surface appeared, under the microscope, to consist of granules at least lOO/x^a in diameter. This phenomenon suggests that in the earlier stages of annealing, smaller granules were formed, which, as annealing proceeded, ran together to form larger granules : and the formation of such minute granules, while, according to our analysis, it does not aft'ect the optical continuity of the film, will explain the diminution in density which occurs on heating. It is, therefore, most probable that (c) tl\e diminution in density produced by heating is effected by the passage of metal from the amorphous to the granular phase and the growth of the larger granules at the expense of the smaller; and the increase in density produced by pressure may be accompanied by the passage of metal from the granular to the amorphous phase. \

The optical properties of the films of gold whicli FARADAY produced by reducing that metal from its solution by means of phosphorus, tend to show that these films are composed of amorphous or granular gold of density less than the normal. The films appeared to consist of pure gold ;|| when first prepared the films appeared of a grey

* 'Phil, Trans.,' A, 1904, p. 415.

t IMC. cit., p. 41.

I Of. the effect of pressure on FARADAY'S " phosphorous " films after heating— see next page.

§ [Note added 31st August, 1905. — Subsequent analysis has, however, shown that a sufficient flattening of the granules would cause the colours of the standard metal (//. = 1) to be exhibited.]

|| FARADAY, loc. cit., p. 408.

ON COLOURS IN METAL GLASSES, ETC. 281

colour, which was frequently resolvable into a mixture of green and amethystine striae. These colours would be shown by an amorphous or granular film for which the density was in parts as low as p. = 7. Moreover, such a structure agrees with the fact that " the films did not sensibly conduct electricity " and that " the films cannot be regarded as continuous."* FARADAY further statest that, though they are certainly porous to gas and to water-vapour, the films have evident optical continuity.

Heating diminished the conducting power and changed the colour to amethyst or ruby, just as with BEILBY'S films, pressure, which we should expect to increase the density of the film, changed the transmitted colour to green and increased the reflecting power ; and these are precisely the changes which would, according to calculation, accompany an increase in /A to the neighbourhood of unity in the case of an amorphous or granular film.

Closely allied to these phosphorous films are the deposits of gold on glass which FARAD A Yj obtained " by deflagrating a gold wire by explosions of a Leyden battery." " There is no reason to doubt that these deposits consisted of metallic gold in a state of extreme division." This method of preparing these deposits is similar to BREDIG'S§ method of obtaining suspensions of gold in water ; it is, therefore, to be expected that the deposits consist of small spheres of gold .together with some large crystallites. The films were so discontinuous as to be unable to conduct electricity ;|| but they were such as to present an optical continuity.il FARADAY sums up their colour changes as follows :—

" Fine gold particles, loosely deposited, can in one state transmit light of a Hue-grey colour [/j. = about • 8], or can by heat be made to transmit light of a ruby colour [/* < • 7], or can by pressure from either of the former states be made to transmit light of a green colour,** all these changes being due to modifications Of gold as gold and independent of the presence of the bodies on which for the time the gold is supported."

It appears, therefore, that the conclusions (a), (b), (c), arrived at on p. 280 for BEILBY'S films, are also applicable to FARADAY'S "phosphorous" films and to FARADAY'S " deflagration " films.

One more experiment of FARAD AY'stt on coloured gold deposits remains to be noticed. When a drop of solution of chloride of gold is evaporated in a watch-glass until the gold is reduced, a portion of the gold is generally found to have been carried by the vapour on to the neighbouring part of the glass. This part has the ruby tint ; and we have seen that a ruby tint is characteristic of the light transmitted by

* Loc. cit., p. 407.

t Loc. cit., p. 439.

J Loc. cit., p. 401.

§ Cf. above, p. 252, and footnote, p. 255.

|| Loc. cit., p. 402.

U Loc. cit., p. 439.

** Probably /A = 1 nearly ; but see fourth footnote on p. 280, above, ft Loc. cit., p. 428. TOL. COV. — A. 2 O

282 MR. -I. C.' MAXWELL GARNETT

amorphous or granular gold, the density of which is in the neighbourhood of '6 of that of normal gold.

The similarity of this method of preparing a metallic film with \L < 1 to that by which Professor R. W. WOOD prepared the sodium and potassium films, described in § 12 of the former communication,* is deserving of notice, and, from a different standpoint, tends to confirm the view there expressed as to the physical nature and structure of Professor WOOD'S films.

The conclusions (a), (b), (c) arrived at above (p. 280) as to the effect of heat and pressure on metallic films do not apply only to gold, as the following observations on silver films show. FARADAY! obtained silver films by reducing silver from a solution of the nitrate. The thinner parts of these films transmitted light of a " warm brown or sepia tint [//, < '8]. Pressure brought out the full metallic lustre and converts the colour from brown (ju, < '8] to blue [p. > '8]." The behaviour ot these films corresponds to that of the gold films obtained with phosphorus. Again, ft. W. WOOD| prepared films by chemically depositing silver on glass. These films, as originally prepared, show the same reddish-brown colour by transmitted light, and have a good blue-green reflection. It has been shown above that both these colours are characteristic of amorphous or granular silver, for which p. is appreciably less than -8. These films showed no electrical conductivity ; § so that, as in the case of BEILBY'S gold films, || the evidence of a loose structure afforded by the colours exhibited is confirmed by the evidence from conductivity.

12. The Exceptional Case of Beaten Metal Leaf.

There is one class of metallic film which, when heated, does not exhibit the colour changes that, according to our calculations, correspond to a gradual diminution in the density of the film. To such films the conclusions (a), (b), (c) of p. 280 do not directly apply. Instead of being obtained from finely divided metal by chemical deposition, deflagration, &c., the films in question are prepared by beating sheets of the solid metal into thin leaves.

FARADAY IF observed that heat caused gold leaf to lose its olive-green colour and silver leaf to lose its deep blue colour, the films at the same time becoming more

* Loc. cit., p. 412.

t Loc. cit., p. 409.

| 'Phil. Mag.,' August, 1903. The silver was prepared by the method of CAREY LEA (' Amer. Journ. of Sc.,' 1889). A further memoir on WOOD'S silver films is now in course of preparation.

§ Of. BARUS and SCHNEIDER, ' Zeitschr. f. Phys. Chem.,' VIII., p. 285, 1891, who attempted to measure the conductivity of a silver film prepared by CAREY LEA'S method, and found that, so soon as a drop of the silver suspension dried, so that the charged particles of silver could no longer move about, the conductivity of the drop vanished.

|| See above, p. 280.

f Loc. cit., p. 395 et seq.

ON COLOURS IN METAL GLASSES, ETC. 283

transparent and tending to shrink during the process.* Thus a silver leaf which before heating was opaque, or only able to transmit deep blue light, and that very feebly, was so altered by heating that the light of a candle could be seen through forty thicknesses.! But in every case the original colour of the leaf, whether of gold or of silver, returns when the leaf is subjected to pressure.

The differences between the effect of heat on chemically prepared films and on beaten leaf correspond to differences between the laminatedf structure of the leaf and " the closer and more horn-like texture of the films deposited by chemical agents.''^

The optical properties of a laminated metal leaf may be estimated and compared with the corresponding properties of an amorphous or granular film of the same metal, if the optical constants of a plate built up of a number of flat spheroids§ with their polar axes normal to the plate can be calculated. The general problem of the transmission of electromagnetic waves by a medium composed of a number of minute similar and similarly situated ellipsoids, distributed at random many to a wave-length, has now been solved, and it is hoped that the discussion of the optical properties of gold and silver leaf, of the change in those properties which is produced by heat, and of the relations of metal films (spheroidal, granular, and amorphous) to polarised light, may form the subject of a future memoir.

With these exceptions, namely, the properties peculiar to beaten leaf and the relations of metal films to polarised light, all the experimental relations of gold (and other metals) to light, which FARADAY described in his Bakerian Lecture have now been discussed, and we are led to the conclusion that the phenomena exhibited— whether by chemically or electrically deposited films, or by particles of gold diffused in glass, jelly, or water — are due to different groupings of the metal molecules and to variations in the mean distance between adjacent molecules, and in no case are they due to allotropic modifications of the molecules themselves.

13. CAREY LEA'S "Allotropic" Silver.

In the former communication] | it was suggested that CAREY LEA'S "allotropic" silver was in reality only finely divided silver, the division being sufficiently fine to admit of the films being optically continuous. 11 He advances** two principal arguments

* Loc. cit., p. 396.

t FARADAY, loc. cit., p. 399.

t BEILBY. loc. cit., p. 43. The difference in structure is shown by the fact that while mercury will diffuse slowly and uniformly in the compact film, in the leaf thin streams of mercury may be seen shooting rapidly in all directions.

§ BEILBY (loc. cit., pp. 48 et seq.) has shown that a layer of exceedingly flat spheroids is generally found on the surface of a metal.

|| 'Phil. Trans.,' A, 1904, p. 419.

U It is not necessary to suppose the microstructure of the finely divided silver to be granular, as was done in the former paper. It may be in part granular and in part amorphous.

** Fide ' Amer. Journal of Science,' 1889, and 'British Journal of Photography,' March, 1901. Also 'Phil. Mag.,' vols. 31, 32 (1891).

2 O 2

284 MR- J- C. MAXWELL GARNETT

for the allotropy of silver in the form in which he prepared the metal. We proceed to examine these arguments.

In the first place, then, all CAREY LEA'S silver films were prepared from silver suspensions. He claims that these suspensions were "true solutions," and that the ability of the silver to remain in solution in water was evidence that the molecules of the silver in question differed from those of normal silver, or, in other words, the silver was in an allotropic form. We are now, however, familiar with the fact that particles of normal silver, as of many other metals, are able, in consequence of mutual electrostatic repulsions,* to keep themselves in suspension in quite pure water. Again, we have seen that, when a silver solution is prepared by BREDIG'S method, its refractive index is that which is possessed by a suspension of small spheres, but not of molecules, of silver in water, f and in the same case there is a strong absorption band at exactly that point of the spectrum at which small spheres, but not molecules, of silver in water would produce a maximum ;J so that in this case the greater part of the silver is certainly present in the form of small spheres. Further, if, when prepared by deflagration, silver in suspension in water takes the small sphere form, it is primd facie probable that it does the same when obtained by CAREY LEA'S method,§ and this probability is increased by the fact that CAREY LEA'S silver suspensions exhibited the same red, yellowish-red, and yellow colours which are shown by BREDIG'S suspensions of different densities.

We conclude that CAREY LEA'S " solutions of allotropic silver " consisted of small spheres of normal silver in suspension. ||

We should therefore expect that the films obtained by CAREY LEA would be similar in constitution and behaviour to BEILBY'S " gold paint " filmsH and to FARADAY'S phosphorous films.** This leads us to CAREY LEA'S second argument for the allotropy of his silver ; he states :—

"The brittleness of the substances B and C [blue and gold coloured respectively, by reflected light], the facility with which they can be reduced to the finest powder makes a striking point of difference between allotropic and normal silver. It is probable that normal silver, precipitated in fine powder and set aside moist to dry gradually may cohere into brittle lumps, but there would be mere aggregations of discontinuous material. With allotropic silver the case is very different, the particles dry in optical contact with each other, the surfaces are brilliant, and the material evidently continuous. That this should be brittle indicates a totally different state of molecular constitution from that of normal silver." ft

* See footnote p. 253 above, t See above, p. 258. t See above, p. 257. § Above p. 259.

Cf. also the fact that the silver in a silver-stained glass is in the form of small spheres. || Cf. also evidence given on p. 259 above. H See above, p. 279.

** See above, p. 281. This expectation is verified by a further examination of WOOD'S films. See note above p. 282.

tt 'Brit. Jour. Phot.,' March 1901, p. 21.

ON COLOURS IN METAL GLASSES, ETC. 285

All these properties are shared by FARADAY'S " phosphorous " gold,* so that our expectation is, so far, fulfilled. We are, in fact, perfectly familiar with " mere aggregations of discontinuous material " which are optically continuous — for example, gold ruby glass.

Many of the observations which CAREY LEA has recorded on the colours of his silver films are in accordance with the expectation that these films, like BEILBY'S gold films and FARADAY'S " phosphorous " gold, should behave according to the laws (a), (6), and (c) stated above. But two difficulties arise in the way of this accordance, for, in the first place CAREY LEA'S recorded observations do not sufficiently distinguish between transmitted and reflected light. For example he recordst that his freely precipitated silver dissolves to a blood-red colour, and proceeds

"When the substance is brushed over paper and dried rapidly it exhibits a beautiful succession of colours. At the moment of applying it it appears blood red| ; when half dry it has a splendid blue colour and lustrous metallic reflection;! when quite dry this metallic effect disappears and the colour is matt blue."||

Lastly, in the case of the films discussed in § 11 above, the colour depended on the fact that the density of the film was less than that of the metal composing the film when in its normal state ; but pressure increased the density to its normal value, at the same time bringing out the normal colour, both by reflected and by transmitted light, of the metal. And CAREY LEA'S silvers " show a lower specific gravity than that of normal silver ; "1 and pressure "instantly converted gold-coloured allotropic silver into normal silver."**

We conclude from the above evidence that this silver was not " allotropic," but consisted of normal silver in a finely divided state.

14. HERMANN VOGEL'S Silver. Before leaving the consideration of these discontinuous forms of silver, reference

O

must be made to a paper by HERMANN VoGEL,tt in which the author describes how

* " The least touch of the finger removed the film of gold. . . . These films, though they are certainly porous to gas .... have evident optical continuity " (FARADAY, loc. cit., p. 439). Of. also the facts that films analogous to CABBY LEA'S did not conduct (BARUS and SCHNEIDER, loc. cit., p. 285), and that the phosphorous films did not sensibly conduct electricity (FARADAY, loc. cit., p. 407).

t 'Brit. Journ. Phot.,' March, 1901, p. 19.

\ This is the colour by transmitted light when /* is fairly small. Cf. figs. 9 and 11.

§ This is the reflected colour for values of p from zero to nearly -8. Cf. fig. 13.

|| Professor R. W. WOOD repeated this experiment, using glass instead of paper to support the silver film. The metallic effect, then, does not disappear, but remains after the film has become quite dry. Cf. above, p. 282.

H 'Brit. Journ. Phot.,' March, 1901, p. 21. ** 'Phil. Mag.,' vol. 31, p. 244, 1891. tt 'Pogg. Ann.,' CXVIL, p. 316, 1861.

286 MR, J. C. MAXWELL GARNETT

he prepared silver of less specific gravity than that of normal silver, by depositing that metal on the platinum electrode of a platinum-zinc battery. He also prepared silver in suspension in water by chemical means, observing the characteristic amber colour and noticing that precipitation could be accelerated by the addition of salt to the water.

VOGEL concludes (loc. cit., p. 337) that there are three forms of silver, (1) regular dendritic silver [crystalline] ; (2) granular powdery silver [small spheres] ; (3) mirror silver [amorphous]. He found that the second type " tended to the formation of a coloured powder," but could be changed into the third type by pressure. He adds (loc. cit., p. 441) that the silver precipitated by photography is of the second type, and this is the view suggested in the preceding memoir (p. 417), because of the red- brown transmitted colour and the green colour of the reflection from fogged photo- graphic films, which, according to the analysis given above, § 10, are the colours exhibited by films of amorphous or granular silver,* of less than standard density.

15. Allotropic Forms of Metal.

In the course of the preceding investigations we have been led to recognise that variation of the relative position of the molecules of a metal will cause the metal to change colour, whether it be examined by reflected or by transmitted light. It has been shown, for example, that mere variation in density causes gold in one state to transmit green light, in another blue, in another purple, and, in another again, ruby. Further, this discovery has led us to the conclusionf that, in order to account for the properties of CAREY LEA'S anomalous silvers, it is not necessary to assume the existence of an " allotropic " molecule of silver. The question thus arises : Are there any other cases in which an allotropic molecule has been unnecessarily postulated ?

EGBERTS- AUSTEN^ has collected particulars of a large number of supposed cases of allotropic § states of metals. We proceed to the examination of these particulars in order to determine whether the effects, for the explanation of which the allotropic molecule was postulated, are not merely those which, according to the analysis of § 10 above, would be due to a decrease in the density of the metal in a granular or amorphous state.

In the first place, then, the discovery that metals in different states, corresponding to different methods of preparation, possessed different densities and had widely different properties, although chemical analysis could detect no change in the

* Cf. figs. 9, 11 and 13, and also p. 282 above, where the same colours, exhibited by one of E. W. WOOD'S silver films, are discussed.

t Above, p. 285.

J ' Metallurgy,' pp. 87 el seq.

§ ROBERTS- AUSTEN defines "allotropy" as follows (loc. cit., p. 89): "The occurrence of elements in .... allotropic states means that .... the atoms are differently arranged in the molecules."

ON COLOURS IN METAL OLASSES, ETC. 287

composition,* does not require those different states to have been allotropic. Again, it is unnecessary to suppose that BOLLEY'S lead,t prepared by electrolysis, and similar in composition to sheet-lead, is allotropic because it oxidises rapidly in air while sheet- lead does not : for the electrolysis gives the essential fine division, and the consequent large amount of surface exposed to the air greatly accelerates oxidation.

Lastly, SCHUTZENBERGER| supposed that the copper deposited on the platinum electrode of a copper- platinum cell was allotropic because it was very fragile, its density was only about '9 of that of normal copper, it oxidised rapidly in air, and it could be converted into normal copper by prolonged contact with dilute sulphuric acid. Here, too, the supposition of allotropy is not required to account for the facts. For the low density, the fragility and the rapid oxidation are all accounted for by the loose structure which we should expect in such a deposit of copper, while CAREY LEA found that his silvers, which, if our conclusion at p. 285 is correct, were only finely divided silver, could be transformed to normal silver by contact with sulphuric acid. Similar remarks apply to SCHUTZENBERGER'S silver.^

Consider now MATTHIESSEN'S important generalisation^ that metals may sustain change in their molecular condition by union with each other in a fused state. ROBERTS- AUSTEN points out|| that the evidence that metals ever assume allotropic states, when they enter into union with each other, is difficult to obtain. When obtained, the evidence is generally composed of the facts that the specific gravity of the normal metal is greater than that of the metal in the state alleged to be allotropic ; that the chemical activity is less in amount, although the same in kind, for the former than for the latter state ; and that the appearance of the metal is different in the two states. Reference is also sometimes made to a difference in physical properties which is accounted for by lack of continuity, and consequently of electric conductivity, in the supposed allotropic state. IT Occasional reference is also made to a readiness to form hydrates which the metal in the latter state exhibits. Setting this last property aside, as not yet established, the remaining evidence is not conclusive, for all the facts in question are also characteristic of optically continuous granular (or amorphous) pieces of metal. Increase of chemical activity, for example, is a consequence of the enormous effective surface in a medium built up of independent granules.

Further, when one metal is united with another in a fused state, a chemical compound is not, in general, formed, but the molecules of the two metals freely mix. Thus one metal is in solution in the other. So long, therefore, as the temperature remains sufficiently high to permit the molecules to move about freely, the molecules of each metal tend to segregate, and to group themselves into separate crystals as the * JOULE and LYON PLAYFAIR, 'Memoirs of the Chem. Soc.,' vol. iii., p. 57 (1846).

t EOBERTS-AUSTEN, loc. tit., p. 90.

J 'Bull. Soc. Chim.,' XXX., p. 3 (1878).

§ ROBERTS AUSTEN, loc. cit., p. 87.

|| Loc. cit., p. 91.

IF Of. PETERSEN on "Allotropic Forms of Metals" (' Zeitschr. f. Phys. Chem.,' 8, pp. 601, 1891).

288 MR. J. C. MAXWELL GARNETT ON COLOURS IN METAL GLASSES, ETC.

temperature is slowly lowered. It is, however, probable that, as in the case of gold and copper ruby glasses, the molecules of each metal first group themselves into small spheres. If the temperature were rapidly lowered at this stage, this granular structure would be fixed in the alloy. If, then, one metal— that, suppose, of which the larger volume is present — were suddenly annihilated, the other metal would remain in a granular form, possessing a colour* quite different from that exhibited by the normal form of that metal.

Now when an alloy of potassium and gold containing about 10 per cent, of the precious metal is thrown on to water, the potassium is, in effect, annihilated, t and the gold is released as a black or dark brown powder. It will be seen from fig. 12 that granular gold, with a density slightly over '6 of that of normal gold, would reflect light of a brown colour, while the reflecting power would not exceed '5. A granular structure is thus in accordance with the dull appearance and with the colour of the powder. Similarly when a silver-gold alloy containing two parts of silver to one of gold is treated with nitric acid the silver is removed, the gold remaining in the form of a dull brown powder, which can be converted into bright metallic gold by slight pressure or by heating to redness. It appears, therefore, that this brown powder is probably granular gold, the component particles being small compared with a wavelength of light ; so that, once more, the evidence J does not require us to suppose this form of gold to be allotropic.

Finally, it seems unnecessary to assert that iron released from its amalgam by distilling away the mercury is in an allotropic form because it takes fire on exposure to the air. For this burning of the iron would be the consequence of the large surface exposed to the air by an extremely finely divided form of the metal.

We conclude, therefore, that in none of the cases of supposed allotropy, which we have examined in this section, has the existence of an allotropic form of metal been established.

* See § 10 above.

t Of. ROBERTS-AUSTEN, loc. dt., p. 91. The potassium does not catch fire, but combines with the water to form KHO (which immediately passes into solution and is thus removed) and H which catches fire.

t We must except that of the alleged formation of auric hydrate, but I have been unable to obtain any confirmation of the existence of such a compound.

VIII. On the Intensity and Direction of the Force of Grarity in India.

By Lieut.-Colond S. G. BCJRUABD, H.E., F.R.K.

Received March 30, — Read April 13, 1905.

['PLATES 14-20.]

(1.) The Pendulum Observations of /.sv/,-7-7'.^.

BETWEEN 1865 and 1873 observations were taken at 31 stations in India by Captains BASEVI and HEAVISIDE with the Royal Society's seconds pendulums. The results were published in Vol. V. of the ' Account of the Operations of the Great Trigonometrical Survey of India,' and have been subsequently discussed by many authorities.*

Captain BASEVI expressed his results in terms of N, the number of vibrations of the mean pendulum observed in a mean solar day. The International Geodetic Association show their results in dynes, and it is desirable that we should follow their example. We have, therefore, to change the notation employed by our predecessors.

The fundamental formula, expressing the relation between the length of a pendulum, its time of vibration and the accelerating force g, is t = TT \/(l/g)- If N be the number of vibrations, which a pendulum of length / makes in a mean solar day of 86,400 mean time seconds, then

M _ 86400 _ 86400 /g

T~ ~ \f 1 '

v 77" r '

where t is the time of vibration.

If N becomes N + c/N, when g becomes g + dg, then

>

By this formula, if certain values of N and g be adopted for a Standard Station, the results of the older pendulum observations can be converted, and the symbol g substituted for N.t

The pendulum observations in India were undertaken, and are now being extended, with the object of determining the difference between the force of gravity as observed

* See 'Phil. Trans.,' A, vol. 186, 1895; HELMERT'S 'Die Schwerkraft im Hochgebirge ' ; HELMERT'S 'Hdhere Geodiisie'; CLARKE'S 'Geodesy'; FISHER'S 'Physics of the Earth's Crust.' t dg = 0'0226(/N is a rough rule, sufficiently accurate for many purposes. VOL. CCV.— A 394. 2 P 12.10.05

290

LIEUT.-COLONEL S. G. BURRARD ON THE

at the standard stations of Europe and as observed in India ; the determination of the absolute value of the force of gravity did not and does not form any part of the operations.

The values of gravity exhibited in Table I. are taken from Professor HELMERT'S Report to the International Geodetic Conference, which was held at Paris in 1900.

TABLE I. — -BASEVI'S and HEAVISIDE'S Results Expressed in Dynes.

	Station. Latitude.	Longitude.	« I 3 E	Observed value.	Correction for unevenness of ground.	i 0-*. II i£ + ^	<70 — attraction of the mass above sea-level = ga".	Tlleoretical value.	-£ I "*	A 1 ft
			Metres.	.'/ eentims.	3' -9-	eentims.	eentims.	Yo eentims.	eentims.	eentims.
	0 / Punnae .... . . . ' + 8 9 '5	+ 77 37'- 7	15	978 -095	•000	978-100	978-098	978*105	-0*007	-0*005
		+ 77 41 '5	51	978 *090	0	978-108	978-100	978*105	5	+ 1
		+ 73 0 '0	2	978 '191	0	978-192	978-191	978*108	+ 83	-r 84
<J		+ 76 17 '6	2	978 •!«>	0	978-167	978-166	978-141	+ 25	+ 26
o	• Mangalore +12 51 '6	+ 74 49*6	8	978-231 978 '237	0 0	978-235 978 -239	978 -234 978 -239	978 -257 978*266	- 23 - 27	- 22 - 27
			3	978 '4 17		978 "448	978-447	978*441	+ 6	+ 7
	Colaba Observatory (Bombay) . . +13 53 -8	+ 72 48-8	11	978-605	0	978-608	978-607	978 *545	+ 62	+ 63
	Mallapatti + 9 28 '8 Pachapaliam . +10 59 '7	+ 78 O-H + 77 37 '5	88 296	978 -091 978 -084	0 0	978-118 978-175	978-108 978-140	978*141 978 *189	- 33 - 49	- 23 - 14
	Bangalore South . . + 13 0 '7	+ 77 35 '1	950	977-998	0	978 -289	978-179	978*263	— 84	+ 2«
	Bangalore, North + 13 4 '9 Namthab&l . . . . . +15 5 '9	+ 77 39-3 + 77 36 '5	917 358	978 -018 978 -"07	0 0	978-299 978-318	978 -193 978 '275	978*266 978 -352	- H	+ at - 34
		+ 77 38 '5	584	978 '213	0	978*461	978 '394	978-451	— "7	+ 10
	Damargiila +18 3 '3	+ 77 40'1	593	978 -283	0	978'484	978-396	978-499	- 103	- 35
			522	Q78 -402		978 '563	978 "502	978 -555	— 53	-r 8
i			342	978 "539	o	978'642	978 • ••(•3	978*651	— 48	— 9
M	Cak-utta, Survey Office .... +22 32 '9	+ 33 21T> + 77 40 '9	6 516	978 -776 978 "674	0 + 2	978-778 978 "832	978-777 978*774	978*764 978 *833	+ 13 — 59	+ 14 - 1
			538	478-7*3'{				978*867	42	+ 21
							978*835		88	— 30
	Usira + 26 57 -1 Datairi + '•'8 44'1	+ 77 37'9	247 *J18	978-972 q;q -QQ5	+ I	979 "048	979-021 979-137	979*067 979*200	- 46 63	- 19 — 38
	Kalh'tna + 29 30*9	+ 77 T-t'2	247	979-107	I		979*154	979*260	— 106	— 77
	Nojli . . +29 53 '5		269	979-110	o	979-198	979-167	979*290	— 123	— 92
	Meean Meer . . +31 31 •&	+ 74 23 '3	•>15	979 -273		979 -339	979*314	979 *420	— 108	— 81
is	Dehra Dun Observatory .... + 30 19'5	+ 78 3-3 + 78 4*4	683 2109	978 -962 978 '751	+ 7 + 27	979-172	979*100 979*181	979*324 979*335	- 224 154	- 152 + 65
X*'	More +33 ljj'7	+ 77 52*0	46%	978-137	+ 9	979*580	979 *044	979 *562	— 518	•*. 18

EXPLANATION OP SYMBOLS EMPLOYED. For fuller details as to the manner in whirl* these numbers are derived, see the explanation of Table II.

/ 2H\ /H + R\2

?V' + ~K I ''ppresonts g \ % / ; the third place of decimals in expressions for gravity at high stations may differ by two or

three units according to the form of the formula used.

g = the value of the force of gravity as observed at the height H, the value at Kew being assumed 981 -200. g' = the observed value of gravity reduced to an infinite horizontal plain of height H. g' - g = topographical correction due to the irregular distribution of mass in the vicinity of the station. ga = the observed value of gravity reduced to sea-level for height only.

ga" = the observed value of gravity reduced to sea-level both for height and for mass above sea-level.

y0 = the theoretical value of gravity computed from HKLMERT'S formula of 1884, namely, 978 -000 eentims. (1 + 0 -005310 sin" •(>). 9o" - Yo = load variation of gravity from the normal, as computed by BOUQUKR, and as used for the determination of mountain-compensation. ?o - Yo = 'oca' variation of gravity, as used by HELMKST in his determination of the Figure of the Garth.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 291

The differences between the observed and computed values in Table I. correspond very nearly to the differences between the observed and computed values of N, as formerly given by General WALKEK. That the correspondence is not exact is due to the adoption by HELMERT and WALKEK of different constants in CLAIRAUT'S law.

The physical meaning of BASEVI'S pendulum results was for many years the subject of controversy.* The deficiency of gravity which he had found to exist in Himalayan regions was attributed by some authorities to the elevation of the level surface above the surface of the mean spheroid, and by others to the defective density of the under- lying crust; by the former the surface of the geoid was held to depart largely in certain places from that of the spheroid, and by the latter the two surfaces were assumed to be almost identical. In his ' Schwerkraft im Hochgebirge,' published in 1890, Professor HELMERT gave a mathematical solution of the problem, and his writings have closed the controversy.

A graphical interpretation of the results of Table I. is given in Plate 14, the method by which the several ordinates are computed being explained in Table II. below. The first figure of the Plate shows the height above sea-level, as determined by spirit levelling, of the surface of India along its central meridian. The second figure shows the deficiency of matter in the underlying crust, as deduced from BASEVI'S pendulum results. The third figure gives the differences between the ordinates in the two upper figures, and shows the surface of India as it would be if the crust were everywhere of equal density. An examination of the figures of this Plate brings to light four significant facts :—

(1) That there exists in the earth's crust throughout India a general deficiency

of matter as compared to Europe ; t

(2) That the apparent excess of matter above sea-level, which the eye observes

at More (Station 43) under the form of mountains, is largely compensated by subjacent deficiencies ;

(3) That an extraordinary deficiency of matter underlies the stations of Dehra

Dun, Kaliana and Nojli (Nos. 38, 37, 36), stations situated not in the Himalayas, like Mussooree (No. 41), but in the plains at the foot of the Himalayas ; this deficiency leads one to beliere that the pressure of the Himalaya Mountains upon the crust is diminishing the density of the latter under the surrounding plains ;

(4) If we disregard the evidence of fig. 1, and if we consider only the distribution

of mass in the surrounding crust, we see that stations in the plains of

* See preface to Vol. V. of ' Account of Operations of the Great Trigonometrical Survey of India.'

t The peninsula of India is composed of crystalline and volcanic rocks ; the great age of the former and

the great weight of the latter would lead us to expect a high value for g ; that g should be abnormally

small is, from a geological point of view, surprising.

2 P 2

292 LIEUT.-COLONEL S. G. BURRARD ON THE

Northern India, such as Nos. 36 and 37, are situated in a deep wide valley between two ranges of mountains, one of which, the Himalayan, is visible, the other, with its summit at Station 24, invisible.*

The northern end of the section in Hg. 3 conveys the idea that the Himalayan mass is pressing upon the crust and producing a dimple, such as that described in Chapter VII. of Professor GKOIWSK DARWIN'S work on 'Tides and Kindred Phenomena.'

The sections given in tigs. 2 and 3 of Plate 14 are based on Professor HKLMERT'S condensation theory and have been constructed by means of his formulas from the data in Table II. The numbers of the stations are not continuous, because pendulum observations were not taken at all the astronomical stations.

After 1874 no pendulum observations were taken in India, but the deflection of the plumb-line continued to be determined in different parts of the country. By the year 1900 the astronomical latitude of 159 stations, the astronomical azimuth at 209, and the amplitude of 55 arcs of longitude had been observed, and thus a large amount of evidence relating to the direction of gravity had accumulated. A discussion of the datal; then available showed that it would be desirable to associate determinations of the intensity of the force of gravity with observations of the plumb-line, and in 1902 the Indian Government sanctioned the re-opening of pendulum observations and the purchase of a new apparatus of VON STERNECK'S pattern.

(2.) Tin1 Pendulnin Observations of 1003-04.

The new apparatus was standardised at Kew and Greenwich in the autumn of 1903, and was taken to India by Major LENOX CONYNGHAM in November of that year. Upon its arrival he thought it advisable to commence work at some of BASEVI'S stations. The accuracy of BASEVI'S results, as given in Tables I. and II., had been questioned by Professor HELMERT in his report to the International Geodetic Conference of 1900. It had been there pointed out that the observer had had no means of measuring the flexure of the pendulum stand, that during his standardisation at Kew his pendulums had not been supported on the stand subsequently used in India but between a stone pillar and a wall, and that when he visited the high Himalayan station of More he had substituted a light portable stand for that belonging to the Royal Society's apparatus.

* Fig. 1 of Plate 14 shows that the altitude of Station 38 above sea-level is 145 metres greater than that of Station 24; fig. 3 shows that if the underlying crust were brought to a uniform density of 2-8 the altitude of Station 38 would be 1430 metres less than that of Station 24. The visible fall of nearly 500 feet from Station 38 to Station 24 is converted by the pendulum diagrams into a rise of nearly 4*700 feet.

t 'Professional Papers of the Survey of India,' No. 5 of 1902. "The Attraction of the Himalaya Mountains upon the Plumb-line in India,"

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 293

TABLE II.

			u
S			Hal -3. §5
11	Station.	Latitude.	11*5
5 - •5 °			._Q^..._ ..
a			Millinis.
i	Punnap	0 / 8 9 '5	0
	Kudankolain . . .	M Kl-4	1
4	Mallapatti. . . .	9 29 '0	108
5	Pachai>aliain . . .	10 59-7	227
1)	Bangalore, South .	13 0-7	388
7	Bangalore, North .	13 4-9	391
10	Namthabad . . .	15 5 -II	555
12	Kodangal ....	17 8'0	718
13	Damargida . . .	18 3-3	791
15	Somtana ....	HI 5-0	874
17	Badgaon ....	2ll 4I'I	1006
20	Ahmadpur. . . .	23 36-4	1238
21	Kalianpur ....	21 7'2	1276
29 31	PahArgarh. , . .	24 56'1	1342 1503
	Datairi	28 4 1 ' 1	16(6
38		29 30 '9	17O9
37	Nojli . . .	29 53 '5
38	Dehra Dun . . .	30 HI '5	1771
41 43	Mussooree. . . . More	30 27'7 :t3 15 -7	1 783 2008

+3 1)	S-J	l| M_.|	11 -D.
j H = Hei ^ ' above sea- 1 i	II to scale ill millims. 0-1	£3 i :1 ~ta '-- a! £~^ M<1-- ",nii!i±
Metres.	To scale in millims. for fig. 3.
15	7 +61 + 0-5	- 46	- o-l
51	n -1	5 + 43 + H '3	+ 8	+ o-l
88	0'7	— :13 + 2xi| + 2 '3	- 1118	- 1'6
29fl	2'1	- 49 + 121 + 3'l	- I2«	- I'd
950	7-6	-84 + 727 + 5'8	+ 223	•f 1 '8
917	7-3	- 73 + 032 + 5-1	+ 2X3	+ 2 '2
355	2-11	- 77 + 6H7 + 5-3	- 3011	- 2-1
581	4'7	- 57 + 193 + 311	+ HI	+ II 'S
5113	4'8	- Iii3 + S02 +7M	- 2911	- 2-3
522	1 '2	- 53 + 159 + 3 '7	4- l'3	-f n "5
312	2'7	- 48 + 111. + 3-3	- 71	- M 'H
516	I'l	- 511 -f 511 + I'll	-t- 5	+ ii'l
538	4 '3	— 12 + 361 -I- 2-H	+ 171	+ 1-1
500	4'0	- 88 + 762 + 6-1	- 2li2 - 2-1
217	2-0 - 46 + 398 1- 3'2	- 151 - 1 -2
218	1-8 - 63 + 515 + I'l	- 327 , - 2'6
217	2'll - lilt. H- HIS + 7'3	- 671 - 5'3
261)	2'1 - 123 + 1(165 + 8-5	- 7H6	- «'l
883	5-5 - 221 + 19311 + 13-3	-1256	- lil'O
2109	16-9 - 154 + 1333 4- 10-7	+ 876	+ 6 '2
469H	37 '6	- 318 1 + 4484 ; + 3.V9 1	+ 212	+ 1'7 l

EXPLANATION OF TABLE II.

Given the amount of matter in tlie crust at a stauJaril station, we wish to find from pendulum observations, the excess or deficiency of matter underlying any other station ; from observation we find ilrj, the local variation of gravity from the normal, and we wish to determine the mass whose attraction at sea-level is equivalent to dg. From its attraction only we cannot determine both the height and density of a hidden mass, but If we assume that the density is equal to 2 '8, the normal density of surface rocks, we can then ascertain the height; by this assumption we. mean that the density of a hidden disturbing mass is 2 -S in excess of the normal density of the surrounding crust . The problem to lie solved is, therefore : given a small attraction dij, what is the height of the attracting mass, its density being 2-8?

It is necessary to consider how dij is obtained ; by observations taken at a station of height H we find the value of gravity to IK; y. To oblain the corresponding value of gravity at sea-level, <70, we have firstly to correct for the amount H, by which the distance of the station from the

centre of the earth exceeds the earth's radius, 2l' =

fj

R-

0 = ij (\ + —\. \ R /

This correction would be sufficient if the obser\ ing station were in mid-air and over the wean, but when we observe at a station on land, wr have to consider the attraction of that portion of the crust that lies between sea-level and the station ; this attraction tends to increase Un- observed value of g, and the correction for it is ncgat ive. The attraction of a horizontal plateau of height H and density S upon a pendulum situated at the centre of its upper surface is A = 2*611. The force of gravity at sea-level is g = JirRA, where A is the mean density of the earth.

Then if ya" be the value of gravity at sea-level corrected Iwth for height of station and for the attraction of the intervening mass, we get the

well-known formula of UOUGUEK, ya" = ya — A = ij ( 1 + — - ^-|f ). (/„" gives then the obsen-eit value of gravity at an ideal station, situated upon

\ R 4 K /

a continent, whose surface is level with the sea. Now <tg = ga" — fa, where •/„ is the theoretical value of gravity. To lind the height of a plateau whose attraction would be sufficient to

increase the observed force of gravity by 0-001 centim., we have — . — . y = dg = O'OOl. II = O'OOl x ; x — . Assuming the earth txi be a

4 K y

sphere with a mean radius of 6367000 metres, and the mean value of the force of gravity to lie 980-0, we get II = O'OOl x J x ** l( _ _ = S'S573 metres.

The attraction thus of a plateau of height 8 '6373 metres wilt increase the observed value of gravity by O'OOl, and vice versa ; if the observed value of gravity at sea-level differs from the theoretical value by -t-'O'OOl there is an excess of matter in the underlying crust equal to a disc 8 '6573 metres thick of a density 2 '8.

If we imagine that from the surface to a depth 1), the density of the crust underlying the station is less by 2 '8 than the normal surface density, then D = - (#," - yj 8 '8573 metres. The visible excess of matter will be equal to H (see fig. 1), the hidden deficiency will be equal to D (see fig. 2), and the actual disturbing mass, shown in the section of fig. 3, will be (H — D).

Prom Table II. it appears that at MorA the value of ga" is 0 '518 less than y0 ; therefore the hidden deficiency = D = 518 x 8 '«573 = 4484 metres (fig. 2). The height of the visible mountain at Mor6 is H = 4696 (fig. 1) ; the actual excess of matter in the crust at More = (H - D) = 212 metres (fig. 3).

At Dehra Dun <y0" - yu> = - 0'224, hidden deficiency = D = 224 x 8-8573 = 1939 metres (fig. 2). The altitude of Dehra Dun is «S3 metres (fig. 1) ; at this station, then, the hidden deficiency exceeds the visible excess, and the resultant is (H - D) = 683 - 1939 = - 1256 metres (fig. 3).

At the important station of Kalianpur (</„" — ya) = — 0 '042, the hidden deficiency = D = 42 x 8 '6573 = 364 metres, the visible excess at Kalianpur = H = 538 metres. There exists, therefore, at Kali:in pur a resultant excess of matter in the crust equal to a disc of density 2 '8, and pf height 174 metres. The existence of this excess has been questioned, and the calculation is therefore giren in detail.

294

LIEUT.-COLONEL S. G. BUERAED ON THE

From the results of observations taken by Austrian observers at some of the coast stations, Professor HELMERT had arrived at the conclusion that BASEVI'S values required a correction of +0'047.* The importance of such a correction cannot be overestimated ; it would have indeed the effect of largely neutralising the negative character of the values of (.(/„"— ytl) and of (H — D) in Tables I. and II., and it would render the value of (#o"— yo) f°r our standard station of Kalidnpur actually positive. Such a correction would lower the line of sea-level as drawn on figs. 2 and 3 of Plate 1 4, but would not otherwise affect the sections in these figures.

Major LENOX CONYNGHAM' s first station in India was Dehra Dun; his results there were astonishing, for they showed that BASEVI'S value was no less than 0'103 centim. too small. t LKNOX CONYNGHAM then visited Calcutta, Bombay, Madras, and Mussooree. At Calcutta observations were rendered impossible by the ceaseless vibrations of the ground, which proved sufficient to cause the pendulums, if left suspended at rest, to oscillate visibly in a few minutes ; this effect on the pendulums was produced in whatever plane the latter were swung. LENOX CONYNGHAM had therefore to abandon Calcutta without obtaining any results ; that lie failed where BASEVI had succeeded was probably 'due to the half-seconds pendulums of the new apparatus being more affected by earth-vibrations than the old seconds pendulums.

* The correction for More was indeterminate, but probably larger than 0-047, owing to the lightness of the stand employed.

t This extraordinary difference could only mean that BASEVI'S final value of N was too small by 4 whole seconds of time. BASEVI'S observations at Dehra Dun lasted four months, and included 234 independent sets of swings taken at pressures varying from half-an-inch to 28 inches, and at temperatures varying from 48° to 102" Fahrenheit.

	H.	9-
BASEVI'S 1st determination	seconds 86,021 '38 86,020 '74	centiins. 978 -973 978 -959
„ 2nd „
Weighted mean
86,020 -86	978 -962
LENOX CONYNGHAM in January, 1904
	979 -063 979-066
„ „ June, 1904 . . . . . .
Mean
!	979-065
	i
Difference between BASEVI'S two determinations =0 '014 centiin. „ „ LENOX CONYNGHAM'S two determinations =0 '003 „ „ BASETI and LENOX CONTNGHAM =0 '103 centim.	centim.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 295

THE Force of Gravity in Dynes as observed by —

	BASEVI and HEAVISIDE in 1866-73.	LENOX CONYNGHAM in 1904.	Difference.
Dehra Dun .	centims. 978-962	centims. 979-065	centims. + 0-103
Madras ...	978-237	978-281	+ 0 ' 044
Bombay ....	978-605	978-632	+ 0-027
Mussooree	978-751	978-795	+ 0-044

LENOX CONYNGHAM'S observations confirm Professor HELMERT'S prediction that BASEVI and HEAVISIDE'S results would be found too small. The sections in figs. 2 and 3 of Plate 14 of this paper have been based on their results, and it may be asked what purpose has been served by the construction of sections from impugned data ? The answer to this question is that BASEVI'S results have been accepted by geodesists and have formed the basis of controversies and theories ; they have, too, been rendered historic by the difficulties and death of the observer at More", and by the great light they undoubtedly threw upon Himalayan formation. Now that pendulum observations are being re-opened, I have thought it advisable in an historical retrospect to give a graphical summary of the results that were formerly obtained, and that have so profoundly influenced the ideas of geodesists.

In figs. 2 and 3 the deficiency underlying Dehra Dun (38) will be reduced by almost one-half if LENOX CONYNGHAM'S value be substituted for BASEVI'S. Similarly the height of Mussooree (41) in fig. 3 will be almost doubled.

In the near future BASEVI'S other stations will possibly be visited ; it seems certain that his results will everywhere be found too small, that throughout fig. 2 the curve of deficiency will have to be raised, and that in fig. 3 the line of sea-level will have to be lowered.

From LENOX CONYNGHAM'S observations at Bombay and Dehra Dim, it appears that BASEVI'S and HEAVISIDE'S results are not in error by any constant quantity, and that the error of each will have to be separately determined ; it is not easy to account for the variation in the magnitudes of their errors ; their observations were taken with a care that it is difficult for us to equal ; in assuming that flexure could be prevented by the employment of a rigid stand, the old observers were following the highest authorities of their time ; the only faults that have been found with their work are such as would tend to produce constant error. That their errors vary so largely can only, I think, be explained on the supposition that the flexure of the wooden stand of the Eoyal Society's apparatus was influenced by temperature and humidity.

The idea that gravity is exceptionally weak throughout India as compared to

296 LIEUT. -COLONEL S. G. BURRARD ON THE

Europe can no longer be upheld ;* the so-called " marked negative variation" of many writers has been found to rest on erroneous data.

The theory of the compensation of the Himalayas has been based to a large extent on the old pendulum results at Mussooree and More". The sections in figs. 2 and 3 show that a hidden deficiency of matter underlies the station of Mussooree (41) equivalent to about three-fifths of the visible excess ; LENOX CONYNGHAM'S recent result reduces this hidden deficiency to one-third only of the visible excess.

Figs. 2 and 3 might lead to the belief that the Himalayas at More' (43) are almost entirely compensated. The height of the visible excess is 4696 metres, the depth of the ideal deficiency 4484 metres. But LENOX CONYNGHAM has not visited More, and, as BASEVI employed there a special and lighter stand, it is impossible to gauge the error introduced into his result by its flexure ; we have lately gained some idea of the effects of the flexure of the lloyal Society's heavy stand, and we can only suppose that the light More" stand was less rigid. That the Himalayas at More are compensated to a considerable extent is certain ; that the error due to flexure could have affected BASEVI'S result to the extent of 22 seconds of time is out of the question. On the other hand, it is more than probable that the compensation, that does exist, lacks that completeness, which has hitherto been considered among its most remarkable features.!

(3.) Deflections of the Plumb-line.

In 1895 General WALKER published an admirable classification of the deflections of the plumb-line that had been observed in India.j His object was to present the data in the form of arcs of meridian and parallel for the use of mathematicians investigating the values of the earth's axes.

In 1898 Great Britain joined the International Geodetic Association, and Professor GEORGE DARWIN, F.ll.S., was nominated to represent her at International Conferences. These steps have brought India into touch with modern European ideas, and have shown vis that the aims of geodesy are no longer limited to the measurements of arcs of meridian and parallel, and to the determinations of the axes of a mean spheroid. At the International Conference, held at Copenhagen in 1903, the following resolution was passed : —

" II est desirable qu'on fasse dans les Indes anglaises une etude approfondie de la repartition de la pesanteur, tant dans les contrees montagneuses que dans les plaines.

* No standard value of g has as yet been adopted by the International Geodetic Association. When the absolute values of gravity at European standard stations have been finally determined, it may be found that the values at Kew and Greenwich, which we are now accepting as o'ur standards, are not themselves normal. Both BASEVI'S old and LENOX CONYNGHAM'S new values will then have to be corrected by a constant quantity.

t CLARKE'S ' Geodesy,' p. 350.

| 'Phil. Trans.,' A, vol. 186, 1895.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 297

" Attendu que c'est seulement par cette etude qu'on pourra obtenir une repre"- sentation exacte de la distribution des masses dans 1'ecorce terrestre et de la forme du ge"oide dans ces contre"es."

In India itself our view of the subject has been modified by our recent discoveries that the direction of gravity is liable to a constant deflection throughout large regions, and that the density of the earth's crust may differ constantly from the mean surface value throughout great areas. In 1895, when General WALKER'S paper was written, it was believed that deflections of the plumb-line were accidental and due to small local pockets of exceptional density studding every part of the country. It was considered proper to treat deflections by minimum squares,* and it was held that the true direction of the normal to the mean figure could be discovered by grouping stations round a centre, and by assuming that in the mean of the group the effects of local attraction are cancelled.

There are now grounds for believing that the direction of gravity may be deflected through 8 seconds of arc or move over an area of thousands of square miles. To assume, therefore, that its mean direction as deduced from a group of contiguous stations coincides with the normal, is seen to be hardly more justifiable than to assume that the mean direction of the magnetic needle, as observed at several stations in Surrey, gives the true direction of north.

The investigation of the laws governing the deflection of gravity in India has been impeded by many difficulties. Political considerations have erected a barrier round Nepal and Bhutan, which geodetic operations have been unable to pass. Nepal and Bhutan include almost the whole of the central and southern Himalayas. Geodesists wish to approach the Himalayas from the south, and, by working gradually towards their centre of mass, to discover their influence on the plumb-line. Being excluded from Nepal and Bhutan, they have had to attack the mountainous area at its south- west salient at Dehra Dun (see Plate 16).

They have, moreover, been generally confined to deducing the direction of gravity from latitude observations, which give only the meridional component. It is true that our longitude observations show the direction of gravity in the prime vertical, and if we could observe both the latitude and longitude of points on the Himalayan snows, it would be possible to calculate the actual direction of gravity from its two measured components. But until wireless telegraphy can be utilised for longitude determinations, our longitude stations will have to be located near telegraph offices instead of on mountain tops. We have observed astronomical azimuths at numerous stations and their results will in the future be available for plumb-line discussions, but

* When arcs of meridian are employed to determine the figure of the mean spheroid they are not regarded as fixed in latitude. Their most probable positions in latitude are found by the method of minimum squares. Each arc is moved up or down its meridional ellipse until a position is found for it in which the squares of the deflections of the plumb-line are a minimum ; by this method large deflections may be eliminated that exist in nature.

VOL, CCV.— A. 2 Q

298

LIEUT.-COLONEL S. G. BUEEAED ON THE

the geodetic azimuths are at present affected by the errors accumulated in the tri- angulation, which have not as yet been determined. Whilst, then, we are endeavouring to discover the influence on the plumb-line of a mountainous mass situated to the north- east, we are limited to observations which give the north and south component only.

The other difficulties attending plumb-line research are, that our deductions are based upon an assumed figure of the earth and upon an assumed direction of gravity at a station of origin. We have to imagine a mean spheroid, and we then assume that the angle of inclination between the surface of this spheroid and the actual level surface at any place is equal to the deflection of the plumb-line ; we have also to select some station as an origin, and to assume that the surfaces of the spheroid and geoid are there parallel. We have finally to decide from the results accumulated over wide areas, whether the fundamental assumptions on which those results are based — the assumptions of spheroid and origin — are correct.

In the publications of the ' Survey of India ' the deflections of the plumb-line have been always based (1) on the mean spheroid of EVEREST, and (2) on the assumption that gravity acts normally at Kalianpur, our geodetic origin. In his paper on ' Geodesy,' published in 1895, General WALKER gave the deflections of the plumb- line in terms of the spheroid of CLARKE.

EVEREST'S spheroid had agreed closely with BESSEL'S ; but the objection had been raised to both that their values of the ellipticity, 1/300'80 and 1/299'15, differed too seriously from the value 1/289 derived by CLAIRATJT'S theorem from pendulum observations. In 1880, in his work on ' Geodesy,' Colonel CLARKE deduced an ellipticity of 1/293 '4G5 from measures of arcs, and of 1/293 from pendulum results; and his removal of the hiatus gave great weight to his spheroid. Professor HELMERT'S investigations have, however, shown that modern pendulum work has not borne out CLARKE'S result, and that BESSEL'S ellipticity was after all nearer the truth.

Recent geodetic measurements have tended to confirm the accuracy of CLARKE'S value of the major axis, and to indicate that BESSEL'S value was too small.* Until a new determination of the dimensions of the mean spheroid has been made under the authority of the International Geodetic Association, it is advisable for us to adopt for computations a spheroid that has the major axis of CLARKE and the ellipticity of BESSEL.

ELEMENTS of Spheroids.

	Major axis in metres.	Ellipticity.
BESSEL	6 377 397	1/299-15
EVEREST ....	6 377 193	1/300 '80
CLARKE . .	6 378 190	1 /29V 47
CLARKE-BESSEL . . .	6 378 190	1/299 '15

'United States Coast and Geodetic Survey.' "The Transcontinental Triangulation, 1900 ;" "The Eastern Oblique Arc of the United States," 1901.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 299

In Tables III. and IV., given hereafter, the deflections are shown in terms of the Everest, the Clarke, and the Clarke-Bessel spheroids.

If we compare the deflections of the plumb-line as referred respectively to the Everest and Clarke spheroids, we find that the values are almost identical at all stations. The agreement between the two series, though very remarkable, is a mere coincidence ; the influence of CLARKE'S increased ellipticity happens always in India to neutralise the influence of his increased major axis.

If we employ the Clarke-Bessel spheroid, the deduced deflections of gravity are appreciably modified.

(4.) The Regional Classification of Deflections.

It was in 1900 that the suggestion was first made that the deflections of gravity in India, which had hitherto been attributed to accidental and local attractions, could be broadly classified by regions. This new theory had as a working hypothesis an advantage over the old in that it could be tested by further investigation in the field. From the classification of results of regions it was predicted that a southerly deflection of gravity would be found to exist throughout a great zone enclosing the main valley of the Ganges and running parallel to the Himalayas for 1000 miles; but that both north and south of this zone northerly deflections would be met with (vide Plate 15).

With the object of testing the correctness of these predictions, Lieutenant COWIE, R.E., proceeded in 1901 to observe several latitudes between Calcutta and Phallut, working across the zone of southerly deflection and up to the Himalayas (vide Plate 15). The results which he obtained were as follows: — In the country immediately south of the zone northerly deflections of 3" and 4" were found ; at Calcutta the inclination of gravity was slightly southerly. In the 200 miles immediately north of Calcutta, COWIE found southerly deflections at four successive stations ; the inclination of gravity then changed to northerly, at Jalpaiguri it was 6" northerly, at Siliguri 23", at Kurseong 51", and at Phallut 37".

In 1902-03, Lieutenant COWIE was directed to work again northwards across the zone and to follow the meridian of 79°. The results which he obtained were as follows : — In latitude 23° 30' the direction of gravity was inclined 5" towards the north ; in the next 200 miles Lieutenant COWIE found a southerly deflection at seven successive stations ; in latitude 27° 47' the inclination of gravity began to be slightly northerly; in 29° 16' its inclination was 12" northwards. At Birond, in the hills, Lieutenant COWIE found a deflection of 44" north.

It can, therefore, now be prophesied with tolerable certainty that on all Himalayan meridians the direction of gravity will be found to follow one general law ; in the neighbourhood of the tropic, as we move northwards, its direction will change from northerly to southerly; it will then remain deflected towards the south for some

2 Q 2

300 LIEUT.-COLONEL S. G. BURKAKD ON THE

hundreds of miles, and it will again become northerly as the Himalayas come into view.

In spite, therefore, of the fact that the true direction of gravity at any one place cannot be determined with certainty, yet it is possible now to classify deflections of the plumb-line in India by regions. A modification of the spheroid of reference may alter values and may move the regional boundaries, but it will not affect the general correctness of the classification. A change in the assumed value of the direction of gravity at the station of origin will alter all deduced deflections of the plumb-line by the same amount, but it will not affect their differences, nor the mean differences between regions.

I propose now to show :

(1) The classification of stations by regions.

(2) The effects on the classification of changes in the spheroid of reference.

(3) The effects on the classification of the existence of a deflection at the origin.

(4) The final values of deflections of gravity, corrected for errors of spheroid and

origin.

In Plate 1 5 India has been divided into four regions :

(1) The Himalayas, (2) the zone of southerly deflections, (3) the Indian Peninsula, (4) North-west India.

In the following four Tables III A., IIlB., IIIc., IIIo., which correspond to the four regions, the direction of gravity at Kalianpur has been assumed to be coincident with the normal to the spheroid, and has been adopted as the datum. The deflections of gravity are given in the columns headed (A — G) ; the symbol A denotes the astronomical or observed value of latitude, G denotes the geodetic value of latitude, which has been calculated through the triangulation extended from the origin over the spheroid.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 301

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INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 303

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INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 305

TABLE H!E. Summary of the four preceding tables.

Region.	Number of stations.	Mean deflection of the plumb-line.
Everest spheroid.	Clarke spheroid.
{mountains . . plains .... Zone of southerly deflection . . . Indian peninsula	19 23 43 85 27	-35 ±1-15 - 5 ± 0-82 +3 ± 0-24 - 3 ± 0-28 • 1 ± 0-23	-34 ±1-19 - 4 ± 0-83 +3 ±0-24 - 4 +0-28 1 ± 0-23
North-west India

It will be seen that in the Himalayas the deflections are northerly and large, but that as we move southwards from the mountains they decrease rapidly — more rapidly in fact than the law of gravitation requires.

As we recede still further from the Himalayas we enter the positive zone, and here we find a region 1000 miles long and 200 broad running parallel to the Himalayas, throughout which the plumb-line is always deflected towards the south.

As we progress still further southwards we enter the Indian peninsula ; on crossing the boundary line between the 2nd and 3rd regions we find that the deflection of the plumb-line changes its direction and sign ; between latitudes 24° and 18°, from coast to coast, strong northerly deflections averaging 6" now prevail ; as we move south- wards towards Cape Comorin these northerly deflections slowly decrease, and in the extreme south of India change to southerly.

In North-western India the latitude observations have not brought to light any marked characteristic. This region is west of the Himalayas, and longitude deter- minations, if made at numerous stations, would be more likely than latitude observations to yield instructive results.

The opinion had been expressed that the large deflections of 30" and 40", discovered in the sub-Himalayas near Mussooree and Phallut, might prove to be local and exceptional, and that it was unsafe to assume them characteristic of the region. To test the correctness of this view, Captain COWIE observed for latitude in April, 1903, at the Himalayan Station of Birond, and found that the direction of gravity was deflected here 44" towards the north ; in November, 1903, Captain H. WOOD, R.E., observed for latitude at two stations in Central Nepal and met with deflections of 33" and 38". All the evidence that is slowly accumulating tends, therefore, to show

VOL. ccv. — A. 2 R

306 LIEUT.-COLONEL S. G. BURRARD ON THE

that these large deflections of gravity are not confined to exceptional localities, but prevail throughout a vast region.

In October, 1903, Captain COWIE was directed to extend the Great Arc of India northwards across the Mussooree hills to the snowy range, and to observe for latitude in the inner Himalayas. High authorities had expressed the opinion that the large deflections of gravity at Dehra Diin, Birond, and Phallut were due not to the Himalayan mass, but to the peculiar geological formation of its lower and outer range ; that these deflections would be found to disappear when the first Himalayan ridges were crossed, and that large southerly deflections would be met with in the inner Himalayas. Captain COWIE extended the Great Arc of India into the mountains from latitude 30° 29' to 31° 1', a distance of 35 miles, and he observed for latitude at the Himalayan stations of Bahak (9715 feet high), Bajamara (9681 feet), Lambatach (10,474 feet), and Kidarkanta (12,509 feet). Table IIlA. shows that large northerly deflections were met with at all these stations.

The form of the ideal section deduced in fig. 3, Plate 14, from pendulum results rather justified the belief that deflections would be found to decrease rapidly between Station 41 (Mussooree) and Station 43 (More). The northerly deflection of 30" now discovered by COWIE at Kidarkanta* consequently throws doubt on the correctness of that portion of the pendulum section that lies between these two stations, and confirms the opinion that a greater excess of matter exists at More" than has been deduced from BASEVI'S observations.

In Plate 17 is given a cross-section of the Himalayas, drawn by Captain COWIE, through the stations of Kidarkanta and More" ; this section is not ideal but real ; it shows the variations in the actual level of the ground, and illustrates the visible mountain mass separating the two stations ; the vertical scale is twenty times as great as the horizontal.

Plates 18. 19 and 20, drawn by Captain COWIE, give cross-sections of the Himalayas at Kidarkanta, Birond, and Phallut ; they illustrate the increase in elevation between the plains of India and the plateau of Tibet at three different places. In each the vertical scale is ten times as great as the horizontal ; the scales employed in these three last plates are larger than those used in Plate 17.

* As an observer penetrates a mountain range, he leaves more and more of the mountainous mass behind him ; the attraction of the portion left behind is then opposed to the attraction of the masses still confronting him, and tends to decrease the resultant deflection of his plumb-line. To determine the relative effects of the rearward and forward masses a contoured map is necessary.

INTENSITY AND DIRECTION OF THE FOECE OF GRAVITY IN INDIA. 307

(5.) The Adoption of a New Spheroid.

The deflections have so far been deduced from the Everest and Clarke spheroids only. It is now proposed to show the values that will be obtained if the Clarke- Bessel spheroid, as described above, be adopted.

In the Tables IVA., IVu., IVc., IVD., and IVE. is given the inclination at every station between the observed level surface and the surface of the Clarke-Bessel spheroid.

The inclinations are stated, firstly, when that at Kalianpur is taken as zero, and, secondly, when it is taken as +6". The reason for adopting this latter assumption will be explained in section (G) on the zero of verticality.

2 B 2

308

LIEUT.-COLONEL S. G. BURRARD ON THE

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INTENSITY AND DIRECTION OF THE FOECE OF GRAVITY IN INDIA. 309

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TABLE Summary of the four preceding tables.

Region.	Number of stations.	Mean deflection of the plumb-line Clarke-Bessel spheroid.
Referred to Kalianpur as zero.	Referred to Kalianpur as +6".
(" mountains . Sub-Himalayan < L plains . . . . Zone of southerly deflection . . . Indian Peninsula	19 23 43 85 27	// // -33 ± 1-24 - 3 ± 0-87 +3 ± 0-26 - 6 ± 0-23 0 ± 0-25	// // -27 ± 1-25 +3 ± 0-87 +9 ± 0-26 0 ± 0-24 +6 ± 0-24
North-west India

We can now judge of the effects of the substitution of the Clarke-Bessel spheroid for EVEREST'S by comparing the values given in the columns of Tables IVA., IVB., IVc., IVc., headed "Eeferred to Kalidnpur as zero," with the values given in Tables IIlA., IIlB., IIIc., HID. It will be seen that the large Himalayan deflections are slightly decreased, and that the positive tendency of the second region has been accentuated.

There is a marked difference between the values of Table IIIc. and the values "Referred to Kalianpur as zero" in Table IVc. The progressive decrease in the observed deflections, from latitude 24° to latitude 8°, as exhibited on the spheroids of EVEREST and of CLARKE, had led me to believe that the direction of gravity throughout Peninsular India was being influenced by some external excess or deficiency of mass, such as the Himalayas or the Indian Ocean.* The southerly deflections, shown in Table IIIc., at the extreme south of India, were attributed by General WALKER to the condensation of submarine strata, t The introduction of the Clarke-Bessel spheroid eliminates at once both the progressive decrease and the supposed southerly deflections in South India, and substitutes for them throughout the peninsula a large apparent northerly deflection averaging G". The introduction of the Clarke-Bessel spheroid shows that the progressive change exhibited by Table IIIc. in the inclination of the level to the spheroidal surface from latitude 24° to latitude 8° was due, not, as I had supposed, to the deformation of the level surface, but to the abnormal curvature of the surface of EVEREST'S spheroid.

* Professional Paper No. 5 of 1901, "Survey of India;" Monthly Notices, 'Royal Astronomical Society,' January, 1902.

t 'Phil. Trans. Roy. Soc.,' vol. 186, 1895.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 313

(6.) TJie Zero of Vertically.

Tables IVA., IVs., IVc., IVo., and IVE. show in terms of the old datum, namely, with Kalianpur as zero, the angles of inclination in the meridian that have been determined in different parts of India between the level surface and the surface of the Clarke- Bessel spheroid. The difference between any two of these angles of inclination is affected only by changes of spheroid, but the absolute value of every angle is based on the assumption that the level and spheroidal surfaces are parallel at Kalianpur. Any alteration in the assumed inclination of the two surfaces at this our initial station will affect the inclinations as deduced at other stations by a constant quantity. The direction of gravity at Kalianpur has been adopted by the Survey of India as the datum, from which deflections of the plumb-line at all stations are measured ; the direction of gravity is, we know, always perpendicular to the level surface ; at Kalianpur it has been assumed to be perpendicular to the spheroidal surface also. I propose now to deduce a new value for the deflection of the plumb-line at Kalianpur, and to exhibit the values of the deflections of the plumb-line in India that will be obtained, if the deduced direction of gravity at Kalianpur be substituted for the original one — in other words — if our zero or datum be corrected.

The direction of gravity throughout the first, second, and fourth regions appears to be under the influence of abnormal attractions ; there is, I believe, no other area in the world in which the deflection of the plumb-line undergoes at once such large and such systematic variations as it does in the two first regions of Plate 15 ; these observed peculiarities, too, have been discovered to exist in the neighbourhood of extraordinary mountain masses, and though the connection between the observed phenomena and the visible protuberances is obscure, there can be little doubt that the latter are in some indirect way the cause of the former.

The direction of gravity in the fourth region also is probably influenced by the high mountains of Central Asia, though their effects are not directly perceptible. We will, therefore, omit from present consideration the results obtained in the first, second, and fourth regions, and we will confine our attention to those of the third region only.

The third region is in the form of a trigon with its apex at Cape Comorin ; its length from north to south is 1100 miles, and its greatest breadth 1300 miles; its area is 750,000 sq. miles. This trigon is one of the oldest portions of land surface now existing on the earth ; it is mostly composed of ancient gneiss, and though a large part was covered in the cretaceous period by volcanic overflows, it suffers now but slightly from earthquakes and is exceptionally stable. This trigon appears to be as free from abnormal sources of disturbance and to be as suitable for the determination of the absolute direction of gravity as any area of land can be. If we examine the results "Eeferred to Kalianpur as zero" in Table IVc., we find that out of 85 determinations of the direction of gravity made within the third region, 80 show a

VOL. CCY. — A. 2 S

314

LIETTT.-COLONEL S. G. BURRARD ON THE

northerly deflection, two show a southerly deflection, and three show the direction of gravity to be vertical. The mean deflection throughout the trigon is 6"'4 North.

Now if we are to accept these results as final, we shall have to believe that throughout the third region the level surface is always inclined by 6"'4 to the spheroidal surface. We know of no cause tending to produce such an extraordinary deformation, and we are led to suspect the reasoning by which its existence has been inferred. The only certain fact that has been brought to light by observation is that the plumb-lines in the trigon have a northerly deflection greater by 6"'4 than the plumb-line at Kalidnpur. We have, however, taken a step in advance of this safe ground, and have assumed that the direction of gravity at Kalidnpur is vertical, and that consequently the plumb-line throughout peninsular India is deflected 6"'4 towards the north. Would it not be more reasonable to assume that the mean direction of gravity throughout the third region is vertical, and that the plumb-line at Kalidnpur is deflected G"'4 towards the south ? The assumption of a southerly deflection of G"'4 at Kalianpur will lead then to the conclusion that throughout the third region the level surface remains generally parallel to the spheroidal surface.

From visible evidence Kalidnpur, situated as it is in flat plains, would be adjudged a suitable datum station, but it unfortunately lies in the zone of southerly deflection, and its plumb-line is thus exposed to the horizontal attractions of hidden masses.

If our geodetic operations had been confined to the third region, and if our datum station had been originally selected within this region, we should not have been led to suppose that its whole area of 750,000 sq. miles was abnormally affected. If we had subsequently extended our operations to Kalidnpur, we should have discovered there a southerly deflection of about 6", and this we should have adopted without question.

TABLE showing the Number of Observed Deflections in the Third Region —

If we assume that the meridional deflection at
Lying between —	Kalianpur is —
	0 .	+ G" (south).
-12-5 and -14-5	2	0
-10-5 -12-5	7	0
- 8-5 -10-5	13	0
- 6-5 - 8-5	18	2
- 4-5 - 6-5	23	5
- 2-5 - 4-5	10	15
0 - 2-5	8	22
+2-5 and 0	4	24
+ 4-5 „ + 2-5	0	9
+ 6-5 „ + 4-5	0	6
+ 8-5 „ + 6-5	0	2
Total ....	85	85

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 315

The southerly deflection at Kalianpur of G"'4. which has been deduced from plumb-line observations, is to a certain extent corroborated by the section drawn in Plate 14, fig. 3, from pendulum observations ; if we assume that the errors in BASEVI'S pendulum results will be found constant, the section will be raised with reference to the sea-level, but will not be otherwise affected ; and if we regard the distribution of mass exhibited by this section, and calculate the deflection at Kalidnpur by means of CLARKE'S formula (' Geodesy,' p. 298),

f /' l\<i&nW /,,\c«iu2<7 /7/\2A-|

A = P log. {(I) . (y . (~) \+2p{c'<j>' sin2 c/-c4, sin' cr},

we obtain a value of +5"'!.*

In Tables IVA., IVu., IVc., IVD., IVE., in the columns headed " Referred to Kalidnpur as 4-6"," the values of deflections have been exhibited on the assumption always that the plumb-line at Kalidnpur is deflected G" towards the south.

(7.) Summary.

A comparison of the two values given to each deflection in Tables IVA., IVs., IVc., IVD., and IVE. will illustrate the effects of the adoption of the corrected datum ; the large Himalayan deflections, it will be seen, have been slightly decreased ; they amount now to about half the theoretical values derived from an application of the law of gravitation to the visible mountain masses ; the sudden diminution of the large deflections at the foot of the mountains is still very remarkable.

The great zone of southerly deflection has been expanded both to the north and to the south, and it now includes many of the stations classified in the first and third regions; for instance, in the sub-Himalayan region (Table IVA.), the stations of Kaliana, Bansgopal, Jarura, and Jalpaiguri exhibit southerly deflections when the corrected datum is used; in the Indian Peninsula (Table IVc.) the positive zone has been extended southwards to Thikri, Ladi, Hathbena, and Chandipur ; in North-west India (Table IVo.) every station now presents a marked southerly deflection; and the positive character of the deflections in the positive zone itself (Table IVs.) has been strongly accentuated.

To the north of the second and fourth regions stand the mountain masses of Central Asia, but throughout those regions the direction of gravity is systematically deflected towards the south. That the direction of gravity should be deflected everywhere towards the south with a mean inclination of 8" throughout an area of half a million square miles (Tables IVB. and IVo.) is an extraordinary phenomenon of nature, and this phenomenon has been observed on flat low-lying plains bounded

* Attraction at Station 24 of mass lying north of Station 38= 1"-61 24 „ „ south „ 38 = 6" -67

5" -06

2 S §

316 LIEUT.-COLONEL S. G. BURKAKI) ON THE

on the north by mighty mountain ranges and tablelands. Deficiencies of density underlying and compensating the highlands, on whatever assumptions of depth they may be based, will be found insufficient to account either for the prevalence or mao-nitude of these southerly deflections; that the mountains and deflections are, however, in some way connected can hardly be doubted. The section in fig. 3 of Plate 14 perhaps justifies the inference that the general deflection of gravity towards the south is being caused by deficiencies, underlying not the mountains themselves, but the plains in the immediate vicinity of the mountains.

All our pendulum and plumb-line stations situated actually in the Himalayas have so far been located on peaks ; the results deduced have therefore been obtained from the highest points in the several Himalayan districts visited. It is important that observations should be taken at stations situated in the deep valleys of the inner Himalayas. The difficulty of fixing such stations by triangulation has hitherto limited observations to summits, but it is necessary now that we should ascertain whether the subterranean deficiencies underlying the Himalayas vary in amount with the heights of the superincumbent mountains, or whether in their compensation of the mass as a whole they remain independent of the altitudes of the alternating ranges and valleys above (see fig. 2, Plate 14).

Another question of interest has arisen, namely, whether the southerly deflections of the second region merge gradually along the border line into the vertically of the third region, or whether there does not exist an intermediate longitudinal area in which northerly deflections prevail. A study of Table IVc. will show, I think, that throughout a strip immediately south of the dividing line the deflections have a tendency to be uniformly northerly.* The cross-section in fig. 3 of Plate 14 shows an excess of mass to underlie Kalianpur (Station No. 24), and this excess is possibly a contributory cause both of the southerly deflections of the second region and the northerly deflections in the parallel strip. The continuance of similar deflections both to the east and to the west of Kalianpur lead me to think that the pendulum observations of the future will furnish on all Himalayan meridians cross-sections similar to that given in the figure.

Geodetical observations have shown that the density of the earth's crust is variable, but they have not given any positive indication of the depths to which these observed variations extend. All calculations of the effects of subterranean variations in density and of mountain-compensation have, therefore, to be based on arbitrary assumptions of depth. The fact that the plumb-line seems generally to respond readily to results given by the pendulum, perhaps justifies the inference that the observed variations in the density of the earth's crust are not deep-seated. If an abnormal amount of matter exists in the crust near the surface, it will exercise direct effects upon plumb-lines and pendulums in the vicinity, but if it lies at a great depth, its effects, especially on plumb-lines, will be less perceptible.

*Vide stations Chaniana, Valvadi, Badgaon, Aiikora and Mai.

INTENSITY AND DIRECTION OF THE FORCE OF GRAVITY IN INDIA. 317

We have not at present sufficient pendulum stations to warrant definite conclusions, but we can make use of those we have to test whether the observations of the intensity and direction of gravity tend to corroborate one another.* The cross-section in fig. 3 of Plate 14 gives the result of pendulum observations at stations on the meridian of 70° 30' ; the direction of gravity at these and intermediate stations is shown referred to Kalitlnpur as +6" in Table IV A. for all places north of latitude 27°, in Table IVfi. for places near latitude 25°, and in Table IVc. for all places south of latitude 24°. We can therefore institute the following comparisons :—

(1) The pendulum section would lead us to expect a large northerly deflection at

Station 38, and Table IVA. shows that at this station (Dehra Dim) the deflection is 29" north, f

(2) From Station 38 to Station 24 the pendulum indicates a gradual increase in the

density of the crust ; the plumb-lines confirm this increase in a remarkable manner.

(3) At Station 24 (Kalianpur) the pendulum indicates the existence of a greater

amount of subjacent matter than underlies the stations on either side of it. Tables I. and II. show this more clearly than the section. Now, if we look in Table IVi3., we find that from Kesri to Tinsia the plumb-lines are all deflected south towards Kaliiinpur, whilst if we look in Table IVc. we see that from Takalkhera to Badgaon the plumb-lines are deflected north towards Kalianpur. Thus the existence of an excess of matter in the crust indicated by the pendulum at Kalianpur is confirmed by the action of the plumb-lines on both sides of it.

(4) The pendulum section shows a considerable excess of matter to underlie

Stations G and 7 (Bangalore). Now, if we look at Table IVc., we see that the direction of gravity is much disturbed in the neighbourhood of Bangalore. At the two base-line stations near Bangalore, the deflections are northerly ; sixty miles north at Bommasandra the deflection is 8" southerly. The inference is that an intermediate excess of matter exists, and that a station could be found north of Bangalore at which the pendulum would indicate a greater excess than at Bangalore itself. (The section had to be drawn from station to station, but if intermediate observations were to be taken, it is certain that the maxima and minima of the section would be slightly moved.)

(5) Table I. shows that the force of gravity was below normal at all BASEVI'S

inland stations but Calcutta ; for this reason Calcutta has hitherto been

* It is true that the results of the old pendulum observations are not correct, but their errors are mainly systematic and though affecting absolute values do not vitiate differences. Differences are sufficiently accurate to justify the comparisons instituted.

t The names of the numbered stations of the section are given in Table II.

318 THE INTENSITY AND DIKECTION OF THE FORCE OF GKAVITY IN INDIA.

classified as a coast station. It is, however, 100 miles from the coast and is in truth less of a coast station than Kew or Greenwich. It was probably included amongst coast stations because BASEVI obtained there a positive result which accorded with his results at Madras and Bombay. But his positive result will, I think, be found in the future to be due not to Calcutta's proximity to the coast, but to her situation over the long chain of excessive density that is believed to run parallel to the Himalayas from west to east, and that is indicated in fig. 3 of Plate 14 by the position of Station 24.* If we examine the last columns of Tables IVB. and IVc., we see that the deflections are south at Calcutta and Dariapur, but north at Cuttack and Khundabolo.

(G) If an observer working over the plains of Northern India were to trust only to his eye and his level, lie would record the existence of a great mountain range to the north and of low hills or flat plains to the south ; if, however, he were to disregard the evidence of the eye and of level, and were to believe either his pendulum or plumb-line, he would come to the conclusion that he was standing between two mountain ranges, one of which, visible to the north, was rising abruptly out of the plains, whilst the other, invisible to the south, was slowly gaining in elevation for 300 miles.

I have taken several instances of abnormal pendulum results from Table I. and have found in each case a direct response from the plumb-lines at neighbouring stations. This conformity could hardly ensue if the variations in density extended to greater depths than 30 or 40 miles. Our results do not justify us in asserting that HO deep-seated variations in density exist, but they do justify the belief that the variations in density which have been discovered are apparently superficial.

* When I write of the excessive density of the earth's trust, I am judging from local observations only. I mean, therefore, " excessive " compared with surrounding portions of the crust, and not with the mean

surface density of the earth.

[ 319 ]

IX. On the Refractive Index of Gaseous Fluorine.

By C. CUTHBERTSON and E. B. R. PRIDEAUX, M.A.. JB.Sc.

Communicated by Sir WILLIAM RAMSAY, K.C.B., F.K.S.

Received June 5, — Read June 8, 1905.

THOUGH fluorine was isolated by M. MOISSAN as long ago as 1886, no attempt has hitherto been recorded, so far as we are aware, to measure its refractive index in the gaseous state. This omission is the more to be regretted since great interest attaches to the determination. Not only is fluorine the first member of an important group of elements, but its power to retard light, calculated from the refract ivities of its compounds, appears to vary within unusually wide limits, so that the estimates of its refraction equivalent are singularly discordant, and agree only in shmving that it must be remarkably low.

Thus, Dr. J. H. GLADSTONE* originally gave the refraction equivalents of fluorine and chlorine as 1'4 and 9 "9 respectively, figures which correspond to a refractive index for fluorine of TOOOIOS, or considerably less than that of hydrogen (1 '(100139). In 1885 1 he placed it at I'G. In 1886 G. GLADSTONE^ put down the refraction equivalent at between 0'3 and 0'8, and in 1891 the same observer, with Dr. J. H. GLADSTONE,§ estimated it as "extremely small, in fact, less than TO.'' More recently MOISSAN and DE\VAR,|| judging from the appearance of liquid fluorine, recorded their belief that the index would be found to be higher than had previously been supposed, though still low in relation to its atomic weight.

In these circumstances it seemed desirable to attempt to measure the index of the element in the gaseous state, and with this object Mr. CUTHBERTSON visited Paris in January, 1904, and, by the kindness of M. MOISSAN, was enabled to observe the index of a current of fluorine passing through a small hollow prism of copper, the apertures of which were covered by plates of fluor spar. A summary of this work has already

* 'Phil. Trans.,' vol. 160, p. 26, 1870. t 'American Journal of Science ' [3], XXIX., p. 57, 188-j. J 'Phil. Mag.' [5], XX., p. 483, 1885.

§ J. H. GLADSTONE and G. GLADSTONE. 'Phil. Mag.' [5], XXXI., p. 9, 1891. || MOISSAN and DEWAR, 'Proc. Chem. Soc.,' XXXI., p. 175, 1897. VOL. CCV. — A 395. 19.10.05

320 iMESSRS. C. CUTHBERTSON AND E. B. K. PRIDEAUX

been published,* but the following details are added in order that the value of the experiment may be criticised.

Table I. exhibits the results obtained in five experiments, performed on two

occasions.

TABLE I.

Number.	Date.	Refractivity 01-1)10".	Time of flow of gas.
1	January 13, 190-4	232	25 minutes.
')	13, „	228 and 226 Not recorded. About half-aii-hour additional.
3	20, „ 243	About half-an-hour.
4	20, „ 2-41	An additional quarter of an hour.
5	20, „	227	An additional half-hour, with another electrolytic tube.

These figures require some explanation and comment. The refractivity of fluorine is certainly much lower than those of oxygen and nitrogen, while that of all other elements (except hydrogen, helium, and neon, the presence of which need not be suspected) and, tt fortiori, of all compounds,! is higher. Consequently, when air is displaced from a prism by fluorine, the lower the index observed the nearer do we approach to that of the latter.

The first experiment recorded in Table I. may be discarded. During its progress an unaccountable change of zeros took place, which makes it doubtful whether the reading given above, or a lower one ('215), should be accepted. The balance of probability is in favour of the higher value.

When the prism had been swept out with dry air the second experiment was performed, and gave two trustworthy readings of 228 and 22G for the refractivity of the contents of the prism.

After making some improvements in the stability of the apparatus and substituting tubes of finer bore (about '2 millims.) for the old leads, a second attempt was made on the 20th January.

On this occasion a very trustworthy experiment gave a refractivity of 243, or 16 points worse than that of the second experiment of the 13th January; while a second trial, made after recovering the zero by sweeping out the prism with air, gave an almost identical result, 241.

It was then suspected that the electrolytic tube had developed a leak, and a new one was substituted.

The fifth experiment, performed with this apparatus, at once gave a refractivity of 227, which is nearly identical with the best experiment of the 13th January.

* 'Phil. Trans.,' A, vol. 204, 1905, p. 323.

t A molecule of HF probably retards light less than a molecule of fluorine, but since the molecule of this vapour, under normal conditions, is at least as complex as HoF-j, its presence in an atmosphere of fluorine would probably raise the refractivity of the mixture.

ON THE REFRACTIVE INDEX OF GASEOUS FLUORINE.

321

In all these trials a singular fact was observed. The index slowly decreased to a minimum, and, after remaining steady for a few minutes, retrograded by several points, indicating the presence of an increased proportion of some gas of higher refractivity. This effect was observed in nearly every subsequent experiment performed with the prism, and its significance will be referred to hereafter.

But, in spite of the concordance between the lowest values obtained on the first and second days, these experiments could not, for several reasons, be regarded as satisfactory. All previous estimates of the refractivity of fluorine, based on the refraction equivalent, point to a much lower value than 227 ; and, though this expression cannot be relied upon to give a very close approximation, its agreement with the refractivity is usually fair, and there is no other instance of so wide a discrepancy between the two as these figures would show.

In the second place, it was not certain that the current of gas employed, which was at the rate of l£ to 2 litres per hour, would completely displace, from the train of purifiers, a volume of 100 cub. centims. of air, whose density is not far removed from its own. And, thirdly, the retrograde motion of the index after reaching a minimum, and subsequent slight variations, definitely proved that the contents of the prism were not homogeneous.

For these reasons the authors determined to undertake a further investigation of the subject, using the apparatus with which Mr. PRIDEAUX had, by this date, succeeded in obtaining fluorine, by the method of M. MOISSAN. The form of the electrolytic tube employed may be seen in fig. 1. It was kept cool by means of a

Fig. 1.

mixture of alcohol and solid carbonic acid, in such proportions as to form a paste. The current ranged from one to two amperes. In two or three minutes the voltage usually ran up to its steady value, and, soon after this, a piece of blotting paper, wetted with alcohol, when held near the exit tube gave abundant fumes of HF and then burst into flame. When this was observed the U-tube was connected to the train of purifiers and prism or refractometer tube by means of a well-fitting platinum junction.

Many experiments were made with this apparatus, but the results were not more

VOL. ccv. — A. 2 T

322 MESSRS. C. CUTHBERTSON AND E. B. R. PRIDE AUX

concordant than those obtained in Paris, and suggested, as in that case, that the current of fluorine was not sufficiently rapid to displace completely the air in the coolers and tubes.

It was, therefore, decided to liquefy the gas as it was produced, and, when sufficient had been collected, to allow it to boil off rapidly through the prism. This was done with the arrangement shown in fig. 1, but the results showed no improvement, and it seemed probable that some oxygen, either from the air or some other source, was condensing with the fluorine, the boiling-points of the two elements being very nearly identical.

These experiments led to the conviction that it was practically impossible to obtain fluorine in a state of absolute purity.* And since no means could be devised for removing the gases by which it was accompanied without affecting the fluorine, it was decided to have recourse to an analysis of the mixture of gases whose refractivity was observed, and to correct for the impurities detected. For this purpose the prism method was found unsuitable and JAMIN'S refractometer was substituted for it.

The plan at first adopted was to displace the air in the refractometer tube by a current of fluorine, counting the interference bands as they passed across the field, owing to the change of refractivity. When a steady state was reached the tube was disconnected from the source of fluorine and its contents collected over dry mercury, by filling it with mercury from a reservoir. t It was anticipated that the fluorine, on being bubbled through mercury, would instantly be absorbed as fluoride, and that the residual gases could be measured and analysed. In a second set of experiments glass tubes were used, and the residuals were collected over a standard solution of soda. Here the reaction expected was the absorption of fluorine and production of half its volume of oxygen, according to the equation

2F2 + 4NaOH = 4NaF + 2H20 + O2.

The principal figures connected with these experiments are given in the following table : —

* Even in his most recent density experiment M. MOISSAN, after a prolonged trial with a current of 5 amperes, found 5478 cub. centims. of nitrogen still left in a density bulb of volume 159-2 cub. centims., or 3| per cent, of impurity. ' C. R.,' 138, p. 731, 1904.

t A new refractometer tube was made for each experiment, when copper tubes were used.

ON THE REFRACTIVE INDEX OF GASEOUS FLUORINE.

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324 MESSES. C. CUTHBEKTSON AND E. B. R. PEIDEAUX

These results are not concordant, and the causes of the discrepancies were not completely disentangled. It will be sufficient, therefore, to indicate, briefly, their probable nature, without attempting detailed criticism which the figures will not bear. The first four experiments are rendered nugatory by the absence of any means for destroying the ozone produced with the oxygen, which, as will be shown later, invariably accompanied the fluorine (see Column 9).

No correction can be introduced into the figures for this source of error ; for the proportion of ozone to oxygen, produced under the conditions of the experiment, is not known, and the quantity of oxygen present is itself doubtful, since the nature of the reaction between ozone and mercury is not beyond dispute.*

Any correction for ozone would reduce the value found for the index of fluorine.

In the fifth and sixth experiments the measurements of the volumes of residual gases proved insufficiently accurate, and were complicated by the presence of ozone produced by the action of the fluorine on the solution of soda.

In the last three experiments these sources of error had been eliminated, and we are forced to suppose that the method of absorbing the fluorine over mercury is open to some grave objection, possibly the formation of an oxyfluoride of mercury. It is certain that some source of error is to be sought in the process of absorption over mercury, since, in these experiments, the calculated values for the index of fluorine given in Column 10 are more discordant than those given in Column 4 for the observed refractivity of the mixture of gases.

But, though this series did not give values sufficiently concordant to warrant the belief that the true index of fluorine was being measured, some important inferences could be drawn from the results. The number of values which ranged below the lowest figure obtained in Paris confirmed the opinion that, on that occasion, some other gas or gases were present. On the other hand, the absence of any very low value, in spite of the variety of methods employed, indicated that the refractivity of fluorine was to be sought in the neighbourhood of the figure 200, and was by no means so low as students of the refraction equivalents have surmised.

But the most interesting point observed was the presence, in the residuals, of a larger proportion of oxygen than could be accounted for by the amount of air present. This was observed to be the case in all the experiments shown in Tables II. and III., as well as in others specially designed to test the point ;t and it was ultimately proved, beyond reasonable doubt, that the oxygen was produced by the intermittent electrolysis of traces of water in the electrolytic tube, and not by subsequent reactions. Our experience was that the proportions of oxygen and fluorine liberated were not sensibly altered by prolonging the experiment for two or three hours.

Having established this fact, we were enabled to make dispositions for the series of

* ANDREWS and TAIT, 'Phil. Trans.,' 1860, p. 114; SHENSTONE and CUNDALL, ' C. J.,' 51, p. 623; E. C. C. BALY, ' B.A. Reports,' 1897, p. 613.

t It is hoped that the details of these experiments may be published on another occasion.

ON THE REFKACTIVE INDEX OF GASEOUS FLUORINE.

325

experiments by which the index has, we believe, been measured with some approach to accuracy.

Fig. 2 shows, in a diagrammatic form, the arrangement of the apparatus. A is the copper electrolytic tube, from the right side of which issues the fluorine. In order to prevent the escape of the vapour of hydrofluoric acid the exit tubes were

Fig. 2.

carried upwards, and surrounded by vessels, B1( B2, containing alcohol, cooled to — 78° C. As a further precaution, the gas next passed through a length of 73 centims. of platinum tube immersed in a solution at the same temperature (C), and in many of the experiments a guard tube, filled with NaF, as recommended by M. MOISSAX, was added at the point F. The fact that the presence or absence of this salt did not appear to affect the refractivity of the mixture of gases is evidence that the other precautions were effectual.

It was probable that some part of previously observed discrepancies arose from the presence of ozone produced when the oxygen was liberated ; and, in order to destroy this, the gases next passed through a spiral of platinum tube, 49 centims. in length, heated to from 250° to 300° C. (E). The bends, D and F, were immersed in iced water to prevent conduction of heat from the spirals to the condenser on one side and the refractometer tube on the other.

The refractometer tube, H, was of platinum-iridium. 4578 centims. long and 0'65 centim. in diameter. Its volume, with the leads, was 15 '01 cub. centims. Each end was furnished with a collar of platinum 0'25 centim. broad. The plates with which the ends were closed were of fluor spar, and were secured to the tube by a shoulder of shellac, melted round the outside of the circumference of the collar, so as to be as far as possible from the fluorine within. Thus, after leaving the electrolytic tube, the fluorine was never in contact with anything but platinum and fluor spar.

The plan adopted for measuring the volume of fluorine present was to allow it to

326 MESSES. C. CUTHBERTSON AND E. B. K. PRIDE AUX

combine with an element whose fluoride was a solid (lead was chosen), measuring the contraction of volume so produced by means of mercury manometers which were, however, kept as far as possible from the fluorine by a column of air.

In the figure, Lj and L2 are graduated glass tubes of the same bore, having, at their upper extremities, narrow tubes filled with dry lead filings, Ka, K2, and terminating in platinum tubes, Pj, P2, which fitted the leads of the refractometer tube. These closed burettes were connected with open movable burettes, Mj, M2, also of equal bore, which were joined by a wire passing over a pulley. One of the burettes (L2) was in connection with a graduated reservoir of mercury, R, provided with a tap, and all were filled, to the proper point, with dry mercury.

When the air in the refractometer tube had been displaced by the gaseous products of electrolysis as completely as possible, and the number of bands which had crossed the field had been noted, the entry tube was disconnected, and the two tubes P!, P2 rapidly connected with the system of burettes, the junctions being made air- tight by immersing them in mercury. The burette M2 was then raised slowly, while M1; being connected by a pulley, fell by an equal amount. The mercury in LI, L2 followed the motion of that in their respective companions, and the effect was to push the contents of the refractometer tube into the glass tube at Kj, which was filled with lead filings, without appreciably altering the pressure, so as to avoid errors due to possible leaks. As the fluorine combined with the lead, there took place a diminution of the volume of gases in the closed space K1; H, K2, which was indicated by a difference of level between the mercury in the closed and open burettes. As fast as this was observed, pressure was equalised by opening the tap Q and letting in mercury. By continuing and reversing this process the gases in the refractometer tube were pushed back and forward for about an hour. When no further change of levels could be detected it was assumed that all the fluorine present had been absorbed, and that the residual gases consisted of oxygen and nitrogen. The volume of fluorine was measured by observing the change of levels of the mercury in the two closed burettes, and the measurement checked by reading the change of level in the reservoir R.

The amount of oxygen present was found by taking a sample of the total residual gases in the closed system and burning it with phosphorus. The residue was also tested for SiF4, and, finally, was shown by its index and spectrum to be nitrogen.

From these data the index of fluorine could be found. But as the calculation is rather long, a specimen is given below, the figures being those actually observed in the third experiment given in Table III.

ON THE REFRACTIVE INDEX OF GASEOUS FLUORINE. 327

Barometer 741-5 millims. Thermometer (mean) 13° C.

Length of the refractometer tube 45-78 centims.

Number of bands which would cross the field for one atmosphere of air at normal

temperature and pressure introduced into this tube 227- 6

[NX = (p, - 1) x length of tube]

Number of bands which would cross the field for one atmosphere of air at the

temperature and pressure observed 212

Number of bands observed to cross the field as air was displaced by gases

produced by electrolysis 47

Hence, by difference, number of bands which would cross the field for one

atmosphere of these gases at the temperature and pressure of the day . . . 165

Therefore the refractivity of the mixture of gases in the tube (air = 293) is

2-ff*293 = 228

Volume of the refractometer tube and leads containing this mixture of gases . . 15-01 ciil>. centims.

Volume of the gas absorbed by the lead filings 9 • 04

Hence, by difference, the volume of gases in the refractometer tube and leads

which are not fluorine (tube residuals) is 5-97

Determination of the proportion of oxygen in the tube residuals from analysis of the contents of the burettes and refractometer tube (Kb HK2) after the experiment (total residuals) : —

Volume of total residuals 43-62 cub. centims.

Volume of tube residuals 5-97 „

Hence, by difference, volume of air was 37 • 65 „

A sample of the total residuals contained 25 per cent, of oxygen. Therefore the whole contains, of oxygen,

OK

f^x43-62 = 10.90

Of this amount the 37 '66 cub. centims. of air contain, of oxygen,

20-9

100

x37-66 . . . 7-90

And the difference between this and the whole quantity of oxygen found (10'9) is 3-00 „

Hence the refractometer tube, at the moment when its contents gave a refractivity of 228, held —

Fluorine 9-04 cub. centims.

Oxygen 3-00

Nitrogen 2-97 „

15-01

Now, the refractivity of the mixture, multiplied by its volume, is equal to the sum of the refractivities of its constituents multiplied respectively by their volumes.

Taking the refractivity of oxygen as 270 and that of nitrogen as 297, the refractivity of fluorine is given by the equation

15-01x228 = whence

328

MESSES. C. CUTHBERTSON AND E. B. R. PRIDEAUX

The following table shows the results obtained by this method. With the exception of the second experiment, which was discordant, probably owing to the presence of a slight leak afterwards detected, the coincidence is as close as we can hope to attain, having regard to the difficulties of the inquiry.

We believe that the refractivity of fluorine for the D line lies in the neighbourhood of 195, most probably within 2 per cent, of that number.

TABLE III. — Refractive Index of Gaseous Fluorine.

1.	2.	3.	4.	5.	6.	7.	8.	9.
Num- ber.	Date.	Refrac- tivity of mixed	Volume of tube.	Volume of fluorine	Volume of oxygen	Volume of nitrogen	Refrac- tivity of fluorine.	Remarks.
		gases.		present.	present.	present.
			cub.	cub.	cub.	cub.
			centims.	centims.	centims.	centims.
1	February 9,1905	237	15-01	7-45 5-32	2-24 195
2	18, „	225	15-01	8-40 2-43	4-17 177	A slight leak was
							detected, after
								the experiment.
3	D 28, „	228	15-01	9-04	3-02	2-95	192
4	28, „	227	15-01	9-62	2-25	3-14	194
5	March 10, „	236	15-01	8-4	3-37	3-24	198J	The glass tube KI
								(fig. 2) holding
								lead filings was
								replaced by a
								copper tube, to
								diminish the
								chance of ob-
								taining SiF4.
Mean of 1, 3, 4, and 5 . . . 195
Index of gaseous fluorine 1-000195.

The principal difficulties and sources of error involved in the method are as follows :• —

(1) In disconnecting the refractometer tube and connecting with the measuring

apparatus a few seconds are spent, and a small proportion of the contents of the tube may be lost by diffusion. To minimise this the leads were made about 20 centims. long, of platinum tube of less than 2 millims. bore.

(2) The volume of fluorine present is measured by contraction during the com-

bination of the fluorine with the lead filings. The success of the method, therefore, depends absolutely on the assumption that no gaseous compounds are formed, or that if formed (e.g., SiF4) they are allowed for : and, secondly, on the absence of leaks.

ON THE REFRACTIVE INDEX OF GASEOUS FLUORINE. 329

These dangers were met by reducing the quantity of glass used as tar as possible, bringing the lead filings into immediate contact with the exits oi the refractometer tube, drying the whole with scrupulous care, and, finally, by testing the residuals for silicon fluoride. This test was carried out in three cases and only in one of these was a minute quantity of SiF4 found (0*3 cub. centim.), which did not affect the index by more than one or two units.

(3) The presence of ozone in the refractometer tube, especially if afterwards

disintegrated, would introduce a serious error. The device employed to obviate this is described above.

(4) Owing to the form of the apparatus it was not possible to isolate the tube

residuals and measure their index directly, and their volume was so small, in comparison with that of the total residuals, that it would have been unsafe to calculate the refractivity of the former from that of the latter, even after the most careful measurements. It was necessary, therefore, to calculate the value of the refractivity of the tube residuals from their composition, ascertained by the analysis given above. But the calculation was confirmed by comparison with the figures given in the preceding series of experiments when the residuals were collected undiluted with air. Table II. shows that, in that series, the index lay, in nearly every case, between those of oxygen and nitrogen.

It will be observed that the result given here rests on the assumption that the density of gaseous fluorine is 1'319 (air = 1) ; i.e.. that the molecule is diatomic under normal conditions. M. MOISSAN has twice measured this important constant. On the first occasion, in 1889, he obtained four concordant values,* 1'2G4, 1'2G2, 1'2G5, 1'270, the mean of which was 1'265, and he thence inferred that the gas contained a small proportion of molecules of Fj. f

In 1904 a second series of experiments, in glass bulbs, gave values of 1/298, 1'319, I '313, 1'312, of which the mean is 1'310, a figure very nearly identical with the value assumed. We have accepted this later determination, which was made with the precautions dictated by many additional years of experience, and is supported by a priori probability.

In a recent paper by Mr. CUTHBERTSON,^ it was shown that, in four groups of elements, the refractivities of the different members of the same group are related in the ratios of small integers, and it was pointed out that, if this coincidence were not due to chance, the refractivity of fluorine should bear to that of chlorine the ratio of one to four, as those of nitrogen, oxygen, and neon do to phosphorus, sulphur, and argon respectively.

* 'Le Fluor,' p. 87; 'Ann. de Chimie et de Physique,' vol. 25, 1892, p. 131; 'C. R.,! vol. 109, 1889, p. 863.

t 'C. R.,' vol. 138, p. 729 (1904).

t 'Phil. Trans./ A, vol. 204, p. 323, 1905,

VOL. CCV. — A. 2 U

330

MESSRS. C. CUTHBERTSON AND E. B. R. PRIDEAUX

This prediction has been verified. The refractivity of chlorine for sodium light is generally accepted to be 768, or 4x192. That which is now found for fluorine is 195.

Table IV. shows the exact ratios experimentally obtained in all cases. The indices were determined, in the case of the inert gases for white light, in that of iodine for the red and the violet. In all other cases the measurements are for sodium light.

It will be seen that, except in the case of the anomalous red rays in iodine, the discrepancies between the ratios actually found and those of integers do not exceed 3 '2 per cent. A discussion of the possible causes of these discrepancies will be found in Mr. CUTHBERT.SOX'S paper.

The element chosen as standard in each group is indicated by an asterisk, and, to avoid doubling all the other figures, the ratio of helium is taken as one half.

TABLE IV.

Elements.	Reflectivities observed.	Ratios in each group.	Observer*.
Helium	72-6	0-511	RAMSAY and TRAVERS.
Neon	137-4	0 • 968	> »
* Argon Krypton . . . Xenon	568 850 1 ;!78	4 5 • 986 9 • 704	"
Fluorine	195	1-015	CUTHBERTSON and PRIDEAUX.
*Chlorine	768	4	MASCART.
Bromine	1125	5-859	„
Iodine <	1920V. 2050 R,	10 10-68	I HURTON.
Nitrogen *Phosphorua	297 1197	0-992 4	MASCART. CUTHBERTSON.
Oxygen *Sulphur	270 1101	0-981 4	MASCART. CUTHBERTSON.

With the addition of fluorine the table given in the paper quoted above, showing the refract ivities of all the elements whose index has been measured in the gaseous state, now stands as follows. A few additional elements are put down to suggest the framework of the periodic system, and the refract ivities are rounded off.

ON THE KEFKACTIVE INDEX OF GASEOUS FLUORINE.

331

TABLE V. — Relative Refractivities of some of the Elements.

H 139	— He 139 x|
Li	Be B 0 N 0 F No 297x1 270x1 192x1 139x1
Na	—	P S CI A 297x4 270x4 192x4 139x4
K	—	Bi KJ 192 x (i 139 x (i |
Rb	—	I X 192x10 139x10
Cs	—	> |
—	Hg 1857	1

We have pleasure in expressing our thanks to Sir WILUA.M RAMSAY and to Professor TROUTON for assistance throughout the research, to M. MOISSAN for his kindness in supplying the fluorine used in the first series of experiments, and to the Iloyal Society for grants in aid of the expense incurred in the research.

2 u

[ 333 ]

X. Modified Apparatus for the Measurement of Colour and if* Application to tin- Determination of the Colour Sensations.

Ht, Sn- WILUAM I.K W. ABXEY, K.C.K., f'.It.S.

Received April 17,— Head May 18, 1905.

PART 1. (I.) Introductory.

IN a paper contributed to the ' Phil. Trans.' in 1.899 on Colour Vision, the colour sensations in terms of luminosity were given in detail. Since that date a large part of the leisure which 1 could command outside my official duties has been occupied in revising the measures there given. To effect this revision, a modification of the apparatus I previously employed was carried out. Some slight alteration in the sensation curves was the result, and. though small, ought to be recorded.

The principal alteration that lias to be made is in the amount of what may be called "inherent white" which exists in the spectrum colours. The white is due, at all events in part, to the overlapping of the three sensations. It will in the first instance lie necessary to describe the change that has been made in the colour-patch apparatus with which my previous measures had been made, since a good deal of the alteration in the blue sensation curve between X. 5000 and X 5100 is dependent on it.

(2.) The Colour-patch Apparatus.

The colour-patch apparatus is now arranged to enable two spectra formed by the same source of light to be used either separately or together. This arrangement allows a comparison of any differing mixtures of spectrum colours to be made, and it also allows the addition of any desired quantity of white light to the colours formed by the aid of either of the two spectra,

In the original apparatus the intensity of the white light used for comparison with the colours varied with the intensity of the spectrum. The mode adopted to secure this result was to use the light reflected from the first surface of the first of the two prisms used in forming the spectrum. The beam of white light so obtained was reflected by a mirror on to the screen, on which the patch of colour was thrown. In the modified apparatus this principle of reflection has been still further utilized. The white light is used as before to form the spectrum to the comparison light, but, in addition, the light, after passing through the two prisms, passes through a half-

VOL. ccv. — A 396. 17.11.05

334 SIR AV. DE W. ABNEY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

silvered mirror, inclined at about 45° to the axis of the lens. The reflected beam is again reflected so as to pursue a course roughly parallel to the main spectrum, so that two similar spectra are placed side by side. The accompanying diagram will show the arrangement.

55

B \

Diagram of mo'litiud apparatus for the colour patch.

As in the apparatus described in " ( 'olour Photometry," Part 111. (' Phil. Trans.,' A, L892), K is the soui'ce of light used outside a darkened room, LI, L> are lenses throwing an image of the source of light on the slit S, of the collimator C. TJie parallel beam passes through the prisms I1,, P., and is received on a colour-corrected photographic lens, Lh of sufficient diameter to take in the whole of the light coming through the prisms.

The lens forms a spectrum on a focussing screen at D,, which can be removed and slits S2 placed in the image. L(i collects the colours and gives an image of the face of the prism P, on the screen B. When slits are placed at Du the image is of the mixed colours passing through them.

Behind the lens L, is placed a semi-silvered mirror M]; reflecting, as nearly as may be, the same amount of light as is transmitted through it. If the mirror be on a plate of glass with parallel sides, it should be as thin as possible, to avoid any „ serious mixture of colour in the second spectrum clue to the reflection of the unsilvered surface. If a plate be made up of two thin prisms, as in margin, with the surface AB of one of them silvered, the transmitted beam is not deviated, and the beams reflected from DB and AC are diverted and not used.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 335

The reflection from the semi-silvered mirror M! falls on a silvered mirror, M2, which reflects the beam in such a direction that it falls on B, the image of the spectrum being thrown on D2, in which are slits, S3. The image of P, is thrown on B by the lens L.v A beam of white light is reflected from the face of P] by M:, (\vhich may be either a silvered mirror or plain) and is also focussed on B, so that we have the patches from both spectra and from the white light falling over one another on B. By means of rods correctly placed, a colour or colours from either spectrum can be isolated and be mixed with anv proportion of white by using sectors as shown. There are slides carrying the slits at D! and I)2, and to them are attached trans- parent scales, [n the case of 1)^ a beam of white light falls on the mirror M5, as shown, and passes through the transparent scale at "<•«," and a lens X throws a magnified image of the graduation on a distant white screen, on which a zero mark is drawn. This enables the transparent half-millimetre scale to be read to a tenth of that unit. In a similar way the scale at " <t " is magnified by X' by a beam of light falling on M,. When the scale readings are not required, the sources of light illuminating them are covered up.

Again the small lenses A1 and A" are mounted in a sliding arrangement and can be moved in front of lenses L., and Lti. When a sl.it is drawn in front of A1 or A1' the image of the aperture is magnified on a distant screen, carrying a scale, and the width of the slits can be accurately ascertained by noting on such scale the reading of the breadth (say) of -J- millim. width of slit. This is the instrument with which the following measures were made.

(:j.) I* t/icri' <i 4t/i Xcitx'ttinii. in tin- Vinli-t .'

As in my previous investigations, the red at the red lithium line was used as exciting only the red sensation, and the violet at X4100 was also employed as a .provisional sensation, since it excited only the blue and the red sensations.

Since my last paper on the subject was published, Brucn, in his paper in the 'Phil. Trans.' (B, vol. 191, 1899) has given it as his opinion that besides the red, blue, and green sensations there is a 4th sensation excited by the violet. Before using the violet as a provisional sensation, it became necessary to ascertain if this 4th sensation really existed, and various experiments were made with this object. From the first I was sceptical as to the 4th colour sensation, as it appeared to me to be unnecessary, and was a departure from the simplicity with which nature usually works. Amongst the experiments tried was that of fatiguing one of my own eyes with strong red light, and by a simple artifice immediately afterwards viewing a patch of violet light, keeping the uufatigued eye closed. The violet became a bright blue, whilst to the unfatigued eye it was of its natural violet hue. Not satisfied with my own vision, 1 got several unbiased persons to repeat this experiment, and they invariably stated that the patch became blue. A red-blind

33fi SIR W. DE W. ABXEY: MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

person matched without any difficulty the blue lithium line* with the violet near H, though he described the former as rather paler than the latter, a description which the colour- vision theory indicates as probable. Using my two eyes, one fatigued and the other not, I endeavoured to obtain a measure of the amount of red sensation destroyed, but owing to the mixture of white in the blue the match was never perfect, as the fatigue passed away before the match was made, and when white was added to the blue, it too had lost part of its red, and the " fatigued " violet appeared

too green.

These experiments and others went to prove the absence of the 4th sensation, and if further proof were required, it would be found in the ease with which the violet, when white is added to it, can be matched with a mixture of red and blue near the blue lithium line. 1 have therefore felt justified in using the violet as a temporary sensation in all my measures, reducing it to its components of red and blue in my final results.

(4.) Fi, ''ft! f'niiifs in flic Spectrum.

Several points in the spectrum could be readily f'ouzid. Thus the complementary colour to the red in the blue-green is a fixed point, as is the complementary colour to the violet. The complementary colour to the blue (near the blue lithium line) is also known. For other preliminary details a reference should be made to my previous paper.

(5.) DeteriuiiMtioti of the \Vhitc in f/ic Colour which only excites the Green

Sensation u-ith \\'hite.

In my previous investigations 1 was unable to match spectral orange to which white could be added with mixtures of red and the green, but had to use the light transmitted through a solution of bichromate of potash placed in the path of the white (reflected) beam as representing an orange. By a suitable arrangement white could be added to it, till the mixtures were of the same colour. A small quantity of white had then to be added to the spectral orange to match the colour and the bichromate solution. From the two amounts of white added, the amount of white necessary to add to the spectral orange in order for it to match the mixture of red and green was deduced. With the apparatus now employed the determination of the amount of white to be added to the orange was made direct. There was also an advantage in these direct determinations with the spectrum colour, as more than one shade of orange could be used as checks to one another.

The results of the many measures made show that a slight correction has now to be made to my previous determination.

* It will be noted further ou that the blue lithium excites only the blue sensation and that of white.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 337

(6.) Amount of Blue Sensation in Yelloiv and Yellow-yreen.

In reviewing my previous measures of the amount of blue sensation in the yellow and yellow-green of the spectrum, I was struck with the variation of results obtained on different occasions, and though every care was taken at the time, I am led to think that the amount of this sensation was under estimated, though at the most the quantity is but small. This part of the spectrum has occupied my attention for a considerable time, and the determination of the blue sensation in this region has been conducted on perfectly different lines to that formerly employed, which was to make an equation by mixing the colour under consideration with red and violet in sufficient quantities to form white, and then to equate it with the standard equation. The equations were formed, but uo great stress is laid on the correctness of the blue sensation found, but only on the correct proportion of red and green sensation. The corrected value of the blue sensation was found by the following- plan :—

Slits were placed in the red and green at the standard positions red Li and SSN 37 '5 (standard scale number) in the green and as good a match as possible was made by mixtures of the two colours with the intermediate spectral colours, to which a little white was added. The amount of white added was not considered, but only the white inherent in the green. This last was deducted from the green and the percentage of red and green sensations in each colour calciilated without taking into account the white which was due to the presence of the blue sensation. From the equations were obtained the percentage of red and green when the white present in each colour -was included, and by the last measiires the percentage of red and green when such white was excluded. From these different percentages it was easy to calculate the amount of blue sensation present, for it only exists in the '' inherent white." On subsequently considering the sensation curves of equal stimulation as given by KCENIG and myself, my attention was called to the fact that at the place where the red and blue curves cut a large and very sudden increase of white inherent in the colour should be seen, so large indeed that it would never escape notice. The colour at that point ought to be much paler than colours close to it, but such is not the case. The new measures show that there is no sudden rise in the amount ot white present in any colour, and that the maximum of white is at SSN (standard scale number) 43 (X 5427) and not at 37'5 (X 5150). This point will be referred to later.

(7.) Measures from the Blue-green to the Violet.

The measures taken from the blue-green to the violet were made by the same method as described in the paper above mentioned, but the process was much simplified. The colour whose percentage composition had to be measured was isolated by a slit placed in one spectrum, and a slit in the other spectrum moved till a

VOL. ccv. — A. 2 x

838 SIR W. DE W. ABNEY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

complementary colour was found which, when mixed with the former, gave a match to white. The luminosity of each was taken separately when the match was complete. The composition of the part of the spectrum from the yellow to orange was already known, and the complementary colour found was converted into the percentage components of red and green. This enabled an equation in two standard colours and the unknown colour to be formed. When equated to the standard equation, the percentage composition of the last was found. When the colours in one spectrum approach the violet, the movement of the scale in the other spectrum to find the complementary colours is very small, and the magnified scale, formed as described, was of great assistance, since a vernier could be used when required.

(8.) The Composition of the Violet.

This remains as given in my previous paper of 1899. Many measures were made in this region of the spectrum, and the percentage of red to blue remains as before. The following are some of the principal measures : —

(9.) Inherent White Light in SSN 37 '5.

To find the amount of inherent white in SSN (standard scale number) 37 '5. Taking an orange below D at SSN 50'2, it is found that in luminosities

RS. 37-5. Orange. White. 41 + 55 = 57 + 39.

As there is no white in RS (red sensation), it follows that the 39 white is in

55 (37'5), or that the

Orange. RS. GS.

57 = 41 + 16,

and that there is |f of GS (green sensation) in 55 (37 -5); that is, there is 29'5 per cent, of GS in (37 '5).

Taking another orange near D at SSN 50 '05, it was found that

RS. (37-5.) Orange. White.

487 + 45-8 = 63 + 31'5, as before

RS. GS.

487 + 14-3 = 63.

That is to say, there is 31 '2 per cent, of GS in (37'5). Other measures, and they were many, gave

31 per cent., 3T8 per cent., 30'8 per cent., 29'8 per cent, 32'4 per cent., &c.,

and the mean gave 31 per cent, (very closely) of green sensation in the colour, and this number was adopted.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 339

(10.) The Standard Equation.

Four separate series of 50 equations each were made with slits at the red lithium line (37 '5) and (4) in the violet. The mean of each series gave the following results, using 31 per cent, of GS in (37'5) : —

RS. GS. V. White. 68-48 + 30-16 + 1-36 = 100 6818 + 30-50 + 1-32 = 100 68-62 + 29-98 + 1-45 = 100 68-44 + 30-17 + 1-39 = 100.

The mean taken to one place of decimals gave

RS. GS. V. W.

68-4 + 30-2 + 1-4 = 100,

and this equation was adopted as the standard equation for white. It will be noticed that this is somewhat different to the equation given in the former paper, but this is accounted for by the fact that the white of the comparison lights in the two cases are not quite identical, selective absorption by the glass used for reflection being far greater in the former measures than in the latter. The percentage of white in (37 '5) also differs.

(11.) Red and Green Sensations from SSN's 58 to 49.

The red and green sensations in this part of the spectrum were determined by placing one slit in the red lithium colour and another in the yellow-orange of one spectrum, and matching the intermediate colours thrown on the screen from the second spectrum. The composition of the yellows and orange used had been previously determined by mixing the colour of the red lithium with (37 '5) SSN :—

SSN. RS. GS. SSN. RS. GS.

56-2 =95-6+ 4-4 56 = 96"5 + 3'5

55-1 =92-6+ 7-4 55 = 93'!+ 6'9

54-05 = 90 +10-0 54 = 90-5+ 9'5

*54 =90-6+ 9-4 *54 = 90'6+ 9'4

5275 = 86-8 + 13-2 52 = 84'2 + 15'8

*52'6 =86 +14-0 *52 = 83-9 + 16-1

51-6 = 80 +20-0 51 = 78-7 + 21-3

*51'6 =80-2 + 19-8 *51 = 78-9 + 21-1

50-9 = 79-3 + 207 50 = 75-0 + 25'!

50-65 = 78-2 + 21-8 49 = 70'0 + 30'0 *50'55 = 78 +22 *497 = 75-7 + 24-3 *49'3 = 73-6 + 26-4

49-25 = 7T5 + 28-5

The numbers marked * were taken with a slit at D, the others at 49'0 SSN.

2x2

340 SIR W. DE W. ABNEY: MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

As special accuracy was necessary between SSN's 48 to 50, a large series ot measures was taken at this part of the spectrum. This necessity arose from the fact that a great part of the complementary colours between the greenish-blue and the violet lay in this part of the spectrum, ami their composition could only be accurately determined when the exact percentage of RS and GS was known.

(12.) Red and Green Sensations from SSN's 49 to 37'5.

The equations were formed, as stated before, in the ordinary manner, keeping the slits in the red and violet at the standard places and altering the position of the green slit.

As an example, the value of SSN 45"8 was found as follows :—

RS. (45-8.) V.

38-8 + 16-8 + 2-03 = White, or

RS. 45-8. V. W.

18-6 + 80-4 + 1 = 100, but

RS. GS. V. W.

68 -4 + 30 '2 + 1-4 = 100 (standard equation),

.L 1 L* '

therefore

45-8. RS. GS. V.

LOO = 6T9 + 37-6 + -5. Similarly SSN's (40'5), (43), and (47 '5) were found

(40-5.) RS. GS. V. 100 = 5 1-68 + 47 -49 + '83.

(43.) RS. GS. V. 100 = 56-9 +42-5 +'60.

(47-5.) RS. GS. V. 100 = 66-20 + 33-5 +'30.

The more accurate values of the violet were determined as described by matching the intermediate colours between SSN's 49 and (37 '5) of one spectrum by mixtures of these standard colours. Using luminosities, we get

SSN. R. G. (37-5.) 38 = 2-62 + 97-38 40= 9-66 + 90-34 42 = 17-16 + 82-84 44 = 24-45 + 75-55 46 = 31-31+68-69 48 = 39-41 + 60-59.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 341

Deducting 69 per cent, of white from the green (SSN 37 -5), we get the following R and G sensations in luminosities : —

TABLE I.

SSN. RS. GS. 38 = 8 +92 40 = 257 + 74-3 42 = 40-1 + 59-9 44 = 51-1 + 48-9 46 = 59-2 + 40-8 48 = 67-1 + 32-0.

From the plotted curves of the red sensations and green sensations at this part of the spectrum we get the following figures :—

TABLE II.

SSN. RS. GS. 38 = 47-9 + 51-2 40 = 51-0 + 48-5 42 = 54-9 + 44-7 44 = 57-7 + 41-9 46 = G2 +37-6 48 = 67 +32-9.

Any slight corrections due to alterations found in the violet were made in the green sensations. The violet was calculated from Tahle I. and Table II. as follows : — There is a certain quantity of red sensation and of green sensation which with the violet forms white. From the standard equation we know that the luminosity of the red sensation is 2*265 times larger than the green sensation and 49 times larger than the violet in the white. If x be the factor of red in Table I. (which is only due to the excess of red beyond that required to form white), then the same factor must be used with the green. The red sensation in Table II. (which takes into account the white present in the colour) must have deducted from it the red of Table II. x x, and the resulting amount must equal the green in Table II. less the green in Table I. x a; and multiplied by f|4f or 2 '26 5.

Let R be the red in Table I., R! the red in Table II., G the gi-een in Table I., and GI the green in Table II. Then

R-.-eR, = (G-xGJ 2-265.

From this equation we derive x. When x is found, we have a known amount of red on the left-hand side of the equation, which is the amount which combines

342 SIE W. DE W. ABNEY : MODIFIED APPAEATUS FOE MEASUEEMENT OF COLOUE with green and violet to form the white, and - — gives us the amount of violet.

4*7

Take, as an example, SSN (42) :

ES. ES. GS. GS.

54'9-40-la; = (447-59'92a;) 2'265.

From this we get

, 54-9-40-1 x -484 „„-

x = '484 and — = '726,

49

the amount of violet present in SSN 42.

The scale numbers in Tables I. and II. were thus treated and the violet as shown in Table III. was so obtained.

(18.) Colour Sensation* in SSN (37'5).

The amount of white light in (37 -5) has already been determined as 69 per cent. It only remains to add this amount of white to the green in the standard equation and equate it when so altered to the standard equation.

When the luminosity of the GS is increased by 69 per cent, the equation becomes

ES. (37-5.) V. W. ES. GS. V.

40'91-f 58'27 + '84 = 100, the standard equation being 68'4 + 30'2 + r4 = 100.

These give us the composition of

(37-5.) RS. GS. V.

100 = 47-19 + 5T85+-96.

(14.) Determination of SSN's from 36 to 12.

The method described above was adopted to determine the SSN's 36 to 12. The following is an example : —

(54.) (34-9.) W.

53'4 + 46-6 = 100 (j.),

(54.) ES. GS.

But 100 = 90-5 + 9-5 (ii.).

From (ii.), (i.) becomes

RS. GS. (34-9.) W. 48-33 + 5-08 + 46-6 = 100.

Equating with the standard equation we get

(34-9.) ES. GS. V. 100 = 43-07 + 53-93 + 3U

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 348

Another example may be given of SSN 25'5. The equation is

(49-05.) (25-5.)

96 + 4-0 = 100. RS. GS.

In 49'05 there is 70'1 +29'9, and the equation becomes, after equating with the standard equation,

(•25 -5.) RS. GS. V. 100 = 27-5 + 37-5 + 35.

Similarly it was found that

(27-1.) RS. GS V. (18-6.) RS. GS. V.

100 = 30-8 + 45-9 + 23-3, 100 = H-3 + 7'3 + 81 "5,

(23-7.) RS. GS. V. (15-5.) RS. OS.

100 = -24 + 24 + 52, 100 = 4'6 + 2'l +93'3.

Beyond SSN's 14 and 12 respectively, where the red and green sensations vanish, the violet alone remains, but having different intensities.

(15.) Formation of the Sensation Curves.

From the foregoing equations curves of violet-green sensation and red sensation were plotted, and any small irregularity was smoothed. The ordinates thus found are given in the following Table III., in Columns IV., V., and VI.

Columns I., II., and III. represent (i) the standard scale numbers of the prismatic spectrum (the same as used in my previous paper), (ii) the wave-lengths, and (iii) the luminosity of the spectrum of the crater of the electric (arc) light as judged by the centre of the eye.

Columns VII., VIII. , and IX. are the luminosities of the colours in terms of the red sensation (RS), the green sensation (GS), and the violet (V). These are obtained by multiplying IV., V., and VI. by Column III. and dividing by 100.

In Columns X. , XI. , and XII. are given the percentage composition of the different rays in terms of RS, GS, and BS (the blue sensation). These are obtained by reducing the violet sensation to -ffa of its value in Column VI. (which is the percentage of blue which the violet contains), and adding i-0-0- of the violet to the red in Column IV. GS is the same in Columns V. and XI.

Columns XIII., XIV., and XV. are the luminosities of RS, GS, and BS as contained in the different colours, and are obtained, as before, by multiplying XI., XII., and XIII. by the luminosities in Column III. and dividing by 100. Column XVI. is Column XIV. multiplied by 2 '3, and Column XVII. is Column XV. multiplied by 178.

344 SIR W. DK W. ABNEY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

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AND ITS APPLICATION TO THE DETERMINATION OF COLOUK SENSATIONS. 345

The areas of the curves given by Columns XIII., XVI., and XVII. are equal, and represent equal stimuli of all three sensations. A mixture of the three colours, each being represented by ordinates of the same height, makes white. The points where the red and green curves cut the blue curve are the points in the spectrum which the green-blind and the red-blind match with white.

5 10 15 ^0 Z5 30 35 40 45 50 55 60

Fig. 2. Sensation curves having equal areas (equal ordinates at any poin*-- make white).

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100 90

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40 30 20 JO

35 40

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Fig. 3. Percentage composition in luminosities of red, green, and blue sensations of the spectrum colours. VOL. CCV. — A. 2 Y

346 SIR W. DE W. ABNEY : .MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

(16.) Determination of Colour Sensations and White.

The results given in Table III. are carried still further in Table IV. In it Column L, is as before, the standard scale number. Columns II., III., IV., V. are

TABLE IV.

I.	II. III. IV.	V. VI. VII. VIII.	IX.	X. XI.
		' 1
	Luminosity of sensation together with the white.	Percentage composition of the sensations, white being deducted.	Colour mixtures.
SSN.
	RS.	GS.	BS.	W.	RS.	GS.	. BS.	R.	G.	B.
64	•2				100			100
62	2			100	—	=j_	100	—	—
60	7	.		100	—	100	—	—
58	20-79	•21		99 1	—	97-2	2-8 —
56	47-75	•2 • 25		95-5	4-5	—	86-8	13-2	—
54 .	72-40	7-60	.	90-5	9'5	—	74-7	25-3	—
52	80 • 64	15-36		84-2 15-8	—	62	38	—
50	75	25		75 25	—	48-4	51-6	—
48 60	29-5		7-5 67-1 32-9	—	38-7	61-3	—
46 38-2	25-7		23-1 59-9	40-1	—	31-5	68-5	—
44 22-5	21-6	_ _	30-9 51-1	48-9	—	24-3	75-7	—
42	12-8	18-2		31-5 40-1	59-9	—	17-6	82-4	—
40	5-8	15-4		28-8 27-4	72-6	—	10	98	—
38	1-4	11-4		24-5 10-9 89-1	—	3-5 96-5	—
36	—	8	•031	16 99-6	•386	—	99-87	•13
34		5-1	•OS8	9	98-31 i 1-69	—	99-47	•53
32		3-2	•125	5-2		96-24 : 3-76	98-09	1-91
30		2-12	•155	3-43	_,	93-18 6-82	97-78	2-22
28	. — .	1-33	•192	2-48	87-37 12-63	95-71	4-29
26	* -53	•235	2-03	68-8 31-6	87-56	12-44
94. tOTi	•03	•250	1-66		10-5	89-5	—	27-42	72-58
22 -43	.	•245	•73	15		85	15	—	85
20 -54	—	•235	•33	69-7	— .	30-3	69-7	—	30-3
18 -51	—	•201	•15	71-8	—	28-2	72	—	28
16 -49	—	•180	•03	73-1	26-9	72	—	28
14 -39	— .	•154		72 —	28	72	—	28
12 -334	—	•126	—	72 —	28 72	—	28
10 -253	—	•098	—	72	28	72	—	28
8 -187	—	•073	—	72 —	28	72	—	28
6 -130	—	•051	—	72 —	28	72	—	28
4	•101	—	•039	—	72		28	72	—	28
n	•072	—	•028	—	72	—	28	72	—	28
0	•057	—	•022	—	72		28	72	—	28
Areas .	450	192	2-53	187

RS, GS, BS, and (white) W. These are obtained from Columns XIII., XVI., and XVII. of Table III. From SSN (standard scale number) 64 to 49 no white is present

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 347

in the colours, but at 48 some small quantity of white is shown to exist, and it is found to SSN 16. Taking SSN (40) as an example, in Table III. this colour has for its components in the columns showing equal stimuli

RS. GS x 2-3. BS x 178. 25-80, 55-40, 20.

As equal ordinates make white, the smallest ordinate, 20 in that case, is deducted

from the other two and we have

RS. GS x 2-3.

5-80 and 3 5 '40.

35-40 2-3

or

Thus after deducting 28 '8 of white, the amount of ES is 5 '8 and of GS

15'4, so that the colour at SSN (40) is given by the equation

ES. GS. W. SSN (40). 5-8 + 15-4 + 28-8 = 50.

In the same way the equations to the other colours of the different SN's were found, and fig. 4 gives the curves of RS, GS, BS, and W. It will be seen that all the

90

80

70

fiO

SO

W

30

0.S (too r,

V

X

20

10

20

\

30

35

SCALE of

45

so

55

60

Fig. 4. Luminosity curves of red, green, blue and white sensations of the prismatic spectrum of the

crater (positive pole) of the arc light.

curves are smooth, and not one is abrupt, which is the case where the old numbers in my paper of 1898 for the BS are treated in the same way, more especially in the green and white curves.

Columns VI. , VII., and VIII., Table IV., give the percentage composition in terms of RS and GS, GS and BS, and of RS and BS of the different colours. These columns are useful when we are considering the accurate calculation of the colours of pigments either reflected or transmitted.

2 y 2

348 SIR W. DE W. ABNEY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

Columns IX., X.,and XT. give the percentage composition of the different colours of the spectrum in terms of the three colours which best represent the colour sensations when white is deducted from them, viz., red lithium, SSN 37 '5, and SSN 23 '2.

In reference to this table it may be remarked that of the whole spectrum '225 is white and '775 colour. This shows that the white in the colours is by no means a negligible quantity.

(17.) Significance of the Inherent White.

In regarding the table, Column V., for white, it will be remarked that the maximum amount of white is near SSN (42). In ' Colour Photometry,' Part III. (' Phil. Trans.,' 1892), it was shown that in this region the light disappeared last when the intensity was reduced. It was also shown that the maximum luminosity of a very feeble colourless spectrum was near this point, and in the concluding page of my last paper on the colour sensations, I pointed out that the presence of the fundamental sensation of light, which is white, must be taken into account in any theory of colour vision. The fact that in these slightly revised measures we get more than indications that white exists in the region where the fundamental sensation has been shown also to exist, leads one to believe that we are in some way separating this sensation from the three-colour sensations. What seems to confirm this view is that when a very bright spectrum, such as is given by sunlight, is measured, there is a tendency for the proportion of white in the region SSN's 48 to 16 to diminish. This is what we should expect to find, since fixed amplitude of wave colour vanishes at some, as also does the fundamental light at a lesser amplitude, be the spectrum feeble or brilliant.

(18.) The Normal Spectrum Curves.

Table V. gives the sensation curves for the normal spectrum, and is shown in the same manner as it was in my previously quoted paper.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 349 TABLE V. — Normal Spectrum.

Wave- length.	Spectrum lumi- nosity.	Percentage composition.	Luminosity.
RS.	GS.	BS.	RS.	GS.	BS.	GS x 2-38.	BSxl46.
6800	I	100			1
G700	6	100		—	C
6600	10	99-7	•3	—	9-97	•03	— .	•070
6500	17	98-5	1-5	—	16-74	•26		•6
6400	26	97	•30	. —	25-22	•78		1-9
6300	41	95	•50	—	38-95	2-05	—	4-9
6200	59	92	8	— .	54-28	4-72	—	11-2
6100	75	88-5	11-5	—	66-25	8-75		20-8
6000	85	84	16	—	71-4	13-60		32-4
5900	93	78-5	21-5	—	72-93	20-70	—	48-7
5800	99	72	28	—	71-28	27-72	—	66	_—
5700	100	66-2	33-8	•028	66 '20	33-80	•028	80-4	•050
5600	95	62-2	37-7	•104	59-09	35-91	•099	85-4	•145
5500	89	58-5	41-4	• 1 50	52-06	36-84	•133	87-8	19-4
5400	80	55-2	44-6	•185	44-96	34-88	•148	83-1	21-6
5300	70	52-7	47-1	•215	36-89	32-97 -150 78-5	21-9
5200	54	49-5	50-3 -243	26-73	27-16 -131 64-1	19-1
5100	30	46-5	53-1 -400 13-95.	15-93 -120	37 • 8	17-5
5000	18	43-8	55-3	•860	7-88	9-95	•155	23-7	22'6
4900	11	42	56	2-00	4-62	6-16	•220 14-7	32-1
4800	7-5	43	52-4	4-6	3-23	3-93	•345 9-4	50-3
4700	5	50	41-3	8-7	2-50	2-06	•435 4-9	63-5
4600	3-5	62	21-8	16-2	2-17	•76	•567 1-8	82-7
4500	2-7	72	7-3	21-7	1-94	•20	•586 -5	85-5
4400	2-1	72	2-2	25-8	1-51	•05	•542 -1	79-1
4300	1-7	72	—	28	1-22	—	•476	69-5
4200	1-3	72	—	28	•94	—	•367	53-6
4100	1	72	—	28	•72	—	•280	40-9
4000	•75	72	—	28	•54	—	•210	30 '7
3900	•50	72	—	28	•27	—	•140	20-4
3800	•25	72	—	28	•13	—	•070	10-2
				1 1

100 90 80 70 60 kj

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\

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V

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2

30

£0

10

X

4500

5000

5500 LENGTHS.

6000

65CO

6800

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Fig. 5. Percentage composition in luminosities of red, green, and blue sensations of the colours of

the normal spectrum.

.S50 SIR w. DE w. ABNPIY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

90

80

\

z

\

X

I

\

"5

•»!

X

X

\

\

20

10

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4500

5000

5500 6000

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Fig. 6. (Normal spectrum) Curves of equally stimulated red, green, and blue sensations to form white.

PART II.

(19.) A Colour Defined Inj a Ware-length, &c.

In a note " On the Numerical Registration of Colour," which I communicated to the Royal Society ('Proceedings,' vol. 49, 1891), it was indicated that any colour could be accurately defined by a wave-length, its luminosity, and the percentage of white light that it contained. In Table III. we have a very ready means of stating all these with extreme accuracy.

If the percentage of each colour of the spectrum which a coloured medium or a pigment transmits or reflects be known from measurement, then from Table III. we can find the wave-length, the luminosity, and the percentage of white light which the colour contains.

(20.) Measurement of Spectrum Intensity.

I have already described the method employed by myself in measuring the intensity of the light transmitted or reflected. Fig. 7 shows the plan. S is the slit moving in

Fig. 7.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 351

the spectrum, L the lens throwing the image of the face of the prism on the screen G. In the path of the ray X a plain glass mirror is inserted reflecting a proportion of the beam to a second silvered mirror Ma, which in its turn reflects the beam Y to C. Sectors can be inserted in one or both of the beams X and Y.

If the colour to be measured is that of a piece of (say) coloured glass, it is inserted at D in the path of the beam X ; or if it be a pigment whose colour has to be measured, it is placed at C, so that it is illuminated by X, and a white square placed adjacent to it is illuminated by Y, a rod R being placed in a proper position to throw two shadows touching each other at C. I have found that instead of using one plain mirror at M, it is better to have a bundle of glasses, so that the intensities of the beams X and Y are more equal than when a single glass is employed.

The readings are made by equalizing the brightness of the illuminated shadows first with the colour in position and then without. The two measures give the percentage of light reflected or transmitted from the coloured medium or surface.

(21.) Measurement of Emerald-green and Chrome-yellow.

As examples of the way in which Table III. is to be used, the light reflected from emerald-green, Table VI., and from chrome-yellow, Table VII., has been tabulated.

In both tables Column I. shows the standard scale numbers.

In both tables Column II. the relative intensity of the light reflected from the colour compared with that reflected from a white surface.

In both tables Columns III., IV., and V. are copied from X., XL, XII. , Table III.

In both tables Columns VI., VII., and VIII. are III., IV., and V. multiplied by the intensities in Column II.

The areas of the curves of RS, GS, and BS in VI. , VII., and VIII. for emerald- green are taken and found to be on an empiric scale (which is the same as that of the luminosity of the naked spectrum of the crater of the arc light), RS 202, GS 133, BS 1-418.

GS and BS are multiplied by 2 '3 and 178 (the factors for making the sensation curves of equal area) respectively, and found to be 306 and 252 respectively. The lowest of the ordinates is RS 202. This must be deducted from GS x 2'3 and BS x 178, and we have as the remainders 104 and 50. These must be divided by 2 '3 and 178, and from these (which are 45'2 and '28) the percentages of GS and BS are calculated, and are found to be 99'38 and '62. This, from the diagram and from calculation, gives the dominant colour as SSN 35'64 or X 5070.

The area of the spectrum curve is 830 on the same scale, and the sum of the three curves is 336. The luminosity of the emerald-green, when white is taken as 100, is Jrtt x 100, or 40'5. (This is the same as was made by direct measurement.)

The amount of RS and GS and BS used to form the white is 290. The sum of the areas of the three curves is 336. The percentage of white is therefore f~f £ x 1 00,

352 SIR W. DE W. ABNEY : MODIFIED APPARATUS FOR MEASUREMENT OF COLOUR

or 86 '3. The amount of inherent white in SSN 35 '64 is 68 '5, so that there is 38 per cent, more white in emerald-green than there is in SSN (35'64). Emerald-green is therefore represented by

SSN 35-64. W. Emerald-green.

62 + 38 = 100. Luminosity 40'5.

Chrome-yellow was treated in the same manner.

TABLE VI. — Emerald-green Pigment.

I.	II.	III. IV. V.	VI.	VII.	VIII.
		t !
	Intensity	Composition of white in luminosity.	Composition of green in luminosity.
SSN.	colour
	(white
	100).	RS.	GS.	BS.	RS.	GS.	BS.
64	3-5	•5	—	—	•02 f\n	—	• —	To find luminosity —
62	3-5	2	—	—	•07	—	—
60	3-5	7	—	—	•24	—	—	Sum of areas of green = 336.
58	4	20-8	•2	—	•83	—	—	Sum of areas of white = 830.
56	5	47-75	2-25	—	2-39	•11	—
54	8	72-4	7-6	—	5-79	•6	—	Luminosity of emerald-green
52	14	80-64	15-36	—	11-3	2-16	—
50	28 75	25		21	7		= ££°x 100 = 40-5.
48	42 65-16	31-78	•039	27-36	13-34	•017	830
46	53 ; 54-2	32-7	•090	28-73	17-33	•047	To find the amount of white and
44 42	63 43-75 71 | 34-61	30-81 27-75	•118 •122	27 • 56 24-57	19-41 19-7	•074 •086	the dominant wave-length
40	74 25-8	24-09	•112	19-9	17-83	•083	RS = 202,
38	74 17-5	18-43	•091	12-95	13-65	•067	GS=133,
36 34	73 11-09 70 6-22	12-83 7-86	•101 •124	8-10 4-35	9-34 5-5	•074 •087	BS= 1-418,
32	65	3-58	4-77	•145	2-31	3-12	•094	GS x 2 • 3 = 306,
30	61	2-45	3-08	•174	1-49	1-84	•106	BSxl78 = 252.
28	58	1-78	2-03	•202	1-08	1-2	•117
26	53	1-41	1-15	•243	•74	•61	•129	Residue after forming white
24	46	1-15	•53	•262	•51	•24	•121	S06 - *>02
22	40	•91	•24	•247	• Sfi	. -I	.AQQ	OVVJ ft \frnl A r* (\ f~i ci — 1 r) " IT ft
20	32	•77	•1	•234	ou •24	•03	Vi/t7 •075	2-3
18	27	•62	•04	•202	•18	•01	•054	252-202 .oqpsj
16	22	•51	•01	•18	•11	—	•04	178
14	17	•39		•154	•07		•026
12	12	•33	—	•126	•04		•015	Percentage of GS and BS
10	5	•25	—	•098	•01	, —	•005	Emerald-
8	3-5	•19	—	•073	—	—	•002	GS. BS. green.
								99-33+ -62 = 100.
	Areas .	202	133	1-418	This is SSN 35-64 or A. 5070.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 353

TABLE VII. — Chrome-yellow Pigment.

I.	II.	III.	IV.	V. VI.	VII.	VIII.
• Intensity of	Composition of white in luminosities.	Composition of yellow in luminosities.
SSN.	colour
	(white
	100).	RS,	GS.	ES.	RS. GS.	BS.
64	100	•5		_	•5			To find luminosity —
62	100	•2			2
60	100	7			7	.		Sum of areas of yellow — 682.
58	100	20-8	•2	—	20-8	•20	—	Sum of areas of white = 830.
56 54	100 100	47-75 72-4	2-25 7-60	—	47-75 72-4	2-25 7-60	—	Luminosity of yellow
52	100	80-64	15-36		80-64 ' 15-36		68°
50	100	75 25	—	75 25	—	= g2x 100 = 82-2,
48	100	65-16 ! 31-70	•039 65-16 . 31-70	•034
46	100	54-2 '32-70	•090 54-20 32-70	•090	white =100.
44 42 40	84 62 42	43-75 30-81 -118 36 -75 25-88 34-61 | 27-75 -122 21-46 17-20 25-80 24-09 '112 ' 10-83 10-12	•099 •076 •047	To find the white and dominant wave-length
38	26	17-50	18-43 -091 4-55 4-78	•024	RS = 504,
36	19	11-50	12-83 -101	2-11 2-44	•019	GS=178,
34	16	6-22	7-86	•124	1 1-26	•021
32	14	3-58	4-77	•145	•5 -67	•020	BS » 694,
30	12	2-45	3-08	•174	• 29 • 37	•021	GSx2-3 = 409,
28	11	1-78	2-03	•202	•19	•22	•022	BSx 178=123.
26 11	1-41	1-15	•243]
24	11	1-15	•53 '262				Residue after forming white
22 20	11 11	•91 •77	•24 -247 •10 '234				504-123 = 381 RS
18	11	•62	•04 ' -202				409-123
16	11	•51	•01 1 -180			2-3
14 12	11 11	•39 •33	•154 . •126 |	•76 -23 -216	Percentage of RS to GS.
10	11	•25 -098			RS. GS. Yellow.
8	11	•19	•075			75-4 + 24-6 = 100.
6	11	•13	•052
4	11	•10	•039		|	Therefore chrome - yellou- is
2	11	•8	•028				50 SSN, and contains 26 per
0	11	•6	•022J				cent, white.
Areas .	503-9 178	•694

(22.) Principles of Three-colour Photography.

At the present time the accurate determination of colour composition in terms of the three-colour sensations, of pigments, and transparent media, is of great practical importance. There is now a large business carried on in the production of prints by the three-colour process of photography, and up to the present time the colours produced are, with rare exceptions, wanting in truth, probably owing to screens of

VOL. ccv. — A. 2 z

354 SIR W. DE W. ABNEY : MODIFIED APPAKATUS FOR MEASUREMENT OF COLOUR

the wrong colours being used. In order to take the three negatives from which the prints are produced, it is necessary to place screens of different colours (reddish, greenish, and blue) in front of the sensitive plate in order to get distinctive images which will represent the three sensations in the three printings. As to the printing itself, nothing need be said in this communication, but I shall confine myself to the negatives alone. If the negatives are correct, three transparencies from them should give three images, which, if illuminated by the three colours which represent best the three sensations, and superposed, should give the true colours of nature.

Where the three positives are each devoid of deposit at the same part of the image, the mixture of colours should give white, which means that in the negatives the deposits should be equally opaque. This is the starting point of the process.

The deposit being without colour, the different parts of the three component negatives have to be such that the transparencies, when projected on a screen, allow so much of each coloured beam to pass as will give the natural colour by mixture. [It may be remarked that the negatives themselves, if illuminated with the three colours, and the images superposed, should show the complementary colours.] If there were a perfect photographic plate, there would not be much difficulty in calculating directly the colours for the three screens which should be used. As, however, no photographic plate is perfect in one sense, the proper exposing screens have to be ascertained by trial. It is useless to make such trials with the spectrum, and I have adopted a system which allows an accurate determination to be made by trial.

(23.) The Pritidple on which a Colour Sensitometcr is Made.

The principle I have employed, and which has been outlined before, is as follows : — If we have to find a screen to take what we may call the red negative (i.e., one in which the opacities of deposit are proportional to the red components of the objects photographed), we may take a variety of pigments, each of which contains red, and utilize them for the purpose. Such pigments may show a diversity of luminosities, and the relative proportions of red, green, and blue will also be very different in each. If (say) squares of paper are covered with the pigments of different colours and photographed through almost any coloured screen we should be unable to say without measuring the different opacities of deposit whether the screen was correct or not. If, however, by some artifice we are able to make all the red components in each of the pigments identical, and then photograph them, it is evident that the only screen which would be correct would be that which would make the opacities of all the images of the different squares of colour the same. The mode I have adopted of reducing the intensities of pigments and making all the luminosities of red, green, or blue the same, is by making annuluses of the different pigments and filling up parts of them with black pigment (the amount of white light reflected from such black being measured and taken into account), and then rotating them round the centre of the disc on which they are fixed.

AND ITS APPLICATION TO THE DETERMINATION OF COLOUR SENSATIONS. 355

(24.) Practical Application of the above Principle.

The method shown of ascertaining the composition of the colours in terms of the three sensations, and of ascertaining their luminosity, enables us to make an accurate determination of the amount of reduction which the various pigments should undergo. Suppose we wish, for instance, to make the red sensations in the yellow, the green, and the white the same, we should proceed as follows : —

The amount of red, green, and blue sensations in these three on the same empiric scale are —

	RS. i	GS.	BS.
White	i I . . 571	248	3")C4
Chrome-yellow . . . p]merald-green	. . 503 202	178 133	•694 1 • 4 1 8

If we reduce these sensations to colours, from Table IX. then we shall have for (say) the red component in white 342, in chrome-:yellow 284, and in emerald-green 79.

In order to reduce all these to show equal red components, the centre of the disc would be occupied by emerald-green pigment. The chrome-yellow would have to be reduced to -2\a4-, or '278 of its normal luminosity, so that '278 of 360°, or 260° of the annulus, would have to be occupied by dead black.

The white would have to be reduced to -g7^, or '231 of its normal luminosity, so that 277° of the annulus would have to be occupied by dead black.

If a green screen had to be obtained the green sensations reduced to green colour would be white 447, chrome-yellow 298, emerald-green 255. Then, as before, emerald-green would occupy the centre of the disc, and chrome-yellow would have to be reduced to fff, or '856 of its luminosity, and white to f I*, or '534 of its luminosity.

The above will give an idea of the method to be adopted in making what I have called colour sensitometers. Examples have been given only with those pigments which have been considered in the foregoing pages ; but naturally there would be many other colours introduced in order, as far as possible, to imitate the spectrum colours.

2 z 2

[ 357 ]

XI. The Pressure of Explosions. — Experiments on Solid and Gaseous Explosives. Parts I. and II.

By J. E. PETAVEL. Communicated by Professor ARTHUR SCHUSTER, F.R.S.

Received August 18, — Read November 16, 1905. [PLATE 21.]

CONTENTS.

Page

Introduction . . .358

PART I. METHODS AND APPARATUS.

Explosive pressure gauges 359

Maximum pressure gauge . . . 359

Recording manometer 361

Chronograph 364

Explosion chambers 365

Spherical explosion chamber 366

Cylindrical enclosure 368

Firing plug ; . . . . . 368

Standard gauges 368

Valves and connections . 369

PART II. EXPERIMENTAL INVESTIGATION OF THE EXPLOSIVE PROPERTIES OF CORDITE.

General shape of the curves 373

Effect of the diameter of cordite 374

Effect of the enclosure 379

Relation of pressure to gravimetric density 384

Distribution of the explosive 386

APPENDIX : Tables of numerical results 390-398

VOL. CCV. — A 397. 30.12.05

358 MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

LIST OF FIGURES.

No.

1. Maximum pressure gauge . . .

± Recording manometer diagram . . .

3. Recording manometer drawing . • • 363

4. Chronograph camera

5. Spherical explosion chamber • • 366

6. Cylindrical explosion chamber • 368

7. Firing plug

s. Valve and cone connections • • 370

9. Effect of ignition by oxyhydrogen and by gunpowder compared ... . . 372

10. Typical time pressure curve resulting from the explosion of cordite in a closed vessel . . . 373

11. Variation of rate of explosion with the size of cordite used, gravimetric density 0- 10 . . . 374

12. Variation of rate of explosion with the size of cordite used, gravimetric density 0-15. . . 375

1 3. Effect of the gravimetric density and the diameter of the explosive on the time required to

reach the maximum pressure 375

1 4. Rate of rise of pressure for cordite of the smallest diameter 377

15. Effect of the shape of the enclosure on the maximum pressure developed by cordite of large

diameter 380

16. Effect of the dimensions of the enclosure on the rate of cooling of the products of

combustion 381

17. Variation of maximum pressure with gravimetric density .... 384

18. Diagram illustrating the action of the explosion wave which is set up when the explosive

is unevenly distributed 386

19. Variation in the rate of combustion and maximum pressure produced by a non-uniform

distribution of the charge for cordite of 0-475 inch diameter 389

•20. Variation in the rate of combustion and maximum pressure produced by a non-uniform

distribution of the charge for cordite of 0' 175 inch diameter 389

INTRODUCTION.

THE scientific treatment of this question may be said to date from the researches of Count RUMFORD who, at the end of the eighteenth century, devised the first apparatus by which explosive pressures could be estimated with some degree of approximation.

During the past century the natural fascination of the subject, and the importance of the problems involved, attracted many of the ablest scientific minds. Several have made the study of explosions the object of their life work.

In the short space available, an adequate historical epitome is unfortunately impossible. A mere enumeration of the names with which we shall most frequently have to deal must therefore suffice.

Our knowledge of the behaviour of solid explosives is due principally to the brilliant work of NOBLE in this country, and of BERTH ELOT and VIEILLE abroad. With regard to explosive gaseous mixtures, the exhaustive work of LE CHATELIEB and MALLARD in Paris, of DIXON in Manchester, and CLERK in London, is familiar to all.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 359

At first sight it may appear to be over ambitious on the part of the author to attempt to add to the edifice built up by such able investigators. Closer consideration will, however, show that there is a gap in the structure ready to be filled by the small stone which he has quarried out.

In the case of solid explosives, thanks to NOBLE'S crusher gauge, the actual maximum pressure attained can be accurately measured. The mechanism of the explosion itself and the rate at which the pressure rises from the moment of ignition need, however, further investigation.

For gaseous explosives the same criticism holds true, more especially for mixtures which are highly compressed before they are fired. The first case has a bearing on all ballistic problems, the second provides some of the data necessary to the designers of the modern gas engine, and thus both are of considerable practical, as well as scientific importance.

PART I. — METHODS AND APPARATUS. Explosive Pressure Gauges.

At the time this research was started, some six years ago, there was no instrument by means of which the variation of pressure during the course of such explosions could be satisfactorily recorded. Numerous attempts have been made, but without success, to reduce the moment of inertia of the existing types of recording manometers sufficiently to make them of service for this work. The natural period of oscillation, however, invariably proved to be too slow. In consequence, the curves traced out did not record the rise of pressure in the enclosure, but merely the vibrations set up in the mechanism of the gauge by the sudden shock to which it was subjected. To design a satisfactory instrument it was, therefore, necessary to start ab initio. Before, however, the work could be carried out, some further knowledge of the conditions prevailing during the explosion was necessary, and this more especially in the case of highly compressed gaseous mixtures, the behaviour of which was at the time practically unknown.

Maximum Pressure Gauge.

For this work a gauge was employed the construction of which will easily be understood from the drawing given in fig. 1. In principle the apparatus is the same as that used by BUNSEN, and consists of a piston closing an aperture in the explosion chamber, the piston lifting if the pressure of the explosion rises above the load for which it is set.

To reduce the inertia to a minimum, the weights, used in BUNSEX'S apparatus, are replaced by a gaseous pressure. The moving part consists of a double-headed piston

360 MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

(P, p}, the smaller end of which (^>) is exposed to the force of the explosion, while the larger end (P) closes' a cylinder filled with gas at a known pressure. The piston,

Fig. 1. Explosion gauge. (Maximum pressure indicator.)

The gauge consists of a double-headed piston, P, p. The smaller head p is exposed to the pressure of the explosion, which is counterbalanced by a fixed gaseous pressure acting on the larger head P. The ratio of the two areas (and therefore of the two pressures when in balance) is fifty to one in the case of the gauge illustrated in this drawing. The lift of the piston is limited to about one hundredth of an inch, the distance of the stop B being adjusted by means of a fine screw. The piston on lifting closes an electric circuit and works an indicator. S is the stuffing box through which the stop B passes, C the cover of the cylinder in which the piston P works ; it is held clown by the nut N. G is the gas inlet by means of which the space E is connected to a source of supply of gas under pressure and to a gauge. K is the plug through which the electric connection to the insulated contact-piece H is made. To prevent back pressure, which might arise through leakage past either of the leathers, the space X is connected with the atmosphere by means of the vent V.

on lifting, closes an electric circuit and works an indicator. To ensure rapid action, the travel of the piston is limited to about a hundredth of an inch.

Two such instruments were constructed. The first, for pressures up to 100 atmospheres, had a ratio of 4 to 1 ; in the second (shown in fig. 1), intended for use up to 1000 atmospheres, the ratio of the areas of the two sides of the piston was 50 to 1. Fairly satisfactory measurements of the maximum pressure were obtained by means of this apparatus.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 361

With this instrument the work is very tedious, and no information is obtained as to the rate of combustion of the explosive. The experience gained during the course of the above preliminary investigation was, however, of the greatest use in the design of the final apparatus.

Recording Manometer.

The requirements for a reliable recording gauge are somewhat complex. In the case of gases, the explosive pressures to be dealt with range from 100 to 800 atmospheres ; in the case of solid explosives it was desirable to extend the research to pressures of 2000 atmospheres, or above. The combustion of several gaseous mixtures is much more rapid than that of the fastest explosives used in ballistics, and the time period of a recorder designed for this work must, therefore, be exceptionally small.

Before passing on to a description of the instrument it may be well to recall in a few words the law which governs the time period of vibrating bodies.

If A represent the force required to produce unit deflection of the vibrating system, W the weight of the moving parts, the time period will be

We have, therefore, two variables at our disposal, namely, the weight of the moving parts and the controlling force. The former must be made a minimum, the latter a maximum.

In most instruments where a short period is desirable, the strains to which the parts are subjected are very small, and the desired result is obtained by decreasing the size of all moving parts, and using, wherever possible, materials of low density. This method is employed in the case of all oscillographs, telegraph recorders, phonograph receivers, galvanometers, &c.

In the present case, the instrument having to withstand pressures of 20,000 or 30,000 pounds per square inch, applied with extreme suddenness, strength becomes a condition of vital importance, and steel is the only material which will withstand the strain. We cannot, therefore, use materials of small density, neither can we reduce the dimensions of the moving parts below a certain limit.

It is thus evident that we must have recourse to the second variable factor to secure the short time period which is necessary. As we have seen above, the controlling force brought into play per unit length of motion must be as great as possible. In other words, we must use the stiffest spring we can obtain.

The stiffness of a spring will vary with the material of which it is made and with its shape, increasing as the shape approaches more nearly to that of a solid bar subjected to longitudinal strain. This bar can be made as short as may be desired

VOL. CCV. — A. 3 A

362 MR J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

and, in theory, the time period of the system is only limited by the density of the material and by its modulus of elasticity.

In practice, however, the travel of the moving parts cannot be indefinitely decreased, for the deflections must remain of such dimensions as to be accurately

measurable.

The following diagram illustrates the application of the principles we have just established to the construction of a recording instrument (see fig. 2).

Fig. 2. Diagram of recording manometer.

A cylindrical groove is cut half through the walls of the enclosure. The upper part, P, of the cylinder thus obtained represents the piston of our indicator, and the lower portion, S, the spring. Under the pressure of the explosion the piston P will be forced outwards to an amount corresponding with the elastic compression of the material of which the spring is made. This motion is transmitted to the exterior by the rod R.

The lever L, supporting the mirror, rests on the fulcrum F at 3 ; it is kept against the knife-edge 2 of R by the tension of the wire W. The wire W is of considerable length, and is stretched almost to its limit of elasticity. The lever L can, therefore, follow the small advance of the rod R without greatly diminishing the tension of the wire W. The mirror focuses a point source of light on to a rapidly revolving cylinder, thus recording on a magnified scale the motion of the piston P.

It is not impossible that an indicator of this type would work in practice, but the deflection of the mirror, and, therefore, the scale of the records obtained, would be much too small. To increase the deflections, three modifications are necessary — the

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

363

spring S must be made longer, the ratio of its cross-sectional area to that of the piston must be decreased, and the knife-edges 2 and 3 be brought closer together.

In fig. 3 the design of the actual instrument is given, the lettering being the same as in the previous figure.

By means of the thread U the gauge screws into the explosion chamber, the end C of the piston being flush with the inside surface. An air-tight joint is formed by

w

LJ LJ

01 234 56789 10

SCALE /N INCHES.

Fig. 3. Recording manometer.

the ring U on the manometer pressing against a flat ledge in the enclosure (see fig. 5, a). The end of the gauge from D to E is a good fit in the walls of the explosion chamber, and the joint is thus protected from the direct effect of the explosion.

The spring S, about 5 inches in length, is tubular in shape. To prevent any buckling it is made to closely fit the cylinder, in which it is contained, at two places, el and e2. The spring is fixed at the outer end Z, being held in place by the nut K ; at the inner end it is free and supports the piston P. The copper gas check used in the crusher gauge is replaced by a leather washer, attached to the piston by the screw C and to the fixed part of the gauge by the ring E. The end of the piston projects by about one-hundredth of an inch, and it can therefore move back, by this amount, without straining the leather.

The mirror (not visible in the figure) is carried by the lever L. This lever is so designed that the knife-edges 1, 2 and 3 (see fig. 2) are in the same plane, it being at the same time possible to bring the knife-edges 2 and 3 within one-hundredth of an inch of each other, should so great an amplification be found necessary. Up to the present, however, the distance has not been decreased below one-sixteenth of an inch, the scale obtained with this distance being found sufficiently large.

The actual working of this type of recorder has proved very satisfactory. Its time period is sufficiently small to allow records to be obtained not only of the curve of rise of pressure of the fastest cordite, but also of the rapid vibrations which modify the curve under certain conditions.*

* Captain BRUCE KINGSMILL has proposed the application of this gauge to ballistic work with a view to " indicating " a gun in much the same manner as we now indicate a steam engine. This suggestion, which might lead to valuable results, has, as far as I am aware, not yet been carried out in practice.

3 A 2

364

ME. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

Chronograph.

Owing to the high speed required, the chronograph used for this work had to be specially designed. It is unnecessary to go into all the details of its construction. The ordinary methods were used to measure the velocity of the rotating drum and to ensure the constancy of speed during the course of an experiment.

When measuring the rise of pressure during an explosion, a linear speed of between 100 and 1000 centims. per second was used. For measuring the fall of pressure during the cooling of the products of combustion the driving mechanism could be geared down to give a linear speed of 5 or 10 centims. per second.

The drum of the chronograph can be easily detached and taken to the dark room, where the photographic film is wound on ; it is then placed in a light-tight box. As explained in connection with fig. 4, this box is so arranged that the drum can be

SCALE IN INCHCS.

Fig. 4. Chronograph camera.

The drum D is shown fixed on the axle A of the chronograph. To remove the drum without exposing the film which is wound round it to the light, the camera is first moved a little to the right, causing the ring E on the camera to fit into the groove F of the drum. The brass tube G is next forced into the groove H ; its cover, C, can then be removed and the nut N unscrewed. The camera, with the drum firmly held in it, can now be detached from the chronograph (by sliding it to the right) and taken to the dark room, where the film is developed and replaced by a fresh one.

fixed on to the axle of the chronograph in the full daylight without fogging the film. The box surrounding the revolving drum is pierced with a long and very narrow slit ; this, in turn, is covered by a shutter, which is lifted immediately before the explosive is fired and closed again a second later, after the photograph has been taken.

Thanks to the above arrangement it is not necessary for the room in which the experiments are carried out to be absolutely dark. The mirror of the recorder is

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 365

illuminated by a straight-filament incandescent lamp, the image of the filament being focused on to the slit of the chronograph camera, forming a straight streak of light perpendicular to the axis of rotation. The beam of light is deflected to an amount proportional at each instant to the pressure in the explosion chamber and, travelling along the slit of the camera in a direction parallel to the axis of rotation, traces out a curve on the photographic film. The ordinates of this curve represent the instantaneous pressures, the abscissae the times at which the said pressures existed.

A low-voltage high-candle-power lamp is used to illuminate the mirror, the comparatively thick filament of such a lamp giving correspondingly more intense illumination. At the moment of firing, the lamp is switched for a few seconds on to twice its normal voltage, and thus the strongly actinic light required is produced.

The recorder is calibrated by hydrostatic pressure before and after each set of experiments.

Explosion Chambers. .

It is well known that the shape of the enclosure has a considerable effect on the behaviour of the explosive during combustion. On the other hand, the ratio of the internal surface to the total volume of the chamber determines to a large extent the rate at which the pressure will subsequently fall.

With a view of obtaining some further information on these questions, two explosion chambers were constructed having approximately the same volume, but differing largely in shape. The first, a sphere, offers the least possible cooling surface ; whereas the second, a long narrow cylinder, has a surface more than twice as great.

One of the subjects of the research was to study the oscillations of pressure which are set up under certain conditions. In a long cylinder such oscillations are easily started, but in a small sphere the symmetrical shape and the short distance from wall to wall tend to equalise the pressure existing at each instant throughout the enclosure. Thus, in a spherical enclosure, the pressure rises usually without vibration and forms a smooth curve, the shape of which depends exclusively on the nature of the explosive used. In a long cylinder, however, the normal curve is modified by the distribution of the explosive, the method of firing, and various other factors.

Before designing these chambers, the relative advantages of solid metal and wire winding were fully considered. The latter construction, if properly carried out, adds considerably to the ultimate strength. A system of winding suitable for a spherical enclosure is, however, not easy to devise, and this fact, together with the ever important consideration of cost, led to the adoption of solid walls.

Mild steel was chosen as the material best suited to withstand the sudden impact of an explosion. The limit of elasticity, ultimate strength, and elongation of test pieces cut perpendicular to the direction of rolling were carefully determined before the forgings were machined.

36(5

ME. J. E. PETAVEL ON THE PRESSUEE OF EXPLOSIONS. Spherical Explosion Chamber.

The first explosion chamber is a nearly perfect sphere, 4 inches in diameter (see fig. 5). The measurements made in a plane passing through the axis of rotation

Fig. 5. Spherical enclosure.

The recording gauge screws in at A, the firing plug at B, and two valves at C and D respectively. The spigots, which are turned on the forging at either end (A and 13), fit into a cast-iron stand, to which the enclosure is firmly bolted.

when in the lathe (i.e.., in the plane in which any variation from the spherical shape would be a maximum) showed that the greatest divergence from the mean diameter did not exceed one hundredth of an inch.

The cavity was cut out of a solid block of rolled steel through an opening only 1|- inches diameter, a clever piece of engineering, for which I am indebted to Messrs. LEXNOX and Co. Exceptional care was also taken to give the inner walls a smooth polished surface.

The internal volume of the cavity was redetermined by weighing the mercury required to fill it. From these determinations the diameter of the sphere is 10'20 centims. The volume is, therefore, 556 cub. centims. and the internal surface 327 sq. centims.

The minimum thickness of the walls is 2|- inches, and the apparatus would doubtless withstand a pressure of 2000 atmospheres. As, however, the experiments had to be carried out in an ordinary laboratory, under conditions which would have rendered

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 367

the consequences of an accident disastrous, it was decided not to exceed half this limit. The second enclosure was alone used for higher pressures, it being, as we shall see, of stronger construction.

Apart from the effect of actual pressure, that of the sudden impact or blow given by the more rapid explosives has to be considered. As will be seen below, some mixtures of compressed coal gas and oxygen develop their full pressure in something like one ten-thousandth of a second and, in fact, occasionally detonate. It is difficult to estimate the actual strain produced by a force so suddenly applied.* When we consider that the present work comprised the repeated explosion of such mixtures, it will be seen that exact calculation becomes impossible. In all probability, during the course of the first few explosions of this kind the part of the material nearest the inner surface is strained to beyond its limit of elasticity, and therefore yields. In the case of steel, like the present, having a fair elongation, the first effect is actually to strengthen the enclosure ; the inner layers of the steel having been thus permanently elongated are under an initial compression which will help them in resisting further deformation. Aided, however, by the extremely rapid variations of temperature, this effect will in time cause surface cracks. Under successive strains the cracks will deepen to an extent that may become dangerous. Being on the inner surface of the chamber, the extent of the damage cannot be clearly ascertained. In the present work this danger was guarded against by a method which, though perhaps some- what crude, is at least easily carried out and, faute de mieux, may be considered satisfactory. On the outer surface of the enclosure a ring was accurately turned ; the plane through the centre of this ring passes through the centre of the sphere and through the gas and mercury inlets : it therefore encircles the weakest portion of the enclosure. A large micrometer gauge was made, by means of which the diameter of this ring was from time to time measured. This micrometer will clearly show an increase of one three-thousandth of an inch on the 8-inch diameter, or a change of about one two-hundredth of one per cent.

Up to the present no variation of diameter has been detected, and it is reasonable to infer that the apparatus has not been strained to a dangerous extent.

A sectional drawing of the enclosure is given in fig. 5.

The recording gauge screws into A, the steel ring (D, fig. 3) pressing on to the ledge a and thus forming a joint. The end of the gauge fits closely into the neck b and protects the joint from contact with the heated gases. The firing plug fits into the aperture B.

When gaseous mixtures are to be tested, the two valves which screw into C and D are brought into use. The cavity is first filled with mercury through C and the gas is then forced in through D. As soon as the mercury has been driven out, the valve

* It is usual to take an instantaneous load as equivalent to twice the same statical load. In the present case, however, we have to deal with the momentum of the gas itself, which is travelling at an enormous speed.

368

ME. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

C is closed and the pressure and composition of the mixture adjusted by means of the apparatus described below.

After each explosion the sphere is washed out first with a solution of caustic potash, then with distilled water.

Cylindrical Enclosure.

The cylindrical enclosure, shown in fig. 6, is also made of mild steel.

The dimensions are: external diameter 12 '2 centims. ; internal diameter 3 "17 centims. ; length of bore 69'64 centims. It has, therefore, a capacity of 550 cub. centims. and an internal surface of 709 sq. centims. —roughly speaking, the same

D

33" -

Fig. 6. Cylindrical enclosure.

The recording gauge screws in at A, the firing plug at B, and two valves at C and D respectively. The volume of this enclosure is nearly the same as that of the sphere, its surface 2 '17 times as great.

volume as the sphere, but rather more than twice its surface. The various apertures are identical to those of the spherical enclosure and the gauges and other fittings can, therefore, serve for either apparatus. This cylinder has been used up to 2000 atmospheres and would doubtless be safe at a considerably higher pressure.

Firing Plug. The design of the firing plug is clearly shown in fig. 7.

Standard Gauges.

A vast number of measurements of statical pressure had to be made during the course of the work, more especially for the part dealing with gases. For this purpose the connections were arranged so that the gauges could be easily interchanged, each one being used for the range over which it was most sensitive. To determine the initial pressure and composition of the gaseous mixtures, two independent sets of observations were always taken. The pressure was first roughly adjusted to the desired amount by means of direct-reading Bourdon gauges, then accurately measured by a standard gauge. A series of mercury columns were used for the lower pressures and manometers of the Cailletet type for the higher pressures. The various

EXPEEIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

369

small modifications introduced in the construction of the latter instrument, though they added to its reliability, are not of sufficient importance to warrant a more detailed description.

Three gauges of this pattern were in use, the first reading from 3 to 12 atmospheres, the second from 12 to 50, the third from 50 to 200.

EXPLOSION ENCLOSURE

Fig. 7. Firing plug.

The insulation of the central conductor is cone-shaped, to prevent its being forced out by the pressure of the explosion. A small cartridge of fine gunpowder can, when required, be placed round the fine wire W. The gas-tight cone joint D is protected, in the usual manner, from direct contact with the flame by a projecting piece, which closely fits the aperture in the explosion chamber.

Valves and Connections.

The various valves by which the flow of the gas is regulated are of the type shown in fig. 8.

The gas inlet is at A, whereas B is connected to a gauge which indicates the pressures behind the valve. A fine screw-thread is cut on the spindle S. By turning the wheel W the conical end F of the spindle is lifted slightly from its seat and the gas flows to the part of the apparatus connected to C. To avoid any sudden rush of gas the spindle bears a slightly tapered prolongation, which nearly fits the outlet, and, therefore, several turns of the screw are necessary to give the full opening.

The many connections required throughout the apparatus are all cone joints of the type shown at C.

The female connection ends in a hollow cone, the angle being about 100 degrees. The male D is a cylinder of brass, an inch or two long, ending in a hemisphere, which is pressed into the cone by the nut N, the inner surface of which bears upon a ring K.

VOL. ccv. — A. 3 B

370

ME. J. E. PETAVEL ON THE PEESSUKE OF EXPLOSIONS.

A

Fig. 8. Valve and connecting cone.

The numerous valves required during the present research were all substantially of the type shown in this figure, though varying considerably in external shape according to the use for which they were intended. The above design was used for the valves serving to regulate the initial pressure and composition of the mixture in the experiments on gaseous explosives. The apparatus is fixed firmly to the working bench by screws (not shown in figure) passing through the four corners of the metal block.

Into this cylinder the copper tube is soldered for a distance of about three-quarters of an inch. These cone joints are superior to the lead washer joints, inasmuch as they are easily made or disconnected, last indefinitely, and remain gas-tight under all pressures.

PART II. — EXPERIMENTAL INVESTIGATION OF THE EXPLOSIVE PROPERTIES

OF CORDITE.*

The maximum pressure developed by explosives can be measured with considerable accuracy by means of the crusher gauge, which was devised some thirty-five years ago by Sir ANDREW NOBLE, t The classical work since carried out by this investigator is too well known to need a mention here. Attention may, however, be drawn to one of the more recent papers, in which NOBLE publishes the cooling curves of cordite and describes the instrument by which they were obtained.^ The apparatus is in principle not unlike an ordinary steam engine indicator, but the spring is initially compressed by an amount corresponding to nearly the full pressure of the explosion, and is automatically released when this pressure has been reached. By this ingenious contrivance the violent oscillations of the spring, which would be set up by the explosion itself, are avoided, and a clear record of the rate of fall of pressure is inscribed.

* The explosive used in the course of this work was issued, by order of the Secretary of State for War, as representing the service cordite of the year 1902. Samples of three different sizes were included in the issue, the nominal sizes being 50/17, 20/17 and 3f.

t 'Proc. Roy. Tnst.,' vol. VI., p. 282, 1871; see also 'Phil. Trans. Roy. Soc.,' vol. 165, p. 49, 1875, and 'Phil. Trans.,' vol. 171, p. 203, 1880, &c.

| 'Proc. Eoy. Inst.,' vol. 16, p. 329, 1900.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 371

In ballistic tests the total energy imparted to the projectile is calculated from the readings of the Holden-Boulanger chronograph, and, in the case of specially constructed experimental guns, the Noble chronograph gives valuable information on the distribution of pressure within the gun itself.*

With regard to the more destructive explosives, such as blasting powders, dynamite, &c., their power is usually estimated by means of the Trauzlf lead block. At Woolwich this method has, however, been recently abandoned, an apparatus of the pendulum type being now in use.J

By the above methods the maximum pressure and the rate of fell of the pressure, or at least the total energy, can in most cases be satisfactorily estimated.

Comparatively little information is, however, available with regard to the initial part of the explosion; i.e., the behaviour of the explosive from the moment at which it is fired up to the time when it is fully consumed.

This point deserves further investigation, the action of the explosive during this period being no less important than the question of the maximum pressure attained.

It must be borne in mind that any structure, whatever its nature, will behave very differently according as it is exposed to a stress gradually applied, or is subjected suddenly to the same stress, or finally is submitted to violent oscillations of load.

In the case of a gun any abnormally rapid explosion gives rise also to another source ot danger. The time elapsing between the ignition and the complete combustion of the charge may be insufficient to allow the inertia of the shot to be overcome and to move it through an appreciable distance. Should this occur, the products of combustion would be confined in an unduly small space, and the pressure would rise above the safe limit.

The study of the initial stage of the explosion for various powders has formed part of the researches carried out by the Service des Poudres et Salpetres in Paris. The gauge first used by VIEILLE was a modification of the crusher gauge §, while of late years he has worked with a new type of spring manometer. ||

In Germany, BICHEL, BRUNSWIG^ and others have suggested that the properties of explosives should be determined by measurements made at relatively low pressures, the results being deduced by extrapolation. Careful work has been carried out by BLOCHMANN** under these conditions. The gravimetric densities ft used are from O'Ol

* 'Report Brit. Assoc.,' Oxford, 1894, pp. 523-540.

t 'Ber. Int. Kong. Angew. Chem.,' Berlin, 1903, vol. II., pp. 299-303 and 462-465. I Captain DESBOROUGH'S report. See '25th Report of H.M. Inspector of Explosives.' § 'Comptes Rendus,' vol. 112, p. 1052, 1891.

|| ' Memorial des Poudres et Salpetres,' vol. XL, pp. 157-210, 1902 ; see also ' Comptes Rendus,' vol. 115, p. 1268, 1892.

H 'Ber. Int. Kong. Angew. Chemie,' vol. II., pj. 282-299, 1903. ** 'DiNGLER's Poly. Journ.,' vol. 318, pp. 216 ar.d 332, 1903.

tt Gravimetric density is defined as the ratio of the weight of the charge to the weight of that volume of water which would fill the enclosure ; it is, therefore, numerically equal to the specific gravity of the gas produced when the explosive is fired.

3 B 2

372

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

to 0'02 and the maximum pressures recorded below one half ton per square inch. It is necessary to point out that such a method may not infrequently lead to most serious errors.

Finally, it is generally understood that, in connection with this subject, numerous experiments have been carried out at Woolwich under the direction of Major HOLDEN, but no results have as yet been published.

Experimental Work.

A preliminary question to be decided, before starting the series of experiments, referred to the method of ignition. The usual practice is to fire the charge of cordite by means of a small quantity of fine powder, which is ignited either by a percussion cap, or by a metallic wire which is brought to incandescence by an electric current. Some records were taken in this way, but it was soon found that alterations in the amount and disposition of this firing charge, though leaving the actual maximum pressure almost unaffected, caused some variation in the shape of pressure curve (see fig. 9). When a relatively small quantity of the igniting charge is used in an

o.oi

0.0?

0.03

0.01 0.02 0.03

Time in seconds.

Fig. 9. Comparison of the effect of ignition by oxyhydrogen and gunpowder.

Cordite of 0-175 inch diameter in a cylindrical enclosure; charge uniformly distributed; gravimetric density 0 • 1 ; A, fired with oxyhydrogen gas ; B, fired with fine-grained powder.

enclosure of considerable length, only the part of the cordite in immediate proximity seems at first to take fire, and the flame is then propagated from layer to layer of the explosive. When the firing charge is larger, or the dimensions of the enclosure

EXPEEIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

373

smaller, or, thirdly, when very fine cord is used, a more satisfactory ignition is obtained. This point in itself would be well worth more careful investigation, but as the present research refers principally to the properties inherent to cordite itself, it was desirable to be independent of such disturbing factors. The ideal conditions would be realised if a method could be found of igniting every particle of the explosive at the same instant over its entire surface. These conditions are approached by the process used.

After the required quantity of cordite had been filled in and the explosion chamber closed, the air therein contained was displaced by a mixture of oxygen and hydrogen at, or near, atmospheric pressure, and this was fired off in the usual way by an electric current. The velocity of the explosion of this mixture is such that the effect of the gaseous combustion is practically over before the pressure of the burning cordite begins to make itself felt, and each cord, being entirely surrounded by the flaming gases, cannot fail to ignite over its entire surface. On the records the impact of this preliminary explosion is marked by a slight tremor occurring just before the actual rise of pressure occurs. The pressure due to the gaseous explosion is about 10 atmospheres which, when compared with the 1000 or 2000 atmospheres resulting from the explosion of the cordite, does not form a serious correction.

General Shape of the Curves.

All the records exhibit certain general characteristics. The typical curve of rise of pressure is illustrated in fig. 1 0. It consists of three parts : (a) beginning nearly

— »- Time in seconds. Fig. 10. Typical time pressure curve resulting from the explosion of cordite in closed vessel.

asymptotical to the time axis and, gradually rising more rapidly, corresponds to the first stage of the combustion ; (b) referring to the full blast of the explosion, shows a much faster and almost constant rate of rise ; while at (c) the rapid decrease in the surface of the explosive can no longer be counterbalanced by the accelerating effect of the higher pressure. At c, therefore, the curve turns round sharply and merges into the cooling curve. So much for the general shape of the records. As we shall see

374

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

below, a more detailed study shows that, while conserving the same configuration, the actual curve may, according to circumstances, either be smooth (see Plate 21, figs. 1 and 2), or made up of continuous vibrations (see Plate 21, fig. 3), or, thirdly, composed of a series of small but sharp steps corresponding with the successive impacts of the explosion wave (see Plate 21, fig. 4).

Effect of the Diameter of Cordite.

The velocity of the explosion depends primordially on the diameter of the cordite, but is modified to some extent by the distribution, the method of firing, and more especially by the gravimetric density. Fig. 1 1 shows the rise of pressure for three

1 000

o

-^

a s

0.01 0.02

0.03 0.0* 0.05 0.06 Time in seconds.

0.07 0.08 0.09

Fig. 11. Showing variation of rate of explosion with size of cordite used.

Gravimetric density 0 • 1 ; charge uniformly distributed ; cylindrical enclosure used ; A, diameter of cord 0'035 inch; B, diameter of cord 0'175 inch; C, diameter of cord 0-475 inch.

different diameters of cord (0'475 inch, 0'175 inch, 0'035 inch); the gravimetric density is in every case O'lO. The largest size is used for heavy ordnance, the smallest size for the army rifle. The three tests were made under the same conditions and in the same enclosure.

Fig. 12 relates to a similar experiment carried out at a higher pressure. Lastly, in fig. 13, the time required for the complete combustion of cordite of various diameters is plotted for three distinct gravimetric densities.

The relation between the time occupied by the explosion and the diameter of the cordite, as shown in this figure, is practically a linear one, the lines converging

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

375

0.02

0.04 0.06

Time in seconds.

0.08

Fig. 12. Showing variation of rate of explosion with size of cordite.

Gravimetric density 0'15; charge uniformly distributed; cylindrical enclosure used; A, diameter

of cord 0-175; B, diameter of cord 0-475.

% 1" • — A fl

01 5 o Diameter in millimetres.

^

/

X

^

^

iameter of cordite 0 p 0 < — r\> la -I

$

y^

'*

^

^

/

//

/&

^

^

*

,X

r*

^

^

C^

<"

Q O.OI 0.0? 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.IO O.ll O.I2

Time required to reach the maximum pressure (in seconds).

Fig. 13. Effect of the gravimetric density and of the diameter of the explosive on the time required

to reach the maximum pressure.

376 ME. J. E. PETAVEL ON THE PKESSURE OF EXPLOSIONS.

towards the zero of time and diameter. We may, therefore, conclude that the combustion of finely divided cordite is nearly instantaneous. Under such conditions the result of an explosion would be very destructive, and it is possible that some abnormal effects which have on certain occasions been observed may be due to the pulverisation of the explosive at any early stage of the combustion.

However rapid an explosion may be, it remains, in principle, very distinct from a detonation. In an explosion the combustion is propagated from layer to layer without discontinuity. In a detonation the chemical reaction is practically instantaneous and simultaneous throughout the entire mass. The determining cause is, in this case, a compression wave of sufficient intensity to raise the material to its temperature of ignition.

Let us take for the sake of illustration a numerical example, although the values employed can only be rough estimations, and suppose a sphere of cordite 1 centim. in diameter under a gravimetric density of O'l. If this were ignited in the ordinary way, the combustion would travel towards the centre of the sphere at an average rate of 8 centims. per second and the maximum pressure would therefore be reached in 0'063 second. If, on the other hand, the material were to detonate, the detonation wave would travel through the mass at a speed of something like 800,000 centims. per second,* and the total time occupied would be one hundred thousand times less.

In an explosion we have usually to deal with pressures which may be considered as statical as far as their action is concerned ; in a detonation with a dynamical pressure or impact. The impact of the products of combustion travelling with enormous velocity may correspond in effect to an instantaneous pressure five or ten times greater than the normal pressure calculated from the composition of the explosive and its heat of reaction.

A typical case of this kind occurred when working with a compressed mixture of coal gas and oxygen. The total pressure of the explosion should have been some 4 or 5 tons per square inch. The mixture, however, detonated, and the solid steel piston of the recorder, though encased in a steel cylinder over 2 inches thick, was expanded outwards like the head of a rivet, t It is not easy to estimate exactly the statical pressure required to produce a corresponding effect, but it cannot be less than 25 tons per square inch.

To return now to the work on cordite, the results obtained with one of the smallest diameters in use are shown in fig. 14. It will be seen that, though the time occupied by the combustion is small, amounting to less than O'OOS of a second, the shape of the

* ABLE found that the rate of detonation of a train of dynamite or guncotton was about 608,000 centims. per second. See also SEBERT, BERTHELOT and METTEGANG. The latter (' Ber. 5. Int. Kong. Ang. Chem., Berlin, 1903,' vol. II., p. 322) gives 700,000 centims. per second as the detonation rate of dynamite.

t A similar effect is recorded hy NOBLE ('Proc. R. I.,' 1900), as having been produced on the copper of a crusher gauge by a charge of lyddite.

EXPEEIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

377

curve is perfectly normal, showing clearly the three distinct stages of combustion referred to on p. 373.

The law of combustion by parallel surfaces as expounded by VIEILLE* applies well to the case of cordite, t

0.005

0.010

0.015

0.005

0.010

Time in seconds.

0.015

Fig. 14. Showing the rate of rise of pressure for cordite of the smallest diameter. Diameter 0'035 inch (0'89 millim.).

A, spherical enclosure ; charge uniformly distributed ; gravimetric density 0 • 074. B, cylindrical enclosure ; charge uniformly distributed ; gravimetric density 0'075. C, cylindrical enclosure; charge concen- trated in one quarter of cylinder, nearest the recorder; gravimetric density 0'075.

The speed at which the flame travels inwards towards the centre of each cord is uniform and relatively slow. When unconfined, cordite burns at a rate of about 0'5 centim. per second. In a closed vessel the average speed increases to 5 centims. per second for an explosion developing 500 atmospheres, 8 centims. for a maximum of 1000 atmospheres, and 11 centims. per second for 2000 atmospheres.^

The shape of the curve representing the rise of pressure depends essentially on two

* 'Comptes Eendus,' vol. 118, pp. 346, 458, 912; 1894.

t The peculiarly regular combustion of cordite was first noticed by NOBLE, who in 1892 ('Proc. Roy. Soc.,' vol. 52, p. 129) remarks that the pieces of cordite blown from the muzzle of the experimental gun he was using were so uniformly decreased in diameter that they might readily have been mistaken for newly manufactured cordite of smaller diameter.

| The time required for the full pressure to develop is, therefore, proportional to the diameter of the cord. The formula L =«= r/c (where L is the time in seconds and »• the radius in centimetres) gives a fair approximation, though, as we shall see, the actual time varies somewhat, according to the conditions of the experiment. The constant c is characteristic of the explosive and, of course, equal to the above rates of combustion.

VOL. CCV. — A. 3 C

378 ME. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

factors : (1) on the surface of the explosive exposed to combustion and hence on the radius of the cords at each instant during the reaction ; (2) on the radial speed at which the zone of combustion is travelling towards the centre of each cord. This speed may be taken as proportional to the pressure. The formula S = ap (where S is the speed in centimetres per second, p the instantaneous pressure in tons per square inch, and a an empirical constant equal to about 3 '5) may be of use where it is not possible to make a direct experimental determination.

The maximum pressure (P) developed by a given charge is usually well known, and by aid of the above formula the curve of rise of pressure can therefore be obtained The radius of the cordite for successive intervals of pressure (p = O'l P, p = 0'2P, &c.) is first computed, and the time required to burn through the corresponding distance at the average pressure (p = 0'5 P, p = 0'15 P, &c.) is then determined. In calculating the radius, the volume of the unburnt explosive must, of course, be taken into account, and this renders the work somewhat tedious.

The formula does not take into account the fact that under experimental conditions some time elapses while the flame is spreading before the normal rate of combustion is set up. The zero of the calculated curve is, therefore, shifted somewhat to the right, and a sharper slope given to the initial stage (a, fig. 10).

It may with some truth be argued that the error occurring at a very low pressure would not affect the results as applied to ballistics, the calculation arid experimental curves being in agreement by the time the motion of the shot commences. It is hoped, however, that the day is not far distant when we shall be able to obtain an indicator card from a gun with the same ease as we now indicate other heat engines ; approximate calculations such as the above will then cease to be of practical value.

We have explained above the system used for firing the charge. When the key is pressed, the atmosphere of oxyhydrogen, with which the enclosure has been filled, explodes and the cordite is surrounded by a sheet of flame. The time at which this takes place is recorded by a slight tremor of the gauge. The charge does not ignite at once,* for though the explosive is surrounded by an intensely hot flame, a quite appreciable time is required for its surface to rise to the temperature of ignition, t

The ignition begins at the ends of each stick or at other parts, where, for instance owing to a blister, the conductivity has been reduced. The last parts to be attacked are those which were in contact with the walls of the enclosure or with some other portion of the charge. These circumstances, together with a slow rate of combustion which is characteristic of cordite under very low pressures, account for the gentle slope of the first part of each curve.

* In the appended tables and curves, time is counted from the instant the cordite ignites, as marked by the first permanent rise of pressure.

t A stick of cordite may under ordinary conditions be passed comparatively slowly through the flame of a Bunsen burner without igniting. If, however, its surface has previously been scratched or scored, the smaller particles will ignite at once and set fire to the mass.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 379

When fully ignited, each particle is freely suspended in space, being kept from direct contact with other bodies by the rush of flame issuing from its surface. It is to these conditions that the law of combustion by parallel layers accurately applies.

While the combustion is taking place, heat is being continually transmitted to the walls of the enclosure, and the maximum pressure attained will therefore be less for a slow explosion than for a fast one ; the actual effect may be seen by reference to figs. 11, 12 and 15.

The heat loss accounts also, as stated above, for the manner in which the curves of rise and fall of pressure merge together. By the time the maximum pressure is nearly reached the diameter of each particle of explosive is greatly reduced. The weight of substance consumed per unit time begins therefore to decrease, although the flame is actually advancing towards the axis of each cord at an ever increasing speed. Finally, the combustion just counterbalances the total thermal loss, and the curve of pressure remains for an instant practically constant at its maximum value. This will be seen clearly in figs. 1 and 2 on Plate 21.

Effect of the Enclosure.

We have just referred to the thermal loss due to the cold walls of the explosion chamber. The total loss, cceteris paribus, is proportional to the time.

When the diameter of the cordite, and consequently the time occupied by the combustion, is very small, the theoretical value of the maximum pressure is closely approached, and the shape and size of the enclosure have but little effect (compare A and B, fig. 14). These factors become, however, of considerable importance in determining the maximum pressure developed by the slower burning cordite (see fig. 15).

The shape of the cooling curve depends, on the other hand, essentially on the dimensions of the enclosure. In fig. 16 the facts are clearly illustrated by the results of comparative experiments carried out respectively in a sphere and in the cylinder.

It is proposed to reserve the general discussion of the questions of dissociation and rate of cooling for the third part of the present research ; we shall then be dealing with gaseous mixtures of simple composition which will serve as a natural introduction to the consideration of more complicated questions. A few words are, however, necessary with regard to the somewhat unusual conditions under which the cooling of the products of combustion of a solid explosive takes place.

Under ordinary circumstances the convection and conductivity of the gas itself are the ruling factors which determine the rate of cooling.

The thermal capacity of the gaseous mixture and the rate at which heat can be transmitted through it are low compared with the corresponding properties of the enclosure. These facts hold true whether the latter is water-cooled or not.

In such cases neither the inner surface of the enclosure nor the layer of gas in

3 c 2

380

ME. J. E. PETAVEL ON THE PKESSUKE OF EXPLOSIONS.

0.4 0.6

Time in seconds.

0.8

1.0

Fig. 15. Showing the effect of the shape of the enclosure on the maximum pressure developed

by cordite of large diameter.

Gravimetric density 0 • 1 ; charge uniformly distributed ; A and AI, in spherical enclosure ; B and BI, in cylindrical enclosure; A and B, diameter of cord 0'475 inch (12 '07 millims.); AI and BI, diameter of cord 0-175 inch (4 -44 millims.).

contact with it rise much above atmospheric temperature, and the rate at which heat is dissipated depends on the temperature gradient which is set up in the gaseous mass.

In previous papers* I have pointed out how the rate of transmission of heat in a gas varies with the pressure. In the case of air, for instance, the law

E x 106 = 403p°'M+ 1 -63/'21 3

was verified up to 1000° C. and 170 atmospheres, t At this pressure already air transmits heat at the same rate as a substance having twenty times the conductivity of air at atmospheric pressure.

* 'Phil. Trans.,' A, vol. 191, pp. 501, 524, 1898; and vol. 197, pp. 229-254, 1901.

t E is the heat abstracted from each square centimetre of surface of the hot body measured in therms per second per degree temperature interval. 5 is the temperature of the hot surface measured in degrees Centigrade, and p the pressure of the surrounding gas in atmospheres.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

381

0.2

0.3 0.4 0.5

Time in seconds.

0.6

0.7

0.8

Fig. 16. Effect of the dimensions of the enclosure on the rate of cooling of the products of

combustion.

A, cylindrical explosion vessel; gravimetric density 0-15; diameter 0-175; uniformly distributed.

B,

C, spherical

D, „

E, cylindrical

F, " „ G,

H, ,,

0-15:

0-1 ;

0-1 ;

0-1 ;

0-1 j

0-1 :

0-1 :

0-1 ;

0-475;

0-175;

0-475;

0-035;

0-175;

0'175; not uniformly distributed.

0 • 475 ; uniformly distributed.

0 " 475 ; not uniformly distributed.

When considering the products of an explosion, it must be remembered that the effective conductivity of the gas is further increased by its state of rapid motion. It is also augmented by the large proportion of hydrogen and water vapour contained therein.

As a result the temperature of the walls of the enclosure rises rapidly as the cooling of the gas proceeds, and before long the rate of cooling will depend essentially on the conductivity of the walls of the enclosure and not on the properties of the gas. The heat abstracted per unit time will then be simply proportional to the temperature.

If the logarithmic decrement of the latter part of the curve is measured, it will be found that the theory is confirmed in this respect by the results of the experiments.

382 ME. J. E. PETAVEL ON THE PEESSURE OF EXPLOSIONS.

The quantity of heat which is transmitted to the walls of the enclosure during the brief period occupied by the cooling of the gas is much greater than would occur in cases met with in ordinary engineering practice. With a gravimetric density of O'l the amount of heat to be absorbed per unit surface of our cylindrical enclosure is some hundred times as large as that which would be absorbed by the cylinder of an ordinary

gas engine.

In the case of artillery of large calibre the inner surface of the steel probably attains a temperature close to its melting-point and the correspondingly plastic material yields easily under the combined friction and chemical action of any escaping gas. In the case of small arms, the temperature being limited by the relatively small volume and therefore small thermal capacity of the gaseous mass, practically no erosion takes place.

To return now to the experimental work. In the following table the time required for the pressure to fall to three quarters, one half, one quarter of its maximum value is given for a number of distinct experiments, whereas the cooling curves for three different diameters of cordite at gravimetric densities of O'l and O'l 5 will be found plotted in fig. 16. It is noticeable that after the first tenth of a second the curves taken under similar conditions, but for various sizes of explosive, lie closely together, showm'r that the diameter has no material effect on the subsequent rate of cooling.

When we refer, however, to the table, we see that the times required to reach a given fraction of the maximum are different for different diameters.

This apparent discrepancy is explained by the fact that the total quantity of heat absorbed is primordially a function of time. When the combustion is very rapid, the maximum pressure is reached while the walls of the enclosure are still cold and the percentage fall of pressure per unit time is high. With a slow-burning cordite the surface of the enclosure becomes considerably heated during the combustion of the explosive, and after the maximum the percentage fall of pressure is correspondingly lower. Briefly stated, at any fixed interval of time after ignition the total heat absorbed by the enclosure, and, therefore, the temperature of its inner surface, will be nearly the same for all diameters of the explosive. In consequence, the rate of cooling as measured by the rate of change of pressure at any stated time is unaffected by the speed of combustion.

The rate of cooling for a given volume of the enclosure does not vary, as is usually assumed, in proportion to the surface, but nearly as the square of the surface.

It will be noticed that the cooling in the cylinder is about four times as rapid as in the sphere, whereas the ratio of the two surfaces is as 2 '17 to 1.

In such massive enclosures the heat generated by the explosion is at first entirely absorbed by the inner layers of the steel walls. It does not travel to the outside until some time after the explosion is over, A decrease in the surface has, therefore, a double effect. The heat to be absorbed per unit area and the average thickness of metal through which this heat must be transmitted are both increased.

EXPEKIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

383

a

o O

o

03

'S

.9

^-H

o

6

C*H

o

H EH

«1

PH

	>b
	t-O		0 00
	I— 1 r— 1		i—c (M
	O O		0 O
	io
	t- O	e	O OJ t~
	p— » I— t		r-H <M t~
		o
	0 0	F*	0 O 0
	^	'c
	ib	tS
	t~ 0	a	C-l CO
	-* r-H	o	r-H CO
	O 0		0 0
	io lO
	CO £— o o		IO t- IO ^> r— H ^
	0 O		o o o
	io 10
	CO t- 0 0		t- oo 10 O l-H -HH
_•	0 0		O 0 0
BB O •c •73	io t- IO	6 •2	CO -*
1	0 0	'S p	o 6
	io
	t- 10		CO CO
	0 0		0 0
	io
	CO O	fe	rf, CO o
	0 71	S ^S -^r-	o c-i o
	6 o	1 ll	o o o
	t- o	p §'§}	CO IO CM
	r-H l-H	^QT;'1-1	r-H CO t~
	0 0		o o o
	io
	t- o		^5 t— O5
	r-H r-H	a	O <M to
	0 0	E-, o	o o o
	io	'S
	i- O	p	CO -C O
	-^ r-H		r-H CO 00
	0 0		o o o
	io
	I- O l-H r-H		o
	0 O		0
'a	-	g
'E	t~~ f"^	o	<*
Q}	-HH r-H	J*H	IO
ft	0 0	c	o
	io
	t- 10		II
	r-H O		1
	0 0		o
•	• •	•	^oiS ocsocS
*	* •	•
	If	... *	tt'f-i O S-i o 'JH S-, ^ 0
•	o &	•	'cai«4H •'^g^'eg'" •
	*+H O	c	o-jo osoos'g
I 8	23 O IO -3 a QJ C 6 >	0	'^ ^^ J '"- ^S 7r ^"fe J5 g^Sceg^csS-SPs
	c3 ™	•
w	S O	s	S^ H " H ^

384

ME. J. E. PETAVEL ON THE PKESSURE OP EXPLOSIONS.

Relation of Pressure to Gravimetric Density.

The present work was not taken up with a view to specially investigating the above subject, which has already been fully treated by NOBLE. It is, however, of interest to compare the results with the much more complete set published by this investigator.

To minimise the effect of the rapid rate of cooling, which, as we have just seen, is inherent to small enclosures, we must select for comparison the values obtained when using cordite of relatively small diameter. The pressures obtained with cordite of 0'175 inch and 0-035 inch diameter are shown in fig. 17, marked in on the curve representing NOBLE'S results, and are, as will be seen, in close agreement with it.

OJ5

Gravimetric density. Fig. 17. Variation of maximum pressure with the gravimetric density of the charge.

The curve is traced out from the values given by Sir ANDREW NOBLE ; the points marked on it refer to the results incidentally obtained in the course of the present work.

Though the pressure and temperature are exceptionally high, there is no reason for supposing that the products of combustion depart considerably from the law which governs the pressure of gases at ordinary temperatures.

This law may be written

In the present case, where the temperature is very high and constant, we may put RT = c, and for a first approximation neglect cohesion of the gas. The formula then takes the simple form

p (v—b) = c.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

385

The volume to which the gas will be reduced under infinite pressure may be taken as closely approaching the inverse of the density of the solid explosive. Therefore

b =

1-56

= 0-641,

whereas v is the inverse of the gravimetric density p. Thus

c = £ -0'641». P

To minimise the error due to cooling we will take the value of p obtained for the smallest cordite in the spherical enclosure. At a gravimetric density of 0'0744 this is 5'137 tons per square inch (see Table VI.), and therefore

c _

0-0744 The pressure developed by the explosive is

o-641 x 5-137 = 6575.

P =

cp

" "

-bp " 1-0-64 l

The results calculated from this formula are compared in the following table with NOBLE'S values and with those obtained during the course of the present work* :—

		Pressure determined	Pressi
Gravimetric density.	Pressure calculated.	experimentally by	expo
		NOBLE.
0-05	3-40	3-00
0-10	7-03	7-10
0-15	10-91	11-30
0-20	15-08	16-00
0-30	24-42	26-00
0-40	35-37	36-53
0-50	48-38	48-66
0-60	64-10	63 • 33

nitally by PKTAVEI..

2-87

7-01

11-48

In the above table the pressures are expressed in tons per square inch.

The experimental results are influenced by many factors, such as the size of the enclosure, the dimensions of the explosive, and the oscillations of pressure, which are doubtless occasionally .set up. On the other hand, the formula we have used does not

* When the pressure is measured in kilogrammes per square centimetre the constant c becomes 10355, whereas t. = 10021 gives the pressure in atmospheres, the constant b in either case remaining unaltered. A formula similar to the above was used by NOBLE and ABEL in connection with their researches on fired gunpowder. They assumed that the gases strictly followed BOYLE'S law, but introduced a factor (1 -- a.p) to allow for the space occupied by the solid residues left after the explosion.

VOL. COV. — A. 3 D

386

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

take into account the cohesion of the gas, or allow for the possible variation of the value b with temperature and density.

Taking these circumstances into account, the agreement between the theoretical and experimental values may be considered satisfactory.

Distribution of the Explosive.

In a long narrow vessel a certain amount of vibration almost invariably occurs during the combustion of the explosives. If the explosive is concentrated in one part only of the enclosure, the effect is increased and the pressure rises by sharp steps, as shown in fig. 18. With some powders the sudden increments of pressure become

T~/ME v

Fig. 18. Diagram showing the type of vibration set up at the commencement of an explosion when the charge placed in a long enclosure is not uniformly distributed. The successive sharp increments of pressure correspond with successive impacts of the wave.

dangerously large and an abnormally high maximum is reached in one or two steps. This phenomenon seems to be the transition between an explosion and a detonation.

That it is difficult, in fact almost impossible, to detonate cordite has long been recognised as one of its principal advantages. Nevertheless, signs of abnormal explosion were visible whenever the charge was crowded together in one part of the enclosure. A fairly typical case is shown in fig. 4, Plate 21, a similar effect being recorded in many other cases, notably F 68, F 69, and F 70 (Tables XL, XII., XIII.).

The experiments in this direction had to be confined to pressures of about 1000 atmospheres. From these tests it seems probable that by working under similar conditions, but with a higher gravimetric density, cordite would give results not unlike those obtained by VIEILLE* in the case of " B.F." and other powders.

* See "Etude des Pressions Ondulatoires," 'Annales des Poudres et Salpetres,' vol. III., pp. 177-236. VIEILLE, in the course of this work, obtained instantaneous pressures amounting to three times the normal value. Using a method of calculation similar to that given below, he showed that the speed of propagation of the smaller disturbance is in fair agreement with the speed of sound in the same medium.

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES. 387

Unfortunately, for this very reason, the experiments could not be carried out in a laboratory.

The sharp steps which go to make up these records may be accounted for in the following manner : When the explosive, which is packed closely at one end of the chamber, bursts into flame, a pressure wave is sent out which travels to the end of the cylinder and is then reflected back. When this wave, on its return journey, reaches the explosive, the combustion, which in the meantime had been proceeding uniformly, is accelerated in proportion to the increased pressure. The case is one of rmitual reaction between the two phenomena. Any irregularity in the combustion tends to start a pressure wave which in turn enhances this irregularity. The process is cumulative in its effects, and with the high gravimetric densities used in ballistic work it may, and doubtless occasionally does, cause disastrous results.*

Incidentally the present work confirms ViEiLLE'st views as to the discontinuity of pressure set up by wave actions, the successive steps of the curve rising abruptly, if not instantaneously.

The velocity of propagation of the wave is measured directly by the time elapsing between the successive sharp increments of pressure which are recorded.

When a wave is set up at the commencement of the explosion, the impacts on the recording gauge succeed each other at intervals of 0'00125 or 0'00121 second when the charge in the cylinder is at gravimetric densities of O'lO or 0*15 respectively. The path traversed, i.e., the double length of the enclosure, is 139'3 centime, and the corresponding velocities 1114, 1150 metres per second. \

Occasionally, when cordite of the smallest diameter is used, the wave motion is still sharply defined at the maximum pressure. The time interval is then O'OOHO second for a gravimetric density of 0*1 and the speed 1266 metres per second.

From the general formula for the velocity of sound we can calculate the theoretical speed under these circumstances,

V= A/— • P

These factors, with the exception of y, are well known,

When the combustion is complete, the density, p, of the resulting gases is equal to the gravimetric density of the charge.

The elasticity, E, is measured by the rate of change of pressure with density.

* See CORNISH, 'Proc. Inst. Civ. Eng.,' vol. 144, p. 241, 1901. t 'Memorial des Poudres et Salpetres,' vol. 10, pp. 177-260, 1899-1900.

| Theoretically the speed should be the same in either case ; the thermal loss, which is relatively less at higher gravimetric densities, probably accounts for the difference.

3 D 2

388 MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

It can, therefore, be obtained by differentiating the expression

which was given on p. 385.

Carrying out this operation we find

The value of the ratio of the specific heats, y, is somewhat uncertain. For the mixture of gases resulting from the explosion, y may be taken as 1'35 or P21, according as the specific heats are considered constant or variable with temperature.

The following table gives the velocity of sound, calculated according to each of the above hypotheses :—

VELOCITY of Sound in the Gases Produced by the Combustion of Cordite at the • Maximum Pressure of the Explosion, measured in Metres per Second.

Gravimetric density.	Velocity for = 1 • 35.	Velocity for = 1-21.
o-i	1251	1185
0-2	1343	1272
0-3	1450	1373
0-4	1575	1491
0-5	1723	1632
0-6	1903	1801

The limiting value for low densities, which should correspond with the speed of the wave at the commencement of the explosion, works out at 1170 (y = 1'35) or 1108

Although, strictly speaking, the above theory applies only to very small disturb- ances, the calculated velocities are in fair agreement with the measurements given on p. 387.

The oscillations referred to in the preceding paragraph are superimposed on the curve of pressure without directly altering its general shape. Within the limits of the present experiments the wave action, consequent on the uneven distribution of the charge, by increasing the thermal loss slightly lowers the maximum pressure. The rate of combustion is, also, somewhat altered ; usually it is accelerated.

These effects will be best understood by reference to figs. 14, 19, and 20, in which the mean values of the pressure are plotted in terms of the time.

1000

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

0-01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

389

o.oa

0.03 0.04 0.05

Time in seconds.

0.06

0.07

0.08

jLime in suconus.

Fig. 19. Variation of the rate of combustion and of the maximum pressure produced by a non- uniform distribution of the charge.

Cylindrical enclosure; gravimetric density O'l ; diameter of cord 0'475 inch (12 "07 millims.). A, charge uniformly distributed ; B, charge placed in one sixth of the cylinder near the recorder.

0.005

O.oio

0.015

o.oeo

O.OiO

0.035

0.005

0.010

0030

0.035

0.015 o.oeo o.oas

Time in seconds.

Fig. 20. Variation of the rate of combustion and of the maximum pressure produced by a non- uniform distribution of the charge.

Cylindrical enclosure; gravimetric density 0-1 ; diameter of cord 0-175 inch (4'44 millims.). A, charge uniformly distributed; B, charge placed in one half of the cylinder farthest from the recorder; C, charge placed in one sixth of the cylinder farthest from the recorder. This case is somewhat exceptional. The charge was so closely packed that it formed a nearly solid mass, which was probably scattered on ignition by the pressure of the gas produced behind it.

390

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

Generally speaking, the results obtained confirm the remarkable properties ot cordite with regard to its high power and to the regularity of the effects produced. It would doubtless be very desirable to extend the research to higher pressures and carry out, on similar lines, a comparative study of other explosives. Treated, however, in this general way the subject is too vast to be dealt with single-handed, and the writer can but express a hope that others more competent and better equipped will be found willing to take up the work.

Before closing I desire to thank Professor ARTHUR SCHUSTER for placing at my disposal the ample resources of his laboratory.

The cost of the apparatus has to a large extent been defrayed by funds awarded by the Government Grant Committee of the Royal Society, while for the cordite I am indebted to the courtesy of the War Office authorities.

APPENDIX.

In the following tables numerical results obtained from the measurements of the principal photographic records will be found.

Where wave action is set up, the pressure given is the mean value of the instantaneous pressure at the time indicated.

TABLE I.— (Record No. F 55.)

Spherical explosion vessel ; charge uniformly distributed ; gravimetric density 0-0496 ; diameter of cord

0-475 inch (12-07 millims.).

Maximum pressure 404 atmospheres (2 '65 tons per square inch); time required to reach the maximum pressure 0'120 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
o-oio	10	0-130	404
0-020	22	0-200	397
0-030	31
0-040	43
0-050	67
0-060	98
0-070	150
0-080	215
0-090	287
0-100	363
0-110	397
0-120	404

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

391

TABLE II.— (Record No. F 56.)

Spherical explosion vessel ; charge uniformly distributed ; gravimetric density 0 • 0496 ; diameter of cord

0-175 inch (4-44 millims.).

Maximum pressure 438 atmospheres (2-87 tons per square inch); time required to reach the maximum pressure 0 • 045 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-005	24	o-ioo	438
o-oio	48	0-200	421
0-015	86	0-300	390
0-020	131	0-400	364
0-025	187	0-500	339
0-030	271
0-035	383
0-040	433
0-045	438
0-050	438	i

TABLE III.— (Record No. F 57.)

Spherical explosion vessel; charge uniformly distributed; gravimetric density 0'024; diameter of cord

0-035 inch (0-89 millim.).

Maximum pressure 144 atmospheres (0-95 ton per square inch) ; time required to reach the maximum pressure 0-014 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-002	12	0-020	144
0-004	28	0-050	143
0-006	52	0-100	141
0-008	77
0-010	103
0-012	127
0-014	144

392

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

TABLE IV.— (Eecord No. F 59.)

Spherical explosion vessel; charge uniformly distributed; gravimetric density 0-099; diameter of

0-475 inch (12-07 millims.).

Maximum pressure 1069 atmospheres (7-01 tons per square inch); time required to reach the maximum pressure 0 • 065 second.

cord

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-005	10	0-070	1069
o-oio	34	o-ioo	1062
0-015	53	0-200	993
0-020	79	0-300	935
0-025	113	0-400	883
0-030	160	0-500	840
0-035	244	0-600	804
0-040	357	0-700	773
0-045	521	0-800	746
0-050	684	0-900	716
• 0-055	880	1-000	689
0-060	1024
0-065	1069

TABLE V.— (Record No. F 60.)

•Spherical explosion vessel; charge uniformly distributed; gravimetric density 0-099; diameter of cord

0-175 inch (4-44 millims.).

Maximum pressure 1115 atmospheres (7 -31 tons per square inch); time required to reach the maximum pressure 0-022 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-002	29	0-024	1112
0-004	59	0-030	1109
0-006	103	0-100	1062
0-008	150	0-200	986
0-010	229	0-300	927
0-012	370	0-400	874
0-014	522	0-500	821
0-016	754
0-018	971
0-020	1089
0-022	1115

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

393

TABLE VI— (Record No. F 61.)

Spherical explosion vessel; charge uniformly distributed; gravimetric density 0-0744; diameter of cord

0-035 inch (0-89 millim.).

Maximum pressure 783 atmospheres (5-137 tons per square inch); time required to reach the maximum pressure 0-008 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
o-ooi	23	o-oio	783
0-002	85	0-015	783
0-003	202	0-020	772
0-004	381	0-050	769
0-005	616	o-ioo	728
0-006	763	0-200	669
0-007	774	0-300	622
0-008	783	0-400	587

TABLE VII.— (Record No. F 63.)

Cylindrical explosion vessel; charge uniformly distributed; gravimetric density 0-1004; diameter of cord

0-475 inch (12-07 millims.). ; temperature 18-6° C.

Maximum pressure 916 atmospheres (6-01 tons per square inch); time required to reach the maximum pressure 0 • 070 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
o-oio	76	0-075	916
0-020	139	0-080	916
0-030	231	0-090	892
0-040	400	o-ioo	866
0-050	618	0-200	694
0-060	843	0-300	562
0-065	909	0-400	463
0-070	916	0-500	397
		0-600	331
		0-700	291
		0-800	255
		0-900	225
		1-000	198

VOL. CCV. — A

3 E

394

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

TABLE VIII. — (Kecord No. F 65.)

Cylindrical explosion vessel; charge uniformly distributed; gravimetric density 0-1004; diameter of cord

0-175 inch (4-44 millims.). ; temperature 18° C.

Maximum pressure 1041 atmospheres (6-83 tons per square inch); time required to reach the maximum pressure 0-028 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-005	66	0-030	1031
o-oio	192	0-035	1005
0-015	298	0-040	992
0-020	579	0-050	959
0-025	959	0-060	936
0-028	1041	0-070	909
		0-080	879
		0-090	860
		0-100	826
		0-200	645
		0-300	512
		0-400	423
		0-500	347
		0-600	298
		0-700	265
		0-800	235

TABLE IX.— (Record No. F 66.)

Cylindrical explosion vessel; charge uniformly distributed; gravimetric density 0-0753; diameter of cord

0-035 inch (0'89 millim.) ; temperature 19 '0' C.

Maximum pressure 793 atmospheres (5-20 tons per square inch); time required to reach the maximum pressure 0 • 007 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-001	26	0-008	787
0-002	102	0-009	783
0-003	195	o-oio	777
0-004	324	0-015	760
0-005	453	0-020	750
0-006	658	0-025	721
0-007	793	0-030	701
		0-050	655
		o-ioo	539
		0-150	456
		0-200	380
		0-300	281
		0-400	212
		0-500	179
		0-600	152

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

395

TABLE X.— (Record No. F 67.)

Cylindrical explosion vessel ; charge all in half of cylinder farthest from the recorder ; gravimetric density 0-1004 ; diameter of cord 0-175 inch (4 -44 millims.) ; temperature 19° C.

Maximum pressure 1035 atmospheres (6 -79 tons per square inch) ; time required to reach

the maximum pressure 0 • 026 second.

Time in seconds.	Pressure in atmospheres.
0-002	33
0-004	66
0-006	93
0-008	152
o-oio	188
0-012 248
0-014	317
0-016	430
0-018	549
0-020	694
0-022	833
0-024	955
0-026	1035

Time in seconds.	Pressure in atmospheres.
0-028	1031
0-030	1025
0-032	1018
0-034	1015
0-040	992
0-050	959
0-060	925
o-ioo	826
0-200	654
0 • 300	529
0-400	456
0 • 500	387
0-600	340
0-700	298
0-800	258
0-900	222

TABLE XL— (Record No. F 68.)

Cylindrical explosion vessel ; charge all in one-sixth of cylinder farthest from the recorder ; gravimetric density 1-004; diameter of cord 0-175 inch (4-44 millims.); temperature 18'6° C.

Maximum pressure 1002 atmospheres (6'57 tons per square inch) ; time required to reach the maximum pressure 0-030 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-002	33	0-032	1002
0-004	60	0-034	995
0-006	86	0-040	985
0-008	106	0-050	942
o-oio	149	0-060	919
0-012	179	0-100	820
0-014	241	0-200	621
0-016	307	0-300	522
0-018	387	0-400	423
0-020	509	0-500	337
0-022	648	0-600	281
0-024	777
0-026	879
0-028	975
0-030	1002

3 E 2

396

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

TABLE XII.— (Record No. F 69.)

Cylindrical explosion vessel ; charge all in one-sixth of cylinder near the recorder ; gravimetric density 0-1004 : diameter of cord 0-475 inch (12-07 millims.) ; temperature 18° C.

Maximum pressure 906 atmospheres (5 • 94 tons per square inch) ; time required to reach the maximum pressure 0 • 70 second.

Time in seconds.	Pressure in atmospheres.	Time in second
o-oio	43	0-075
0-015	73	0-080
0-020	106	0-090
0-025	145	0-100
0-030	188	0-200
0-035	235	0-300
0-040	374	0-400
0 • 045	539	0-500
0-050	678
0-055	807
. 0-060	879
0-065	899
0-070	906

Pressure in atmospheres.

906 896 869 850 671 562 456 364

TABLE XIII. —(Record No. F 70.)

Cylindrical explosion vessel; charge all in one quarter of cylinder near the recorder; gravimetric density 0-0753 ; diameter of cord 0~035 inch (0'89 millim.) ; temperature 17 '0° C.

Maximum pressure 764 atmospheres (5-01 tons per square inch) ; time required to reach the

maximum pressure 0'007 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-001	76	0-008	760
0-002	162	o-oio	754
0-003	241	0-015	727
0-004	417	0-020	694
0-005	546	0-030	661
0-006	724	0-040	628
0-007	764	0-050	595
		0-100	489
		0-200	357
		0-300	265
		0-400	212
		0-500	175

EXPERIMENTS ON SOLID AND GASEOUS EXPLOSIVES.

397

TABLE XIV.— (Record No. F 71.)

Cylindrical explosion vessel; charge uniformly distributed; gravimetric density 0-1004 ; diameter of cord 0'175 inch (4'45 millims.) ; temperature 17'5° C.

Maximum pressure 1058 atmospheres (6 '94 tons per square inch) ; time required to reach the maximum pressure 0'028 second; charge fired with 2 grammes of fine granulated powder.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
0-002	26	0-035	1058
0-004	43	0-040	1051
0-006	83	0 • 050	1025
0-008	126	0-060	998
o-oio	162	0-100	893
0-012	222	0-200	727
0-OH 291	0-300	612
0-016 417	0-400	503
0-018 545	0-500	413
0-020 698	0-600	331
0-022	843	0-700	291
0-024	975	0-800	248
0-026	1038
0-028	1058
0-030	1058

TABLE XV.— (Record No. F 72.)

Cylindrical explosion vessel; charge uniformly distributed ; gravimetric density 0'1004 ; diameter of cord 0-035 inch (0'89 millim.) ; temperature 17-7' C.

Maximum pressure 1124 atmospheres (7 -37 tons per square inch); time required to reach the maximum pressure 0 • 0050 second ; charge fired with 2 grammes of fine granulated powder.

Time in seconds. Pressure in atmospheres.	Time in seconds. Pressure in atmospheres.
0-001	208	0-006	1124
0-002 519	0-007	1107
0-003	807	0-010	1101
0-004	1025	0-015	1071
0-005	1124	0-020	1024
		0-030	992
		0-040	959
		0-050	925
		0-100	810
		0-200	645
		0-300	529
		0-400	430
		0-500	363
		0-600	324
		0-700	281
		0-800	248

398

MR. J. E. PETAVEL ON THE PRESSURE OF EXPLOSIONS.

TABLE XVI.— (Record No. F 73.)

Cylindrical explosion vessel ; charge uniformly distributed ; gravimetric density 0- 1505 ; diameter of cord 0-475 inch (12'07 millims.); temperature 18° C.

Maximum pressure 1633 atmospheres (10- 71 tons per square inch); time required to reach

the maximum pressure 0 • 058 second.

Time in seconds.	Pressure in atmospheres.	Time in seconds.	Pressure in atmospheres.
o-oio	26	0-060	1633
0-020	76	0-065	1631
0-025	126	0-070	1587
0-030	244	0-080	1547
0-032	311	0-090	1504
0-034	413	0-100 1461
0-036	523	0-150	1322
0-038	665	0-200	1207
0-040	793	0-300	1051
0-042	1015	0-400	935
0-044	1174	0-500	843
0-046	1332	0-600	767
0-048	1461	0-700	688
0-050	1554	0-800	628
0-052	1603	0-900	579
0-054	1620
0-056	1627
0-058	1633

TABLE XVII.— (Record No. F 74.)

Cylindrical explosion vessel ; charge uniformly distributed ; gravimetric density 0' 1505 ; diameter of cord 0-175 inch (4-44 millims.); temperature 17 '6° C.

Maximum pressure 1749 atmospheres (11-48 tons per square inch); time required to reach the maximum pressure 0'023 second.

Time in seconds.

Pressure in atmospheres. j| Time in seconds. Pressure in atmospheres.

)-002	33	0-028	1749
)-004 76	0-030	1742
)'006 99	0-035	1719
)-008	162	0-040	1692
)-010	225	0-050	1646
)-012	354	0-060	1610
)-014	545	0-070	1564
1-016	833	0-080	1527
1-018	1220	0-090	1494
1-020	1507	0-100	1455
1-022	1732	0-150	1316
)-023	1749	0-200	1197
1-024	1749	0-300	1041
		0-400	926
		0-500	853
		0-600	767
		0-700	701

[ 399 ]

XII. Fifth and Sixth Catalogues of the Comparative Brightness of the Stars —in Continuation of those Printed in the ' Philosophical Transact ions of the Royal Society' for 179fi-91>.

By Dr. HEESCHEL, LL.D., F.R.S.

Prepared for Press from the Original MS. Record* by Col. J. HERSOLIEL,

It.K, F.It.S.

Received July '24, — Read December 7, 190;).

IN the 86th, 87th, and 89th volumes of the l Philosophical Transactions of the Royal Society' — for 1796, 1797, and 1799 — there appeared a series of four papers by Sir WILLIAM (then Dr.) HKRSCHEL containing- the description and results of observations made by him of the "Comparative Lustres of Stars" visible to the naked eye in northern latitudes. They were arranged in six " Catalogues," of which four were actually published, as above. Apparently two more were to have followed, containing the remaining constellations. The annexed Tables show the distribution of the constellations among the six Catalogues.

It is not known what prevented the completion of the design at the time. Drafts of the intended Fifth and Sixth Catalogues exist among Sir WILLIAM'S papers, prepared, as the previous four had been, by Miss CAROLINE HERSCHEL, by abstraction from the body of his observations of various kinds, entitled "Abstract of Sweeps and Reviews."

Circumstances which it is unnecessary to detail have now led to the revision (and correction where called for) of these drafts and to their publication in the following pages, in the same form as those in the earlier volumes. To save reference to the latter, the following extract will explain the symbols used to denote relative brightness. These are more fully described and illustrated in the pages immediately preceding those from which the extract is made, viz., pp. 187-9 of vol. 86.

"Introductory Remarks and Explanations of the Arrangement and Character*

" used in the following Catalogue.

" This Catalogue contains nine constellations, which are arranged in alphabetical " order. I have called the present collection the first catalogue. The rest of the VOL. CCV. — A 398. 2.3.06

400 DE. HERSCHEL'S FIFTH AND SIXTH CATALOGUES OF THE

" constellations, which are pretty far advanced, will be given in successive small " catalogues as soon as time will permit to complete them.

" Each page is divided into four columns, the first of which gives the number of " the stars in the British catalogue of Mr. FLAMSTEED, as they stand arranged in the " edition of 1725.

" The second column contains the letters which have been affixed to the stars.

" The third column gives the magnitude assigned to the stars by FLAMSTEED in the " British catalogue ; and

" The fourth contains my determination of the comparative brightness of each star, " by a reference to proper standards.

" All numbers used in the fourth column refer to the stars of the same constellation •' in which they occur, except when they are marked by the name of some other " constellation ; and in that case the alteration so introduced extends only to the " single number which is marked, and which then refers to the constellation affixed '• to the number.

" The numbers at the head of the notes, which will be found at the end of the " catalogue, refer to the stars in the same constellation to which the notes belong.

" Simple Characters.

•' ' The least perceptible difference less bright.

" . Equality.

" , The least perceptible difference more bright.

"' - A very small difference more bright.

" -, A small difference more bright.

- A considerable difference more bright. - Any great difference more bright in general.

" Compound Characters, expressing the Wavering of Star Light.

" '. From the least perceptible difference less bright to equality.

" ; From equality to the least perceptible difference more bright.

" ~ From a very small difference more bright, to the least perceptible difference.

"' =, From -, to - &c.

" -The wavering expressed by the passing of the light from a state of the least ' perceptible difference less bright to equality, and to the least perceptible difference •' more bright.

" T The wavering expressed by the changes from - to , and to . or from . to and <l to - "

COMPARATIVE BRIGHTNESS OF THE STARS. DISTRIBUTION of Constellations in Catalogues.

401

An

Aq

Al

AT

An

Bo

Cm

Cc

Cv

Ca

Ci

CP

Cs

On

Ce

Ct

Co

Cl)

Cr

Constellation.

Andromeda . Aquarius . Aquila . Aries

Auriga . . . Bootes.

Camelopardalus Cancer . Canes venatiei Canis major . Canis minor . Capricornus . Cassiopeia . . Centaurus . Cepheus Cetus . . . Coma Berenices Corona borealis Corvus . Cygnus. Delphinus . Draco . . . Kquuleus .

Erianus Gemini .

Dl Dr

I?

Gm

Hr Hercules .

Hy Hydra . . . .

IIC Hydra et Crater

Lc Lacerta

La Leo major .

Li Leo minor.

Lp Lepus . . . .

Lb Libra . . . .

Lu Lupus . . . .

Lx Lynx . . . .

Ly Lyra . . . .

Mn Monoceros

Na Navis . . . .

Or Orion . . . .

Pg Pegasus . . .

Pr Perseus . . .

Ps Pisces . . . .

Pa Piscis austrinus . Sagitta ....

Sr Sagittarius

So Scorpio.

Ss Serpens

St Serpentarius . .

Sx Sextans . . .

Ta Taurus. . . .

Tr Triangulum . .

Ua Ursa major .

Ui Ursa minor .

Vr Virgo . . . .

VI Vulpecula. . .

Catalogue Number.

III.

1.

I.

II.

IV.

III.

V.

III.

VI.

II.

II.

I.

II.

III.

III.

IT. VI III. II.

1.

I. IV.

I.

II.

11.

I.

V.

V.

in. 11. v. in.

VI. VI. IV. IV. IV.

in.

in.

i.

IV.

v.

VI.

I.

V. VI. VI. VI. IV. IV. IV. VI. V. VI. V.

Number of stars in constellation.

66 108 71 66 66 54 58 83 25

31 ii

51

55

5

35

97

4.",

•21

9

81 18 80 10 09 85

113 60 31 16 95 53 19 51 5

45 21 31 •2-2 78 89 59

113 24 18 65 35 64 74 41

141 16 87 24

110 35

VOL. CCV. — A.

3 F

402

DR. HERSCHEL'S FIFTH AND SIXTH CATALOGUES OF THE NUMBER of Stars Catalogued.

CATALOGUE I.	Number of stars.	CATALOGUE IV. Auriga	Number of stars.
108 71 51 81 18 10 113 89 18	66 80 45 21 31 59 41 141 16
	Draco .
	Lynx
	Lyra
Cygnus	Monoceros
	Perseus
	Sextans
Pegasus
Triangulum
CATALOGUE II. Aries Canis major Canis minor	66 31 14 55 97 9 69 85 95	CATALOGUE V. Camelopardalus	58 60 31 53 113 65 24 35
Hydra et Crater ....
Leo minor
Cctus	Pisces
Sagittarius
Eridanus	Ursa minor Vulpeculu
Leo
CATALOGUE III. Andromeda Bootes Cancer	66 54 83 5 35 21 16 19 22 78	CATALOGUE VI. Canes venatici	25 43 51 5 24 35 64 74 87 110
Coma Berenices
Libra ....
Centaurus	Lupus
Cepheus	Piscis austrinus
Corona borealis Lacerta	Scorpio
Serpens ....
Lepus ....	Serpentarius ....
Navis	Ursa major Virso
Orion

COMPARATIVE BRIGHTNESS OF THE STARS.

CATALOGUE V. FIFTH CATALOGUE OF THE COMPARATIVE BRIGHTNESS OF THE STARS.

403

			Lustre of the Stars in Camelopardalus.
1		6	3. 1
2		5	2.3 7-2
3		6	2.3.1
4		6	7 - 4 , 5
5		6	4,5-8
6		6	8 , 6
7		5	779 Aur 7-2 7-, 8 7-, 4
8		7	7-8 5-8,6
9		4.5	10, 9
10		5.4	33 Aur ; 10 10 , 9
11		5	11 , 9 Aur
12		6	9 Aur -, 12
13		4.5	Does not exist
14		5	17 , 14 , 19
15		6	30 Aur (32) , 15
16		6	16 . 30 Aur (32)
17		6	31 , 17 , 30 17 , 14
18		6	24; 18
19		6	14, 19
20		7	22 . 20
21		6.7	30 , 21 . 23
22		7.8	24 , 22 . 20 28 . 22
23		6	21 . 23
24		6	26 . 24 , 22 20 ; 18
25		7.8	25 . 34
26		5.6	26. 24
27		5.6	Does not exist

3 F 2

404

DR. HERSCHEL'S FIFTH CATALOGUE OF THE

Lustre of the Stars in Camelopardahis — continued.
28		6.7	29 , 28 . 22
29		5.6	29 , 28
30		6	31-30 17 , 30 , 31
31		5	30 Aur (32) - 31 - 30 31 , 17 37 , 31 -, 38
32		5	32 - - 33 16 . 30 Aur (32) , 15 30 Aur (32) - 31 42 . 30 Aur (32)
33		7	32 - - 33 . 34
34		6	33 . 34 25 . 34 34 ; 35
35		5.6	34 ; 35
36		6	42 , 36
37		•5.6	37 , 31 37 , 40
38		7	31 -, 38
39		6.7	40- 39
40		6.7	37 , 40 - 39
41		7	8 Lyn , 41 - 10 Lyn
42		4.5	43 ; 42 . 30 Aur (32) 43 , 42 , 36
43		4.5	43 ; 42 43 , 42
44		6	46 ; 44 7 45
45		7	44 , 45
46		7	47 , 46 ; 44
47		6	18 Lyu , 47 , 46
48		6	56 - 48
49		5	51 , 49
50		6	27 Lyu - 50
51		5	55 -, 51 , 49
52		5	58 - 52 - 54
53		6	53 ,56 57 . 53
54		6	52-54
55		5	55 -, 51

COMPARATIVE BRIGHTNESS OF THE STARS.

405

Lustre of the Stars in Camelopardalus— continued.
56		6	29 Lyn - 56 58 , 56 53 , 56 56 - 48
57		5	58 . 57 57 . 53
58	5	29 Lyn , 58 , 56 58 . 57 58 - 52
Lustre of the Stars in Hydra.
1		4	1 ,2
2		4	1 , 2 - 10
3		6	15 ; 3 17
4	8	4	22 , 4 . 7 4.12 35 ; 4 , 31
5	(T	5	7,5 13 . 5 , 18
6		6	9. 6
7	>/	4	4.7,5 7,13
8		6
9		6	22 . 9 . 6
10		5 2 10
11	€	4	16 , 11 16-, 11 -22 4Ci"it.ll ll,4Civit
12		6	4 . 12
13	P	5	7,13. 5
14		5.6	18 . 14
15		6	15 ; 3
16	£	4	16 -, 11 17 Leo , 16 , 11
17		6	3-17
18	ia	6	5 , 18 . 14
19		6 19-20 27 - 19 - 20 23 ; 19 , 21
20		6	19 - 20 . 24 19 - 20
21		6	19 , 21
22	e	4	11 - 22 , 4 22 . 9
23		6	23; 19

406

DR. HERSCHEL'S FIFTH CATALOGUE OF THE

			Lustre of the Stars in Hydra — continued.
24		6	20 . 24 - 29 24 - 25
25		6	24-25
26		6	27 -26
27		6	27 - 19 27 - 26
28	A	6	28 . 33
29		6	24-29
,30	OL	2	46 Or - 30 - 53 Or
31	T1	5	4 , 31 ; 32
32	T"	5	31 ; 32 15 Sext - 32 32 - 30 Sext
33		6	28 . 33
34		6	27 . 34 -, 36
35	/,	4	35 ; 4 35 , 15 Soxt - 32
36		6	34 -, 36
37		6	37 . 34
38	K	4.5	407 38
39	1)	5	41 -, 39 -, 40
40	0 U"	5	39 -, 40 , 38
41	X	4	41 -, 39 4 Crat -, 41
42	/'•	4	2 Crat - - 42 . 43 42 , 7 Crat
43	p	5	42 . 43 . 1 Crat 1 Crat -, 43 1 Crat - - 43
44		6	44 , 3 Crat
45	*	6	8 Corvi - 45
46	y	3	7 Corvi = , 46 46 7 49
47		6	47 - 48
48		6	47-48
49	TT	4	46 7 49 49 - 20 Lib
50		6	52 , 50 50 -, 1 Lib
51		5	51 , 52

COMPARATIVE BRIGHTNESS OF THE STARS.

407

			Lustre of the Stars in Hydra — continued.
52		5	51 , 52 , 50
53		6	58 . 53 , 56 4 Lib .56 54-4 Lib
54		5.6	54 , 58 6 Lib - 54 - 4 Lib
55		6	57 . 55 . 59 57 7 55 -, 3 Lib 12 Lib , 55 - , 3 Lib
56		6	53 , 56 . 57 4 Lib . 56 . 57
57		7	56 . 57 . 55 56 . 57 , 55
58		5	54 , 58 . 53 6 Lib - 54
59		6	55 .59-60
60		6-7	59 - 60
			Lustre of the Stars in Hydra et Crater.
1	&	6	43 Hy . 1 3 . 1 -, 43 Hy I - - 43 lly 2 -, 1 3,1
2	</>«	5	2 - - 42 Hy 2-3 2,1 2-3
3	Ji	6	2-3.1 3 ,13 44 Hy , 3 2-3,1 3,6
4	J'	4	4 . 11 Hy 4-12 4 ; 9 Corvi 4 -, 12 4 -, 41 Hy 1 1 lly , 4
5	&s	G	6,5
6	i«	6	3,6,5 13-6
7	a	4	42 Hy , 7 15 , 7 . 11
8	i	6	10-8
9	X	5	9 , 10
10		6	9 , 10 - 8
11	P	3.4	7 . 11
12	s	4	4-12-15 4 -, 12 12 -, 15
13	A	5.6	3, 13 13-6 27 , 13 , 30
14	e	4	21 - 14 - 24
15	y	4	12 - 15 12 -, 15 , 7
16	K	5	24- 16
17		6	19-, 17 ,18 31 . 17 . 29

40H

DE. HERSCHEL'S FIFTH CATALOGUE OF THE

Lustre of the Stars in Hydra et Crater— continued.
18		6	17 ,18-26 28, 18
19	£	4	19 - 17
20		6	26 7 20 , 23 25 . 20
21	e	4 21-14
22		7	23 . 22
23		6	20 23 23 . 22
24	<	5	14 - 24-- 10
25	i)	5	25 . 20
26	-	6	18 - 26 , 20
27	i	4	27 , 13
28	ft	4	28 , 18
29'		6	17 . 29
30	v	4	13, 30 . 31
31		5.6	30 . 31 . 17
Lustre of the Stars in Leo minor.
1	1 '	1,4 5,1
2	^	6	3 . 2
3	o -u F-3	6	4-3.2 3.0
4	03 CO c	7	1,4-3
5	X	7	5, 1
6	-4-3	6	3. 6
7	5	6	8,7 19 Urssemaj , 7
8	i"	5	8,7 11.8 8-19Urssemaj
9	0> s	6	9 Leonis maj ,9.13 Leonis imij
10	O	4.5	39 Lyncis-, 10 -, 11
11	5-	6	10-, 11.8 11-13
12		5	13. 12

COMPARATIVE BRIGHTNESS OF THE STARS.

409

			Lustre of the Stars in Leo minor — continued.
13		6	11 - 13 .12
14		6	42 Lyncis — 14
15		6	15 . 42 Lyncis
16		6	17 , 16
17		6	19-- 17 , 16
18	i	6	20 - - 18
19		5.6	19-- 17
20		6	21 __ 20 -- 18
21		5	31 ; 21 - - 20
22	o	6.7	24- 22
23	o V> ee	5.6	23 -, 24
24	$ =	6	23 -, 24 - 22
25	o O E»	6	47 , 25
26	^a +3 g	6	27 - 26 . 29
27	o> >	6	28 . 27 - 26
28	•~L 0> ti ^	6	30 - 28 . 27
29	:Q (H 0) 43	6	26 . 29
30	4?5 O O	5.4	30-28
31	fc.	5	31 ; 21
32		6	38, 32
33		4.5	42, 33
34 I		4.5	34 -, 36 34 - 35
35		5.6	34 - 35 , 36
36		6	34 -, 36 35 , 36
37		3	37 -42
38		6	38, 32
39		6	40-39
40		6	41 -, 40 - 39 40 - 44

VOL. CCV. — A.

3 G

410

DE. HEESCHEL'S FIFTH CATALOGUE OF THE

			Lustre of the Stars in Leo minor — continued.
41		5	41 -, 40 41 - 53 41 - 52 Leonis mcij
42		4.5	37 - 42 , 33 42 , 44
43	2 .2	6	44 ; 43 _ _ 45 44 ; 43 - 45
44	c8 "3	6	40 - 44 , 43 42 , 44 ; 43
45	m C 0	6	43 _ _ 45 43 - 45
46	CO J3 43	4.5	36 Leonis maj - 46 , 24 Leonis maj
47	_g	6	46 Leonis maj - 47 47 , 25 46 Ursse maj -, 47
48	> •a	6	48 , 50
49	^i rt r»	6	51 Leonis maj 49
50	£> 49 4J O	6	48 , 50 50 , 52
51	o g.	6	52 ,51 52 , 51
52		5.6	53 --52, 51 50,52,51
53		5-6	41 _ 53 _ _ 52
			Lustre of the Stars in Pisces.
1		7	2.1-3
2		6	5,2.1
3		6	1 - 3
4	ft	5	4,5
5	A	6	4,5,2 7.5
6	7	4	67 28
7	6	5.6	10, 7 . 5 7-16 19 . 7 7-32 7 ; 34
8	*i	5	9--8
9	K2	7.6	9--8
10	6>	5	10 , 7 18 . 10
11		6	14,11 ,12
12		6	11 , 12 , 13
13		6	12, 13

COMPARATIVE BRIGHTNESS OF THE STARS.

411

Lustre of the Stars in Pisces — continued.
14		6	14, 11
15		6	16- 15
16		6	7 - 16 - 15
17	t	6	28,17 17,18
18	A	5	17 , 18. 10
19		5	19. 7
20		5.6	27 , 20 , 24
21		6	21 . 22
22		C	21 . 22 - 25
23		6	23 - 83 Pegasi
24	6	20 , 24
25		6	22 - 25
26	6 28 - - 26
27		5	29 . 27 , 20
28	CO	5	6 7 28 - - 26 28 , 17
29		5	30 - 29 . 27
30		5	33 . 30 - 29
31	f1	6	32 , 31
32	c-	5-6	7 -32 , 31
33		4 33 . 30
34		6	7 ; 34
35		6	41 , 35 , 36 35 , 51
36		6	35 , 36 , 38
37		6	39 ,37 42 , 37 43 7 37
38		7	36 , 38 - 45
39		6	40 ; 39 40 ; 39 , 37
40		6	40 ; 39 40 ; 39
41	d	6	41 , 35

3 G 2

412

DR. HERSCHEL'S FIFTH CATALOGUE OF THE

			Lustre of the Stars in Pisces — continued.
42		6	43 . 42 , 37 42 ; 43
43		6	43 .42 44 , 43 42 ; 43 ~ 37
44		6	44 ,43 44 - 10 Ceti
45		6	38 - 45
46		6	52 -, 46
47		6	47 . 52 47 - 48
48		6	47 - 48 . 49 48 -, 49
49		6	48 . 49 , 53 48 -, 49
50		6	See note at foot as to this number and 55
51		6	35 , 51
52		6	47 . 52 -, 46 56 , 52 , 54 See footnote
53		7	49 , 53
54		6	56 7 54 52 , 54 , 61 54 , 59
55		6	See note at foot
56		6	56 ,54 56 , 52 See footnote
57		6	58 ; 57 58 ; 57
58		7	58 ; 57 64 - 58 ; 57
59		6	54,59,61 66.59 66-59 61
GO		6	62 . 60
61		7	54 , 61 59 ,61 59 - 61
62		6	63 -, 62 . 60
63	8	4	63 , 62
64		6	64 - 58 64 - 66
65	i	6	65 . 68
66		6	64 - 66 . 59 66 59
67	k	6	68 , 67
68	h	6	65 . 68 , 67
69	o-l	5	83 , 69 . 82

COMPARATIVE BRIGHTNESS OF THE STARS.

41. S

			Lustre of the Stars in Pisces — continued.
70		6	Does not exist 71 - 70
71	£	4	71-86 71- -70
72		6	81 , 72 - 75 72 , 87
73		6	77 , 73 , 88
74	*	5	74 , 84
75		G	72 75
76	<r-'	5	78 . 70
77		6	80 - 77 , 73
78		6	82 - 78 . 76
79	f2	6	84 - 79 ; 81
80	«	5	80-77
81	^3	6	79 ; 81 , 72
82	g	6	69 . 82 - 78
83	T	5	83 , 69 s:i ; 90
84	X	5	74 , 84 - 79
85	4-	5	90 . 85
86	C	4	71 -, 86 86 , 89
87		7	72 , 87
88		6.7	73 , 88
89	/	6	86 , 89
90	U	5	83 ; 90 - 91 90 , 95 90 . So
91	I	6	90 - 91 95 , 91
92		7	97 , 92
93	p	5	93 . 94
94		5	93 . 94 - 97 94 , 107
95		7	90 , 95 , 91 96 , 95
96		6.7	96 , 95
97		6.7	94 - 97 , 92

414

DK. HERSCHEL'S FIFTH CATALOGUE OF THE

Lustre of the Stars in Pisces— continued.
98	/*	5	51 Ceti (106) , 98 106 - 98
99	n	4	99 , 5 Arietis 2 Trianguli - 99 - 5 Arietis
100		6	102 -, 100 101 , 100 101 - 100 , 104
101		6	101 -,104 105,101,103 102-101,100 102-101-100 101.105
102	7T	5	102 -, 100 102 - 101 107 , 102 - 109 102 - 101
103		8.7	101 , 103 105 , 103 , 104
104		6.7	101 -, 104 100,104 103,104
105		6.7	105 , 101 101 . 105 , 103
106	V	5	110-106,98 110-106-98 111,106-112
107		6.7	107 , 102 94 , 107 - - 109
108		6	Does not exist
109		8	102 - 109 107 - - 109
110	O	5	110 - 51 Ceti (106) 5 Arietis -,110 110 - 106
• 111	*	6	111 , 106
112		6.7	106 -, 112
113	a	3	113 , 5 Arietis
[NOTE to 50, 52, 55, 56.— The following entries occur : January 1, 1796, " Either 50 or 52 is wanting. By 46 it is 52 that is wanting "...." 56 is wanting." On the same date are comparisons involving 50 and 55, to which asterisks are affixed, referring to si footnote, in W. H.'s hand and obviously of later date, "* As it appears by Index that 50 and 55 have no observation, put 52 and 56 for them." In drawing up Catalogue V, C. L. H. has evidently done this, adding, however, " does not exist " opposite 50 and 55, which is, perhaps, hardly warranted. With this exception, the same substitutions have been made in this Abstract- -though the reason is not clear. — J. H.] [108 is shown to be (by an error of FI.AMSTEED'S, transferred to the Atlas) the same as 109, but 3° out of place.]
Lustre of the Stars in Sagittarius.
1		6	33 Scorpii , 1
2		6	2 , 52 Ophiuchi
3	P	6	51 Ophiuchi - 3

COMPARATIVE BRIGHTNESS OF THE STARS.

415

			Lustre of the Stars in Sagittarius— continued.
4	h	6.7	7.479
5	i	7	5,7 5 . 12
6		7	54 Ophiuchi -6.8
7	a	6	5,7.4 12 , 7
8		7	6.8 8 does not exist
9		7	479
10	y	3	19- , 10
11		7	Does not exist
12		7	5 . 12 , 7
13	/"'	4	27 , 13, 40 39 - 13- , 15 13 -, 21
14		7	15 , 14 . 16
15	1*	6	13- 7 15 , 14 21 7 15
16		7	14 . 16 - 17
17		7	16 - 17
18		7
19	s	3	38 . 19, 27 22 7 19 ; 20 19 = 7 10
20	€	3	19 ; 20
21		6	21 7 15 13 -, 21
22	A	4	41 . 22 , 38 22 ~ 19
23		7	25-23
24		7	24 -, 26
25		7	26 , 25 25 - 23
26		6	24 -, 26 , 25
27	4>	5	19 , 27 , 40 27 ,36 27 , 13
28		7	28-31
29		6	36 , 29 , 33
30		6	33 , 30 . 31 31 , 30
31		6	30 . 31 28 31 , 30

416

DR. HKRSCHEL'S FIFTH CATALOGUE OF THE

			Lustre of the Stars in Sagittarius — continued.
32	r1	5	32 ; 35
33		6	35 - 33 , 30 29 , 33
34	IT	4.3	34 _. -41 34~_4i 50 Aquilse , 34 . 33 Capricorni
35	l'-		32 ; 35 35 33
36	?		27 , 36 , 39 37 - 36 36 , 29 36 , 39
37	f	6	37 - - 36
38	f	3	22 . 38 . 19
39	0	4	36 , 39 36 , 39 39 . 44 39 - 13
40	T	4	27 , -10 13 , 40
H		4	34 11 •>•> it 41
42	^	5	42, 49
43	rf	6	46-43 . 45
14	P1	5	39 . 44 -, 46
45	r'	(i	43 ; 45 50 ; 45
46	t>	G	44-, 46 - 43
47	X1	5	47 - - 48 47 ; 49
48	X2	5	47 -48
49	x3	6	47 ; 49 42 , 49
50		6	50 ; 45
51	A1	6	52-51 51 , 53 . 53
52	A«	6	52 - 51
53		6	51 , 53 . 53
54	e1	6	55 ; 54 . 61
55	«»	6	55 ; 54
56	/	6	56 -, 57
57		6	56 -, 57
58	(0	5	62, 58 . 60
59	6	5	60, 59

COMPARATIVE BRIGHTNESS OF THE STARS.

417

			Lustre of the Stars in Sagittarius — continued.
60	a	5	58 . 60 , 59
61	9	6	54. 61
62	c	6	62, 58
63		6	63-64
64		6	63 - 64, 65
65		6	64,65
Lustre of the Stars in Ursa minor.
1	a.	3	7 ; 1 - 14 Draconis 1,7 a (1) - /3 (7) Polaris (1) '. 7 1,7 1-7 1-7 1-7 a. (50) UrsiB maj 7 1 7 7 1,7
2		6	Is wanting
3		6	4-3
4	b	5	5-4 4-3
5	a	4	22-5-4
6		7	11 - -6 9-6
7	ft	3	7 ; 1 l,7,y (33) Draconis 1-7 50 Ursie maj , 7 50 Ursce maj f 7 1 f 7 7 , 50 Ursa? maj 1,7 1-7 1-7 50 Ursa: maj '. 1 1-7 IT? 79 Ursse maj ,'7 7-64 Ursse maj 7 - 33 Draconis 1,7
8		6
9		7	9-6 9, 10
10		7	9 , 10, 14 14 . 10
11		5	13__ 11 __ 12 11 --6
12		7	11 -- 12 12 . 8 . 8
13	y	3	13- - 11
14		7	10, 14 14. 10
15	e	5	16-15 16-15 15 --18
16	t	4	16-15 16-15
17		7	19-17, 20
18		6	15-- 18

VOL. CCV. — A.

3 H

418

DE. HEESCHEL'S FIFTH CATALOGUE OF THE

Lustre of the Stars in Ursa minor — continued.
19		5	21 , 19. 20 21 , 19- 17
20		6	21 - 20 19 . 20 17 , 20
21	'/	5	21 -20 21 , 19 21 , 19
22	t	4	22-5
23	S	3	23 -, 24
24	6.7	23 - 24
Lustre of the Stars in Vulpecula.
1		5	1 - 1 Sagittie 1-2 1 -, 2
2		6	1-2,1 SagittiB 1 -, 2 , 1 Sagittse
3		6	6-3,3 Cygni 3 - - 3 Cygni 3 - - 3 Cygni
4		G	9,4.5
5		6	4.5,7
6		4	G-- 8 G- 3
7		5	9-7 5,7
8		G	G - - 8 8.3 Cygni 8 . 3 Cygni
9		6	5 Sagittaj -9,8 SagittiB 9-7 9,4 9-10 14-9
10		6	9- 10, 13 10-- 11 13-- 10 10, 14
11			10-- 11
12		5	13 - 12 -, 14
13		6	10,13 13-12 13 --10 16,13.17
14		5	14-9 12-, 14 10,14
15		4.5	15 , 23
16		5	16, 13
17		4.5	13.17 17.22
18		6.5	19 . 18, 20
19		6	19 . 18
20		5.6	18, 20

COMPARATIVE BEIGHTNESS OF THE STARS.

419

Lustre of the Stars in Vulpecula — continued.
21		5.6	23 , 21 -, 24
22		5	17 . 22
23		4.5	15 , 23 , 21
24		5	21 -, 24 24 . 25
25		6	24. 25
26		6	27, 26
27		5	27, 26
28		6	29 . 28 . 32
29		5	31 , 29 . 28
30		6	32 . 30
31	r	6	31 , 29
32	2	5	28 . 32 . 30 35 . 32
33		6	33-34
34		6	33-34
35		6	35 . 32

3 H 2

420 DK. HEKSCHELS FIFTH CATALOGUE OF. THE

NOTES.

rN.B. A long dash between two notes or remarks under the same number indicates

that they are disconnected, and occur at an interval of time — of days or months even — }n the course of the " reviews." The only connecting link is the number of the star to which they refer. — J. H.]

Notes to Camelopardalus.

8 Is not in the place where it is marked in Atlas : the RA should be + to make it agree with a star that is thereabout, or — to make it agree with another. Either of

them will be 7 -, 8. The star following 7 and 8, observed by FLAMSTEED, p. 286,

is in its place, but is much less than 6m. I should call it 8m.

9 Has no time in FLAMSTEED'S observation. It seems to be placed in Atlas considerably too late, so as perhaps to require a correction —10' in time.

13 Does not exist.— —13 does not exist. My double star VI, 35, is 9 Aurigee.

17 The time in FLAMSTEED'S observation is marked " circiter," but I find that my viewing instrument cannot, for want of other near stars, determine whether it is properly placed in the Atlas and catalogue.

27 Does not exist. FLAMSTEED never observed it.— —27 28 There is an obser- vation by FLAMSTEED, p. 286, on a star S. of 28, but it does not exist, nor 27. — 27 is wanting. A star observed by FLAMSTEED, p. 286, is not in the place where it should be. 27 was never observed by FLAMSTEED.

32 Is the same with 30 Aurigse. The stars 32 33 34, as I have called them,

Oct. 30, are small stars nearly in a line, but I doubt whether my 32 is FLAMSTEED'S star. The Atlas does not give it as it is in the heavens.— —The star taken for 32 Cam. is a small star between 33 and 30 Aurigae, not given in any catalogue.

35 Has no time, but seems to be very properly placed in Atlas and catalogue.

39 My instrument will not determine its place. It is without time in FLAMSTEED'S observation.

42 A star observed by FLAMSTEED, p. 288, who calls it 4m, preceding 42 and 43 is in its place.

45 and 46 By FLAMSTEED'S observations should have their PD reversed, but in the heavens they seem to stand as they are placed in Atlas and catalogue.

49 I cannot determine the time of 49, which FLAMSTEED'S observations have : :

52 54 58-52-54 but I am not quite sure of 52 and 54. There are so many small stars, that it is not possible without fixed instruments to ascertain them

positively. 54 in FLAMSTEED'S observations by strias (screws) requires PD-2°,

but it is not possible to ascertain its place positively.

COMPARATIVE BRIGHTNESS OF THE STARS. 421

Notes to Hydra.

8 There is but one star, which if it be 31 Monocerotis, then 8 is not there. FLAMSTEED never observed it.

36 Is not in the place where it is marked in Atlas. The time in FLAMSTEED'S observations is marked : :

43 Is hardly visible in my small telescope.— -1 Crateris - - 43 . Dec. 15, 1795, it is 43 Hydrse . 1 Crateris, but now (Jan. 26, 1797) it is 1 Crateris - - 43 Hydra. I suppose 43 to be changeable.

Notes to Hydra et Crater. 1 See note to 43 Hydrse, above.

22 I cannot see 22 in the place where FLAMSTEED has given it, but 1° above is a star which I suppose is it ; calling that, therefore, 22, it is 23 . 22.

Notes to Leo minor.

12 Near 12 is a star observed by FLAMSTEED, p. 438. 12 wants a correction + in EA.

17 Kequires -10' in PD.

22 Is not to be seen. 23 -, 24 - 22. There is a star pointed out by 23 and 24 which may be 22, but then its situation is faulty about 30', being too far from 28.

32 The star north of 32 observed by FLAMSTEED, p. 220, is in the place.

41 54 54, /3 Leonis, and the star in Leo minor's tail-end, 41 Leo minor, are in succession of magnitude.

49 Is a very small star, and a much larger between 49 and 60 Leonis major is not down in catalogue and Atlas.

Notes to Pisces.

1 Which has the time " circiter" in FLAMSTEED seems to be placed in Atlas and in the catalogue a little later than it should be ; perhaps 5' or 6' of space.

40 ; 39 A larger star than either is 1° 4' towards a Androm. If this was mistaken

for 39 perhaps it might give rise to the supposition of the loss of 40. 40 is

not lost.

48 Has no time. In the heavens it seems to be nearly in the place where the

catalogue gives it. -48 -, 49. The observation 48 . 49, Jan. 1, 1796, is probably

owing to a mistake of the star, as there is one nearly equal to 48 near it which is not in FLAMSTEED'S catalogue nor Atlas.

50 52 56 Either 50 or 52 is wanting. By 46 it is 52 that is wanting ; 56 is wanting. [Note by W. H. : "As it appears by Index that 50 and 55 have no observations, put 52 and 56 for them."]

59 FLAMSTEED has no observation of 59, but there is a star in the place where the catalogue gives it.

422 DE. HERSCHEL'S FIFTH CATALOGUE OF THE

70 Does not exist. 70 is a very small star. FLAMSTEED observed it, p. 406.

71 Is so small that it may, perhaps, not be FLAMSTEED'S star, but there is no other.

72 A star between 72 and 78, observed by FLAMSTEED, pp. 149, 180, is in its place. It is = 72 nearly.

104 Is 8° lower than 1 Arietis (which does not exist) is marked ; perhaps it was by mistake placed 8° more north and called 1 Arietis.

108 Does not exist, or is invisible.— -There is a large star l£° from 6 Arietis and 2| from 107, not in Atlas.— -108 does not exist. 109 is just 3° south of it and is, perhaps, the same. On p. 332 of FLAMSTEED'S observations the number is cast up 3° wrong, which has produced 108 Pise. The observation belongs to 109.

Notes to Sagittarius.

I FLAMSTEED has no observation of 1, but there is a star exactly in the place where 1 is marked in the Atlas.

8 Does not exist. There is a small star at rectangles to 17 15 13 towards the place where 8 is marked in the Atlas, but it is much too near 13 to be 8.

I 1 Does not exist.

12 The RA of 12 requires a correction of about 1° minus, for in the place where 12 is marked in Atlas is no star, but 1° before there is one which answers to it.

14 The star observed by FLAMSTEED, p. 171, is in its place ; it is lj° S. of 14.

181 see many small stars north of 19, but cannot see 18 south of it.— —18 is not in the place assigned by FLAMSTEED'S catalogue, but about 1° more in RA is a star which is probably the one intended. It was observed by FLAMSTEED, p. 115.

23 24 The star between 25 and 26 north of them observed by FLAMSTEED, p. 374, is in its place. 23 does not exist. There is a star that answers pretty well to 23. It is a little farther from 25 than it is laid down in Atlas. 24 should be nearer to 25 than in Atlas. The observation of FLAMSTEED, p. 532, gives it right.

53 Is double, and I cannot say which is FLAMSTEED'S star.

Notes to Ursa minor.

1 « appears uncommonly bright.— -The pole star seems to be decreased, or ft is

increased. The place of the moon may possibly influence appearances -a , ft The

night is not favourable. Very clear, a. - ft.

2 Is not as in Atlas, or rather it exists not. FLAMSTEED observed a star,

pp. 213, 214, 215, which has been misplaced and called 2 Ursa minor. It should be 2° further from 1, and it is in the place where it was observed.

4 By FLAMSTEED'S observation the RA of 4 should be - 3° 50' in time ; but without a fixed instrument I cannot perceive that 4 is misplaced, being so near the pole.

8 Either exists not, or is at least not in the place marked in Atlas. 8 is

COMPAEATIVE BRIGHTNESS OP THE STABS. 423

misplaced in Atlas: there are two small stars about 1° from 7 towards 15: one of them is probably 8. They are equal, 12 . 8 . 8.

1014 There is a larger star than either 10 or 14, between but following these two, which is not in FLAMSTEED.

12 Appears two small for 7m. It is 8 or 9m. FLAMSTEED has no observation of 12.

14 Has no time in FLAMSTEED'S observation, but it seems to be placed very justly in the Atlas.

15 Eequires PD - 10'.

16 19 There is a large star between 16 and 19 not in FLAMSTEED. The mistake of Sept. 14, 1795, is owing to the large above-mentioned star.

18 There are seven stars about the place of 18. FLAMSTEED has no observation

of 18.

19 . 20 Sept. 14, 1795, I suppose this to be a mistake of the star. 24 Requires + 10' or 2|° in RA.

Notes to Vulpecula.

Vulpecula in Atlas is laid down so confusedly and erroneously that it is impossible to ascertain the stars without a fixed instrument.

2 Is misplaced. It requires a correction of |° minus in HA and 30' + in PD.

3 The observation of Sept. 17, 1795, 6-3,3 Cygni does not agree with this 3 _ _ 3 Cygni [i.e., of this date, Nov. 3, 1795]. Nov. 15, 1795, 3 - - 3 Cygni.

7 Requires a correction, — near f° in RA.

11 10- -11, but 11 is very small and FLAMSTEED has no observation of it. I suppose therefore that this is not the star which is given in Atlas and catalogue.—

11 is forgot in Atlas.

13 9 - 10 , 13, Sept. 17, 1795, but 13 is further from 10 and nearer to 14 than it is marked in Atlas.

12 Is placed too far north in Atlas — at least 15' by 12 Sagittse.— —Large star in the breast near 14-9.

13 The expression 10 , 13, Sept. 17, 1795, cannot be right; it is 13 - - 10.— Dec. 4, 1796, I have my doubts about the expression 10 , 13 used Sept. 17, 1795. I could hardly mistake the star 10 as [? and] there is none in the neighbourhood that exceeds 13.

14 A large star in Atlas preceding 14 is not in the heavens, nor do I know how it comes into the Atlas, as FLAMSTEED has it nowhere. This constellation must be reviewed again, when it is higher.

16 A considerable star near 16.

24 25 A star larger than either, north of 24, observed by FLAMSTEED, p. 64, is in ts place. 31 32 Are contrary iu magnitude to what they are in Atlas.

424

DE. HERSCHEL'S SIXTH CATALOGUE OF THE

CATALOGUE VI. A SIXTH CATALOGUE OF THE COMPAEATIVE BRIGHTNESS OF THE STABS.

Lustre of the Stars in Canes venatici.
1		6	5--1.7
2		5	10. 2
3		6	3,7
4		6	9 . 4
5		6	5-- 1 875 , 14
6		5	8 -, 6 , 10
7		7	1,7 3, 7-, 11
8		4.5	25-8 8 -, 5 876
9'		C.7	10, 9. 4
10		6	10, 9 6, 10. 2
11		6	7 -, 1 1 Note
12		2.3
13		4.5	41 Com Ber , 13 (= 37 Com Ber)
14		5	5, 14
15		6.5	15. 17
16		6	17 - 16
17		6	15 . 17 - 16
18		6	19- 18
19		7	23 . 19 - 18 Note
20		6	20- 23
21		6	24-21
22		6	Does not exist
23		7	20 - 23 . 19
24		5.6	24-21
25		5	25-8 Note

COMPARATIVE BRIGHTNESS OF THE STARS.

425

Lustre of the Stars in Coma Berenices.
1		7	2.1,3
2		6	5,2.1
3		6	1,3
4		6	13, 4
5		6	5 , 2 Note
6		5	6, 11
7	h	4.5	14 . 7 -, 8 7-20 24 . 7
8		7	7-8 20 , 8 25 , 8
9		6	9 . 10
10		6 9 . 10
11		4.5	6,11
12	e	5	15 , 12 . 16
13	f	4.5	17 . 13, 4
14	b	4.5	16 . 14 . 17 14 . 7
15	C:	4.5	15 , 12
16	a	4.5	12 . 16 . 14
17	8	4.5	14 . 17 . 13
18		5	21, 18 --22 18.26
19		6	Does not exist
20		G	7 - 20 , 8
21	9	5	23- 21 , 18
22		7	18 --22
23	k	4	23-21
24		5	24 . 7 24 ~ , 27
25		6	25, 8
26		5	18.26
27		5	24 = , 27 , 29 27 , 36
28		6	29-28

VOL. CCV. — A.

3 I

426

DE. HERSCHEL'S SIXTH CATALOGUE OF THE

Lustre of the Stars in Coma Berenices — continued.
29		5	27 , 29 - 28 Is the same with 36 Virginia
30		6	31 , 30
31		4.5	41 . 31 , 30
32		7	38 , 32 . 33
33		7	.32 . 33
34		5 4.5	Docs not exist
35		35 - - 39 Note
36		5	27 , 36 - 38
37		5.6	41 , 37 or 13 C;m venut
38		G	36 - 38, 32
39		5	35 - - 39 39 , 40
40-		6	39 . 40
41		5.4	43 -, 41 . 31 42 -, 41 41 , 37 Note
42		4.5	5 Boot , 42 , 4 Boot 43 , 42 -, 41 Note
43		5.4	43 -, 41 43 , 42
Lustre of the Stars in Libra.
1		5.6	Docs not exist Sec note
2		7	2-96 Yirginis Note
3		6	55 Hyd -, 3 55 Hyd -73. 14
4		6	54 Hyd - 4 4 . 56 Hyd 4 is 53 Hyd
5		6	5 . 18 5 . 10
6		5	45 . 6 . 7 6-54 Hyd 6 is 58 Hyd
7	F	5	6 . 7 . 21 7 , 19 7-- 15
8		6	24-8-25
9	a	2	27 , 9 - 20 27 . 9 - 20 27 -, 9
10		6	5-10
11		6	105 Virg- 11

COMPARATIVE BRIGHTNESS OF THE STARS.

427

			Lustre of the Stars in Libra — continued.
12		6	12 , 55 Hyd
13	e	6	15 , 13; 18
14		6	3. 14 23, 14
15	?	6	7-- 15, 13
16		5.6	16 , 105 Virginia
17		7	18 . 17
18		6	13 ; 18 . 17 5 . 18 19 . 18
19	8	4.5	44 . 19 . 43 31 , 19 7 , 19 19 . 18
20	r	3	20 ,40 -20 , 51 49 11yd - 20 - 38
21	vl	5	7 . 21 . 41 21 -, 22 21 - 26
22	i/2	6	21 -, 22 26 , 22
23		7	23 , 14 Note
24	ti	4.3	48 , 24 , 37 24-8
25	!?	G	8 - 25 25 , 28
26		6	21 - 26 , 22
27	ft	2	27 , 9 27 . 9 27 -, 9 27 , 24 Serpcutis Note
28		6	25 , 28
29	O1	7	32 , 29 . 34
30	02	6	33 , 30
31	£	4	37 , 31 . 35 37 , 31 , 19 37 . 31
32	f1	6	32 ,34 32 , 29
33	f2	7	35 - 33 , 30
34	<?	6	32 , 34 . 35 29 . 34
35	£4	4	31 . 35 . 44 34 . 35 - 33
36		6	40 -, 36
37		6	24 , 37 , 31 37 ,31 37 . 31 Note
38	7	3.4	39 . 38 . 51 51 , 38 , 46 20 - 38 51 . 38 38 -, 46
39		4	40 , 39 . 38 46 . 39 , 40 39 ; 40

3 I 2

428

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

			Lustre of the Stars in Libra — continued.
40		4	20 , 40 , 39 39 ,40 39 ; 40 -, 36
41		6	21 .41 47 , 41
42		6	1 Scorp - 42 - 4 Scorp
43	K	4	19 . 43 . 45 43 , 45
44	»/	4	35 . 44 . 19 48 , 44 48 , 44 44 , 49
45	A	4	43 . 45 . 6 45-47 43 , 45 T 47
46	0	4	51 . 46 , 48 88 , 46 . 39 4G , 48 46 - 48 38 -, 46 -, 48
47		6	45 - 47 45 7 47 , 41
48	*	4	46 , 48 ,24 48 , 15 Scorp 46 , 48 , 44 46 - 48 , 44 46 -, 48 - - 49
49		G	48 - - 49 44 , 49
50		6	42 Sorpii - 50 50 - 43 Sorpentis
5t	4"	4.5	38 . 51 . 46 20 , 51 ,38 51 . 38
			Lustre of the Stars in Lupus.
1		5	5, 1
2	<5	5.0	2,5
3	7	5.6	5 -, 3 , 4
4		5.6	3,4
5	A	5	5 -, 3 2,5,1
			Lustre of the Stars in Piscis austrinus.
1		5	See Note
2		6
3		6
4		4.5
5		6
6		6
7		6

COMPARATIVE BRIGHTNESS OF THE STARS.

429

Lustre of the Stars in Piscis austrinus — continued.
8		4.5	41 Cap -, 8
9	i	4	10-9
10	6	4	10-9
11		6	13, 11
12	7/	5	12-14 12 - 1G
13		6	14- 13, 11
14	M	4	14 .15 14- 15 12- 14-, 13
15		5.6	14 . 15 14- 15
16	X	4.5	12 - 10
17	0	3	17 - 22
18	£	3.4	88 Aquar - 18 . 86 Aquar Note
19		5	23-, 19 , 21 20- 19
20		G	20- 19
21		6	19, 21
22	7	5	22 . 23 17 - 22
23	8	5	22 . 23-, 19
24	a	1	8 Peg , 24 , 44 Peg 44 Peg is 19 Aquar Note
Lustre of the Stars in Scorpius.
1	b	6	2,1-3 1-42 Libra
2	A1	5	5-2-, 3 2-4 2,1
3	A2	7	2 -, 3 4,3 1-3
4		6	2-4,3 42 Lib - 4
5	P	4	5-2
6	7T	3	8-6 23 , 6 , 20
7	s	3	21 -7,8 7-8 7 ; 8 8.7
8	|8	2	7,8- 20 7-8 7 ; 8 8.7 7 ; 8 -, 6

430

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

			Lustre of the Stars in Scorpius — continued.
9	»i	5	9-10 14 , 9 7 10 14. 9 , 10
10	CO2	5	9-10 9 7 10 9 , 10
11		0	19-11 17,11
12	01	0	13 - 12
13	o2	G	13- 12
14	V	4	14,9 14,20 14.9
15	X	5	15 , 10
10		6	15 , 10 . 18 Note
17		0	17 , 11
18		4	10 . 18
19 .		0	19-11 22,19 24,19
20	(T	5	0 , 20
21	a.	1	21 , 50 Cyg a Cyg - - 21 -, a Ophiuchi Note
22		5.0	22 , 19 22-7 25
23	T	4	23 , G 42 Oph , 23
24		0	24, 19
25		0	22 - - 25 Note
20	€	3	14 , 26 - - 27 26 , 9 Oph Note
27		0	26 - - 27 9 Oph - -, 27
28		0	33 , 28
29		6	30.29,31 29 , 38 Oph (= 31)
30		6	30. 29
31		0.7	29,31 29, 38 Oph (= 31)
32		0	33 .32 32 - 50 Oph
33		7	33 , 1 Sagitt 33 .32 33 , 28
34	V	4	35 -, 34
35		3	35 -, 34

COMPARATIVE BKIGHTNESS OF THE STABS.

431

			Lustre of the Stars in Serpens.
1		7	4,. 1,2
2		7	1,2
3		6.7	3, 5
4		6	6,4.1 4-, 8 4. 11
5		6	5 , 10 3,5
6		6	10-6 , 4 6 . 16
7		7	9-7
8		7	4-, 8
9		6	20 , 9 - 7
10		6	10-34 5 , 10 - 6
11			4 . 11 - 14 25-11
12	T1	7	12 , 17
13	8	3	13-, 27 13.37
14	Ai	6	11 - 14
15		6	22- 15
16		7	6 . 16
17		6.7	19 -, 17 12 , 17 Note
18	T2	6	41 , 18
19	T3	6	19- 17 26-19- 29
20	X	6	20, y
21	1	5	35 , 21 - - 22 21 -, 44
22		6	21 --22 - 15
23	*	6	34, 23
24	a	2	27 Lib , 24 , 27 Here
25	A3	6	25 - 36 25 - 11
26		6	26 - 19 26 , 31
27	X	4	13 -, 27
28	P	3	28 , 37 28 , 41

432

DK. HERSCHEL'S SIXTH CATALOGUE OF THE

			Lustre of the Stars in Serpens— continued.
29		5.6	19-29
30		6	36 - 30 50 , 30
31	V	6	26 , 31 ; 39
32	1*	4	32 - 37
33		6	Does not exist
34	(0	6	34 , 23 10 - 34
35	K	4	35 , 21
36	/;	6	25 - 36 - 30 36 , 50
37	e	3	37 . 10 Oph 32 - 37 28 . 37 - 41 13 . 37
38	p	4.3	44 , 38
39		6	31 : 39
40		7	46 - 40 . 45
41	7	3	10 Oph - 41 37 - 41 28 , 41 , 18
42		6	Does not exist Note
43		6	50 Lib - 43
44	TT	4	21 - 44 , 38
45		6	40 . 45
46		6	46 - 40 40 7 47
47		6	467 47
48		6	8 Here ; 48 48 -, 49
49		6	48 - 49
50	IT	5	36 , 50 , 30
51		6	51 , 25 Oph
52		6
53	V	4
54		6	47 Oph - - 54
55	$	4
56	o	5	56 -, 57 Oph

COMPAEATIVE BRIGHTNESS OF THE STARS.

483

Lustre of the Stars in Serpens — continued.
57	c	3	57 - 69 Oph 57 -, 69 Oph
58	V	3	58 - 64 Oph
59	8	6	59 -, 61
60	0	6	61 . GO 60 -, 47 Oph
61	e	6	59 -, 61 . 60
62		6	64-62
63	0	3
64		6	64 - 62 Note
Lustre of the Stars in Serpentarius (or Ophiuchus),
1	8	3	35 . 1 , 13 35 , 1 . 13 GO - 1. 1-13
2	£	3.4	13,2 13-2-10
3	V	5	3 , : : 18 Lib Note
4	^	5	4-5 87477
5	g	5	4-5-9
6		6	Does not exist
fj	X	6	477
8	*	4	874
9	OJ	5	26 Scorpii , 9 - -, 27 Scorpii 5-9
10	A	4	37 Serpentis .10-41 Serpentis 2-10
11		6	21 , 11
12		6	19, 12
13	c	3	1 , 13 1 , 13 13, 2 13-2 1-13 Note
14		6	21 , 14 , 19
15		6
16		6	19 . 16
17		6	Is 43 Herculis
18		6.7	22 . 18

VOL. CCV. — A.

3 K

434

DE. HEESCHEL'S SIXTH CATALOGUE OF THE

Lustre of the Stars in Serpentarius (or Ophiuchus) — continued.
19		6	14,19.16 19,12
20		5.6
21		6	21 ,14 21 , 11
22		7	22 .28 22 . 18
23 |	6
24		7	24-26
25	i	4	51 Serpentis , 25
2G		6	26 -, 28 24 - 26 Note
27	K	4	[A number of comparisons of 27 with a (64) Herculis have been printed in the 2nd of these papers on the " Lustre of the Stars " — see ' Phil. Trans.,' 1796, p. 492 — and it is needless to repeat them here. There are others, of 27 with S (65) Here, and with GO (ft) Serpentarii. The former may be represented by 27; 3 Here and 8 Here; 27. For the latter, see below, line 60. -J. H.]
28		6	26 -, 28 .31 22 . 28
29		6
30		6	Note
31		6	28 . 31
32		6	32 , 33
33		6	32 , 33 , 34
34		G	33 , 34
35	•n	3	35 . 1 35 , 1
36	A	6.5	44 , 36 , 51
37		6	66 Here , 37 66 Here - 37
38		6-7	29 Scorp , 38 (or 31 Seorpii)
39		6	39 ;51
40	p	4
41		G
42	e	4.3	42 - 50 Lib 42 , 23 Scorp
43		4.5

COMPARATIVE BRIGHTNESS OF THE STARS.

435

Lustre of the Stars in Serpentarius (or Ophiuchus) — continued.
44	B	5.4	44 -, 51 44 , 36
45		6
46		6	Does not exist Note
47		6	60 Serpentis -, 47 - - 54 Serpentis
48		6	Does not exist
49	<r	5	67-49
50		7	32 Scorp - 50
51	e	6	51-3 Sagitt 44 -, 51 39 ; 51 36 , 51
52		6	2 Sagitt ,52 58 - 52 ; 2 Sagitt
53		6
54		6	54, 56
55	a.	2	55 , a. Corona; 55 - - 60 55 , 5 Coronas 55 - 33 Drac a Cygni - - 55 - a Coronas a Scorp -, 55 y x Corona;
56		6	54, 56
57	P	4	56 Serpentis -, 57 Note
58	D	6	58-52
59		6	Does not exist
60	ft	3	60 - a Here (3 times) 60 7 a Here (3 times) 60 . 27 (y8) Here 60 , ft Here (twice) 55 - - 60 60 , 17 Aquilse 60-1 60727 60 .27 27 , 60 60 - - 62 Note
61		6	66 , 61
62	y	3	60 - - 62 , 67 72 7 62 72 7 62 62 7 72 72 ,62 72 - 62 -, 71 64-62
63		5
64	V	4	58 Serpentis - 64 - 62
65		6	65-6 Sagittarii
66	n	4.5	68 , 66 , 61 66 , 73
67	0	4	62 , 67 , 70 67 - 49 72 ; 67
68	k	4	70 , 68 , 66

3 K 2

436

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

Lustre of the Stars in Serpentarius (or Ophiuchus)— continued.
69	T	5	57 Serpentis - 69 57 Serpentis -, 69
70	P	4	67 , 70 , 68
71	s	6	72-71 72-, 71 62-71
72	s	6	72-71 72-, 71 72762 72762 62 7 72 ; 67 72,62 Note
73	2	6	74 ,73 66 , 73
74	r	6	74 , 73
Lustre of the Stars in Ursa major.
1	4.5	1 -, 23 1 , 69
2	A	5	3,2,4 2,5
3	^	5	3,2 14 , 3
4'	1T~	6	2,4.6 5,4
5		5	2,5,4
6		5	4. 6
7	I	6	7 is lost
8	l>	5	13 . 8, 11
9	i	4	9-25
10	n	4	39 Lyncis , 10
11	(T1	5	8, 11
12	K	4	41 Lyncis - 12 , 39 Lyn 33 - 12
13	0-2	5	13. 8
14	T	5	14,3 14,16 24-14
15	/	5	15 , 18 15 - 24 30 . 15 -, 18
16	0	5	14,16 16 --,20
17		5	18- 17
18	e	5	15 , 18 - 17 26 -, 18 15 -, 18 -, 31
19		6	8 Leo min -19,7 Leo min
20	7	16 - -, 20 Does not exist Note

COMPARATIVE BRIGHTNESS OF THE STARS.

437

Lustre of the Stars in Ursa major — -continued.
21		6	Does not exist Note
22		7	27 -22
23	h	4	1 -, 23 . 29
24	a	4.5	24-14 15-24
25	e	3.4	25 . 41 Lync 9 - 25 - 69
26		5.6	30-, 26 -, IS
27		6	27 - 22
28		5	Does not exist
29	V	4	23 . 29 29 15 Lyncis
30	<;•	5	30 -, 26 30 . 15
31		6	18 -, 31
32		5	32 . 38
33	X	3.4	34 -, 33 - 12 52 -, 33 - 63
34	p	3	34 -, 33 34 - 52 Note
35		6
36		5	36 - 37 45 - 36
37		5	36 - 37 -, 39 37 , 44
38		5
39		6	37 -, 39 , 43 39 , 42
40		6	41 - 40
41		6.7	43 - 41 -, 40
42		5-6	39 , 42
43		6	39 , 43 - 41
44		6	37 , 44 . 45
45	ia	4.5	44.45-36 45-55
46		6	46 -, 47 Leo min
47		6 !	47 . 49

438

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

Lustre of the Stars in Ursa major — continued.
48	|8	2	50 - _ 48 79 - 48 . 64 79 -, 48 .64 48 7 64 (twice) 64 ; 48 48 - 64 (3 times)
49		6	47 . 49 - 51
50	a	1.2	50 - 77 (5 times) 50 - - 48 50 7 77 (twice) 50 ; 77 50 f 77 50 • 77 77 , 50 85 | 50 • 77 50 , /3 Urs min 50 '. /3 Urs min 50 f 7 (/?) Urs min Note
51		7	49-51
52	#	3.4	34 - 52 -, 33
53	$	4	63 - 53
54	i'	4	54 - 63
55		5	45 _ 55 55 - 67
56		6	56 -, 59 57 . 56
57		6	57 . 56 G7 - 57
58		6	59 , 58 58 . 60
59		6	56 -, 59 , 58 61 . 59 , 62
60		6	65 ,60 58 . 60
61		6	61 . 59
62		6	59 , 62
63	X	4	33 - 63 54 - 63 - 53
64	7	2	48 . 64 -, 69 48 , 64 ~ ~ - 8 or 69 48 7 64 7 Urs min - 64 64 ; 48 48 -, 64 48 - 64 (3 times)
65		7	65 , 60
66		6	71.66 70.66
67		6	55 _ 67 - 57
68		7	70 - - 68 73 - 68 . 72
69	3	2.3	69 -, 70 69-74 1 , 69 64 -,69 64 = - - 69 25 - 69
70		6	69 -, 70 - - 68 75 . 70 . 71 74 , 70 . 75 70 771 70 . 66
71		7	70.71.73 71.66 70771.73
72		7	73 -, 72 73 - 72 68 . 72

COMPARATIVE BRIGHTNESS OF THE STARS.

439

Lustre of the Stars in Ursa major — continued.
73		6	71.73-, 72 71.73-72 73-68
74		6	69 - 74 , 75 74 ,70 76 - 74 76 . 74
75		6	74,75.70 70.75 Note
76		6	76 . 76 - 74 Note
77	£	3	77 ,85 50 - 77 (3 times) 50 7 77 ; 85 50 ; 77 (3 times) 50 -, 77 50 ;' 77 - 79 50 ; 77 77 ,50 77 7 85 77-79 1 Urs min . 77
78		6	78 ; 80 Note
79	f	3	85 , 79 - 48 79 | 7 Urs min 77 - 79 -, 48
80	9	5	83 , 80 , 81 80 ,83 78 ; 80
81		5.6	80 ,81 84 , 81 . 86 83 , 81 . 84
82		6	86 . 82 Note
83		6	87 - 83 , 80 83 ,84 80 , 83 , 81
84		6	83 , 84 , 81 81 . 84 . 86
85	n	3	77 , 85 , 79 77 ; 85 , 79 85 | 50 77 ; 85
86		6	81 .86 84 . 86 . 82 Note
87		5	87-83 87-8 Draconis
Lustre of the Stars in Virgo.
1	0)	6	4-1 2,1 2-1,4
2	'£	5	8-2-4 4-2,1 2-11 2-1 8,2
3	V	5	9_3_8 9-3-8 9,3-8
4	2£	6	2-4-1 4-2 1,4-6
5	/8	3	43 , 5 - 15 43 , 5 - 15
6	A	6	43 - 6 . 109 4-6 7,6 12 ; 6
7	b	5.6	8-7 7. 13 7,6 7-, 10 7-11 7 , 13
8	TT	5	3_8-2 3-8-7 9-8,16 3-8,2 8-16 51 ; 8 - 78
9	O	5	9-3 9-3 9-8 9,3
10	r	6	12-10 7-, 10 11,10 10-17

440

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

			Lustre of the Stars in Virgo— continued.
11	s	6	2-11-12 7 - 11 , 10
12	f	6.7	11-12-10 12.17 12:6
13	n	6	7 . 13 7 , 13- 14
14		6	13- 14
15	•>}	3	5 - 15 15 - 51 15 - 93 109 . 15 , 107 5 - 15
16	0	4.3	8,16 8-16
17		6	12 .17 10 - 17
18		6	Does not exist Note
19		6	Does not exist Note
20		6	27 .20 27 . 20
21	'/	6	26 - 21 , 25
22		6	27 .22 31 . 22 Does not exist Note
23		6	Does not exist Note
24		G	Does not exist Note
25	./'	6	21 , 25 -, 28 Note
26	X	5	26-21
27		6	33 , 27 27 . 22 27 .20 30 - - 27 . 20 33 , 27 - - 42 (see note) 41-, 27
28		6	25 - 28
29	7	3	47 - 29 - 79 67 - 29 . 47 29 , 47
30	P	5	30-32 30 - - 27 30 , 32
31	81	6	32 - 31 - 33 32 ,31 32 . 31 32 , 31
32	G2	6	30 - 32 - 31 32 ,31 30 , 32 . 31 32 , 31 See note
33		6.7	31 - 33 33 ,27 33 - 34 33 , 27
34		6	33 - 34 36 , 34 , 41
35		6	37 , 35
36		6	36 , 34
37		6	37, 35

COMPARATIVE BRIGHTNESS OF THE STARS.

441

			Lustre of the Stars in Virgo— continued.
38		6	48. 38
39		6	40-39 40 - - 39
40	*	5	40-39 40 - - 39
41		6	34 , 41 , 27
42		6	27 - - 42 Note
43	8	3	79-43 79,43,5 79.43-6 47-43,5 79-43
44	k	6	46 . 44 , 48
45		6	Does not exist Note
46		6	46 . 44
47	i	3	67 - 47 - 29 29 . 47 - 79 29 , 47 -, 43
48		6	44 , 48 . 38
49	ff	5	49 -, 50 49 - 50
50		6	49 -, 50 ,52 50 , 52 49 -, 50 - - 56
51	6	4	15-51-74 51; 8
52		6	50 , 52 . 62 Docs not exist Note
53		4.5	53 ,61 61 - 53 , 55
54		6	61 , 54 61 -, 54 73 .54 57 - 54 57 - 54
55		6	55 .57 61 - 55 . 57 55 .57 53 , 55 . 57
56		6	58 .56 56 , 58 50 - - 56
57		6	55 . 57 - 61 55 .57 55 . 57 - 54 55 . 57 - 54
58		6	62 . 58 . 56 56 , 58 Note
59	e	6.7	60 , 59 , 64 70 - 59 , 71
60	<T	5	84 - 60 - 78 60 - 64 60 , 59
61		4.5	61 ,69 57 - 61 , 54 53 , 61 -, 54 61 - 55 61 - 53 Note
62		6	62 . 58 52 . 62
63		6	697 63
64		6	60 - 64 59 , 64
65		6	74 - 65 . 66

VOL. CGV. A.

3 L

442

DR. HERSCHEL'S SIXTH CATALOGUE OF THE

			Lustre of the Stars in Virgo— continued.
6G		6.7	65 . 66 , 72 66 , 80
67	a	1	67-47 ft Gem , a. Virg . a Leon 67 - 29 67 7 -, 32 Leon Note
68	I	4	69 - 68 , 75
69		5.6	69 -, 68 61 -, 69 7 63
70		6	70-59
71		6	59 , 71 Note
72	P	6	80 - 72 66 , 72 . 80 70 -, 72 - 77 80 , 72 82 - 72 - -, 77
73		6	73. 54
74	P	6	51 _ 74 _ 80 74 - 65 74 -, 82
75		6	68, 75
76	h	/>	82 . 76 -, 72 76 - 80
77		7	72 - 77 , 81 72 - -, 77 . 81
78		6	60-78 8-78 -, 84
79	i	6	29 - 79 - 43 79 ,43 47 - 79 . 43 79 - 43
80	p	6	74 - 80 - 72 72 .80 76 - 80 66 , 80 , 72
81		6	77,81 77.81.88 Note
82	m	6	74 -, 82 .76 82 - 72
83		6	89 , 83 , 87 Note
84	0	6	93 _ 84 _ 60 78 -, 84
85		6	87 ,85 86 , 85
86		6	87 . 86 , 85
87		6	83 , 87 , 85 87 . 86
88		6	81 . 88 Note
89		5.6	89 , 83
90	P	6	93 - - 90 , 92
91		6	Does not exist
92		6	93 - 92 90 , 92
93	r	5	15 _ 93 _ 84 93 - 92 107 . 93 . 99 93 - - 90

COMPAKATIVE BRIGHTNESS OF THE STARS.

443

			Lustre of the Stars in Virgo — continued.
94		6	95 , 94 - 97 94 , 96
95		6	98 -, 95 , 94
96		5	94 , 96 , 97 2 Lib - 96
97		6	94 - 97 96 , 97
98	K	4	99 - 98 . 100 98 -, 95 98 , 100 98 ; 110
99	I	4	93 . 99 - 98 107 -, 99 Note
100	A	4	98 . 100 98 , 100 110; 100
101		6	20 Bootis -, 101 Note
102	V1	5	105 - 102 -, 103 102 - 104
103	tf	5	102 -, 103 106 , 103
104		6	102 - 104 . 106 104 , 108
105	4>	4	105- 102 16 Lib, 105-11 Lib
106		6	104 . 106 , 103
107	V-	4	15 , 107 . 93 107 -, 99 109 - 107
108		6	104 , 108
109		4	6 , 109 . 15 109 - 107
110		6	98; 110; 100

3 L 2

444 DR. HERSCHEL'S SIXTH CATALOGUE OF THE

Notes to Canes venatici.

July 22, 1797. 11 There are two stars about the place of 11 nearly alike in brightness.

13 Is 37 Comae Berenices.

19 A considerable star sp 19 is omitted : much larger than 18.

22 Does not exist. It was never observed by FLAMSTEED.

25 Is misplaced: the PD should be +10°. It is not in the place where the catalogue has it, but is 10° more south. 25 - 8 A star observed by FLAMSTEED, p. 228, is in its place about f° or 1° north of this 25, and a little preceding it is * , 14.

A star observed by FLAMSTEED, p. 225, from 64 Ursae towards 54 Ursse is in its

place. It is 1 7 *

Notes to Coma Berenices.

5 December 27, 1786. I looked for 5 Comae, but could not find it. 19 April 19, 1797. 19 does not exist. FLAMSTEED never observed it. 29 Is the same with 36 Virginis.

34 Does not exist, nor did FLAMSTEED observe it.

35 39 A star between 35 and 39 observed by FLAMSTEED, p. 165, is in its place. It is 39 =, *

41 A star near 41 observed by FLAMSTEED, p. 165, is in its place. A star south following 41 observed by FLAMSTEED, p. 165, is in its place. It is 41 - - *

42 A star south of 42 observed by FLAMSTEED, p. 164, is in its place. Calling it in general * it will be 38 , *

Notes to Libra.

1 Does not exist : there is a star of a considerable magnitude near 50 Hydrse, but the place does not agree with 1— —1 is not in the place where it is marked in Atlas, but there is a star which FLAMSTEED observed, p. 166, which is probably 1. It is RA-30' and PD + 2° and is in its place. I shall call it 1 and it is 50 Hydrae -, 1

2 There are two about the place of 2, but I suppose the largest, and nearest to 98 Virginis, to be FLAM STEED'S star. It agrees best with the place.

23 Is not in the place where Atlas gives it, nor did FLAMSTEED observe it there. He has a star, p. 531, which is 1° 26' more in EA. This is probably 23, and it is 23 , 14 and is in its place.

27 Does not seem larger than 9, at least not very decidedly, and so as to be denoted 27 , 9, but 9 has a small star near it, not visible to the naked eye, which increases its lustre ; but in my glass it is evident that 27 is a little brighter than 9.

37 North of 37 is a star nearly as large as 37, but 37 is a very little larger in the finder.— -FLAMSTEED'S star observed, p. 45, north of 37 is in its place 37 - *

COMPARATIVE BRIGHTNESS OF THE STARS. 445

Note to Piscis austrinus.

September 22, 1795. This constellation, on account of its low situation, can be of no use for comparative magnitudes. The opportunities of observing it must be so scarce that no discoveries of changes can be made in it. I can see no other star with the naked eye but those I have equated [viz., 24 and 18. The observations of other stars of this constellation were made two years later. — J. H.].

Notes to Scorpius.

16 Should be about 3 or 4 minutes nearer to 15. FLAMSTEED'S observation, p. 197, leaves the ZD doubtful.

21 Is of a very brilliant ruddy light. Is of a pale garnet colour : it seems to be

the most coloured of all the large stars. Its low situation probably contributes to it.

25 Either does not exist or is misplaced. There is a star about 4° from 23 and 2f° from 22, which may be the star if misplaced. In that case the RA of 25 should be —1° and it will be 22 — 7" 25. Several stars of Serpentarius are so small that 25 may exist.

26 Being low it may be larger than 14, for I make no allowance in my observations.

Notes to Serpens.

17 There are two of 17 but little different in brightness. I have taken the brightest of them.

33 Does not exist. FLAMSTEED never observed it.

42 Does not exist. The place where it should be, according to the catalogue, cannot be mistaken. FLAMSTEED never observed it. 50 Libra? not far from it is in its place.

0 (63) = I Aquila3 and less than X Aquilae.

64 Is the largest of two.

Notes to Serpentarius (Ophiuchus).

3 (u) is misplaced in Atlas 1°. It should be about +1° in RA. A star f° north of it, observed by FLAMSTEED, pp. 442, 443, is in its place.

6 Does not exist. FLAMSTEED never observed it.

13 3° np 13 is a star not marked in FLAMSTEED = 20.

26 28 26 has another near it larger than 28.

30 Seems not to be rightly placed.

38 31 Scorpii is 38.

46 A larger star than 46 is just by, but not marked in Atlas. 46 does not

exist.

446 DK. HERSCHEL'S SIXTH CATALOGUE OF THE

48 Does not exist. FLAMSTEED never observed either of them.

57 A large star np 57 observed by FLAMSTEED, p. 442, is in its place. It is 57 *

59 Does not exist. FLAMSTEED never observed it.

60 I suspect 27 Herculis to be changeable, for it is now 60 . 27 Here, or even 27 Here , 60. There is great difference in the weather.

72 Is much too large for 6m.

Notes to Ursa major.

20 There is a very small star about the place of 20, which I can hardly take for

one of FLAMSTEED'S. It is 16 - -, 20. 20 does not exist in the place where it is

marked in the Atlas. There is no star but of the 9th mag. within a degree of

the place.

21 I think does not exist. There is a star not far from the place where the Atlas has it, but it is much too small. 21 does not exist. I cannot mistake the place.

34 The star south of 34 observed by FLAMSTEED, p. 439, is in the place.

35 Is not as laid down in Atlas.

50 a (Oct. 25, 1795) Appears unusually large 8h- 20™-. When I saw it at 6h- I thought so immediately. I suspect it to be changeable, or rather am pretty sure it is so. It is as large as /8 Ursse minoris, but that is so much higher that no fair comparison can be made between them.

Oct. 26, 1795. 50 is not so bright as last night.

Oct. 28, 1795. It is much less than it was Oct. 26. The place of the moon may possibly influence appearances.

Nov. 28, 1795. It would not be proper to compare « Urs. maj. with e and 17, as they are much lower, but a seems to be remarkably bright.

75 Has no time in FLAMSTEED'S observations and is misplaced in Atlas. It is but very little following 74, being almost in the same RA with it.

76 There are two of 76, at a distance of nearly |-° from each other.

77 June 25, 1796, 77 is very bright. July 21, 1796, 77 is decreased.

78 Is missing; at least is not as marked in Atlas. 78 has no time. In the

observation of FLAMSTEED in the Atlas, it is placed about 20' of a degree too far East.

82 Is missing.

86 The place of 86 is not right in Atlas by many minutes, perhaps 15'.

Notes to Virgo.

18 Is lost. 18 does not exist, or is reduced to 9m at least. 18 does not exist.

19 Is lost ; or, as there are 4 or 5 stars about its place, if it is among them, it is at least reduced to the 10th mag.— —19 exists not, or is less than 9m. There are 3 or 4 stars near the place, but extremely small. 19 does not exist, or is at least 9m or

COMPARATIVE BRIGHTNESS OF THE STARS. 447

10m. 19 exists not, but there is a star sp 20 about the same distance as 19 is

marked np.

22 Is in its place and 7m.— — 22 23 are both either 7 or 7 . 8 mag. 22 does not

exist. The observation 31 , 22, April 9, 1796, can not be right. I mistook very

probably a star sf 32 and 31 instead of np, as there is such a one. 22 and 23 do

not exist. FLAMSTEED has no observation of them. There is a pretty considerable

star near the place of 22. 23 is not to be seen. There is no star that can be

taken for it. 23 does not exist. There is no star that can be taken for it.

24 is lost. There is no small star to represent it. 24 does not exist. There is

no star that can be taken for it. 24 does not exist. FLAMSTEED has no observation

of it.

25 By FLAMSTEED'S observations requires —19' in RA and by the heavens it does the same.

42 Does not exist. There is no star nearer than 1° of any size to the place of 42 given in Atlas. FLAMSTEED never observed this star. The star estimated

April 9, 1796, 27 42 is one of these small stars nearest the place, which is rather

larger than 2 or 3 others thereabout.

45 I cannot see 45. There is no star so large as 10 or llm near the place of 45. 45 does not exist. FLAMSTEED never observed it.

52 Does not exist. There is a very small star not far from the place. FLAMSTEED has no observation of 52.

58 The PD of 58 should be +11'.— —58 by FLAMSTEED'S observations requires + 11' in PD and by the heavens it does the same.

56 58 They are very small stars. 58 is double in my finder. There are two other stars situated like 56 and 58 in Atlas, which were probably taken for them, May 2, 1796, when they were estimated 58 . 56. Not knowing then that 58 wants a correction of PD + ll', occasioned the mistake.

61 There seems to be a change in the brightness of 61 since last night.

67 Is of a sparkling bluish white colour : a beautiful star.

71 A star following 71 observed by FLAMSTEED, p. 194, 478 [sic] is in its place * . 71 59 , * . 71.

77 . 81 . 88 The three last are very small stars. About the place of 88 there are two nearly equal. I cannot determine which is FLAMSTEED'S star.

83 The RA of 83 should be +22' by FLAMSTEED'S observations, and it requires the same by the heavens.

91 Does not exist.

99 The star nf 99 observed by FLAMSTEED, p. 41, is in its place. It is 108 *

101 Is misplaced in the British Catalogue : it should be +1° in PD. Then it is 20 Bootis -, 101.

[ 449 ]

XIII. On the Accurate Measurement of Ionic Velocities, with Applications to

Various Ions.

By R. B. DENISON, M.Sc., Ph.D., and B. D. STEELE, D.Sc, Communicated by Sir WILLIAM RAMSAY, K.C.B., F.R.S.

Received October 14, — Read November 16, 1!)05.

ACCORDING to ARRHENIUS' theory of electrolytic dissociation, the conduction of the current in a salt solution is due to the presence of free ions, which, Tinder the influence of an electromotive force, move towards the electrodes with a velocity depending, other conditions being equal, upon the magnitude of the driving force, or fall of potential. This "ionic velocity" can be determined by means of two quite distinct methods, of which one may be termed the indirect and the other the direct method. The former, or indirect method, was evolved by KOHLRAUSCH on his recognition of the law of the independent migration of the ions, which he thus states : " The molecular conductivity, p, of a solution is proportional to the sum of the velocities of the anion and of the cation, p. = constant x (u + v)."

The ratio of these velocities, u/v, had been determined many years previously by HITTORF, whose " Uberfuhrungszahl," or transport number, p = u/(u + r), for any salt represents the fraction of the total current that is carried by the anion.

The knowledge of this ratio enabled KOHLRAUSCH to calculate the ionic velocities from the molecular conductivity. In order to calculate the velocities of the ions by the indirect method it is, therefore, necessary to know both the molecular conductivity of the solution and its transport number. Although the determination of the former is perfectly easy and straightforward, that of the latter by HITTORF'S analytical method is both difficult and laborious, and the method suffers from the great disadvantage that the success or failure of an experiment is known only after the necessary chemical analyses have been completed.

In recent years a method of measuring transport numbers and ionic velocities has been worked out in which the actual rate of motion of the ions is read off by means of a scale and telescope. The first steps in this direction were taken in 1886 by LODGE (' British Assoc. Reports,' 1886, p. 389), whose idea was to make the invisible ion indicate its presence by some characteristic physical or chemical property, such as its colour or the formation of a precipitate with some other ion. For the gradual

VOL. CCV.— A 399. 3 M 12.3.06

450 DE. E. B. DENISON AND DE. B. D. STEELE ON THE

development of this idea reference must be made to the original literature ;* but the following is a brief account of the method in its present form :—

It has been shown, both theoretically and experimentally, that if two salt solutions containing a common ion are placed one above the other, and an electric current passed through the system, a stable margin will be formed between the two solutions, provided that the specifically slower non-common ion follows the faster one. Under these circumstances it has been shown that the boundary moves in the same direction as the non-common ions and measures their velocity. In order to observe and measure the velocity of the boundary, advantage is taken of the slight difference in the refractive index of the two solutions, which difference renders the margin quite visible when viewed through a telescope. The method of determining the ratio U/V, and hence the transport number. U/(U + V), for a given salt will be rendered clear by the following considerations :-

Let us suppose that a current flows through the system,

Anode — solution of lithium chloride — solution of potassium chloride — solution of potassium acetate — cathode.

At the margin between the lithium and potassium chloride solutions the slower lithium ion follows the specifically faster potassium ion, and there results a stable margin, the velocity of which is that of the lithium and potassium ions at that point. This, velocity, however, depends on the potential gradient, and it has been proved that the concentration of the lithium chloride becomes automatically adjusted, so that the potential gradient is just sufficiently increased to make the lithium ions keep pace with the potassium ions. In the same way the specifically slower acetate ion follows the faster chlorine ion, at the margin potassium acetate, chloride, and the motion of this margin gives the velocity of the acetate-chlorine ions ; but whilst the lithium and acetate ions are moving under the influence of an unknown potential gradient, that which is driving the potassium ion is the same as that driving the chlorine ion— the potassium chloride solution being homogeneous — and, therefore, the velocities of these two ions are strictly comparable. Moreover, the potential gradient in the middle electrolyte can easily be calculated, and hence, also, the average mobility of the ions, or their velocity under a driving force of one volt per centimetre.

The ratio U/V and the average mobilities of the ions of a number of salts have been determined in this way,t and whilst the agreement between the results so obtained and those obtained by the indirect method is, on the whole, fair, considerable deviations occur in the case of many salts. In the method used for these measure-

* WHETHAM ('Phil. Trans.,' A, 1893, p. 337; A, 1895, p. 507); MASSON ('Phil. Trans.,' A, 1899, p. 331); KOHLRAUSCH (' Wied. Ami.,' 1899, LXII., p. 209); WEBER ('Sitz. Berliner Akad.,' 1897); STEELE ('Phil. Trans.,' A, 1902, p. 105; 'Zeit. fiir Phys. Chem.,' 1902, XL., p. 689); ABEGG and GAUS ('Zeit. fur Phys. Chem.,' 1902, XL., p. 737); DENISOX ('Zeit. fiir Phys. Chem.,1 XLIV., p. 575).

t MASSON (he. tit.) • STEELE (loc. tit.).

ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC. 451

ments the solution to be measured was separated from the solutions containing the slower indicator ions by two partitions of a gelatine solution of the same indicators, and in this way it was possible to place the solutions in position without any appreciable mixing taking place at the surfaces of contact.

It has been shown by DENISON (loc. cit.) that the deviations referred to are due, largely, if not entirely, to the occurrence of electric endosmose at the gelatine partitions. DENISON measured the amount of endosmose, and applied the correction to the ratio U/V with satisfactory results in the case of many salts.

The present research has been undertaken with the object of devising a method by means of which the solutions could be superposed without mixing, and which would avoid the use of membranes of any sort during the progress of the experiment.

The use of gelatine especially was to be avoided, not only on account of electric endosmose, which would be caused by any membrane, but also on account of the ease with whicli it melts on the passage of even small currents, and on account of the impossibility of obtaining it free from saline impurities. Even the purest obtainable gelatine contains a quantity of saline matter which is an appreciable fraction of the concentration of a moderately dilute salt solution, and this is probably the reason why our previous attempts to measure such solutions have been unsuccessful, and it appears to be impossible to measure transport numbers in dilute solutions in a system containing gelatine. Now we know that it is only in the case of the simple salts of the alkali metals that the transport number is practically independent of concen- tration ; with other salts the general tendency is for the anion transport number to increase with increasing concentration. This gives, of course, different values for the velocity of the same ion at different dilutions, and, moreover, the velocity of one and the same ion is found to be different for the same concentration when measured in different salts. These differences, however, vanish in dilute solution* in which, in accordance with the theory of KOHLRAUSCH and ARRHENIUS, a given ion has the same velocity in different salts. Hence an extension of the direct method of measuring ionic velocities to dilute solutions is much to be desired on account of its great simplicity and ease of manipulation.

Before this could be accomplished, however, it was necessary to devise some means of observing the margins. If gelatine be employed as membrane, the margins become invisible in solutions more dilute than about 0'2 N.

It was thought that an electrical method could be used to indicate the position of the boundary by taking advantage of the difference in conductivity of the indicator and measured solutions. This method, however, proved impracticable. Coloured salt solutions could not be used as indicators, as in solutions of about 0'02 normal the colour of solutions of salts such as copper sulphate, cobalt sulphate, nickel sulphate, &c., is much too faint to be serviceable. The intensely coloured ions of some of the organic dyes were tried, but, although certain acid dyes are excellent anion indicators, * STEELK and DENISON, ' Journ. Chem. Soc.,' 1902, LXXXL, p. 456.

3 M 2

452

DE. R. B. DENISON AND DR. B. D. STEELE ON THE

no basic dye was found sufficiently free from hydrolysis to be of any use as a cation indicator. It was ultimately found that, by the use of very high voltages, distinctly visible refraction margins could be obtained in solutions as dilute as 0'02 N.

Method of Experiment.

The placing of the two indicator solutions in contact with the middle electrolyte without appreciable mixing taking place, and without using gelatine or other membrane during the experiment, presented considerable difficulty. The ordinary method of pouring the one solution upon a thin piece of cork floating on the other solution was found to be quite unsatisfactory. The problem was finally solved by the use of the apparatus shown in figs. 1 and 2. It consists of two parts which are

\

Fig. 1.

Fig. 2.

joined together at G or G'. If both indicators are lighter than the solution to be measured, the apparatus is made up of two halves of the form shown in fig. 1. If both are heavier, then both halves are of the form shown in fig. 2. If one indicator

ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC. 453

is lighter and the other heavier than the solution to be measured, one-half of the form of fig. 1 and one-half like fig. 2 are used in constructing the apparatus.

In fig. 2 a hole with a collar is shown in the bottom of the thermostat H. The apparatus can be passed through this hole and held in position by a rubber ring, L. This was found to be the best way of supporting this half of the apparatus. Both forms of the apparatus consist essentially of three tubes, an electrode vessel, E or E', a vessel, B or B', containing the indicator solution, and the tubes A or A' in which the boundaries move. The tubes A and A' are made from carefully selected glass quill-tubing of about 4 to 5 millims. diameter, and are accurately calibrated. They are sealed into the wider tubes B or B' in such a manner that they form a shoulder projecting inwards. The lower extremity of the capillary tube K is expanded into a cone, C, around which a piece of parchment paper can be tightly stretched. This parchment paper fits over the projecting shoulder, so that when pressed upon it there is no mixing of the liquid poured into the apparatus through the tube K with the liquid previously placed in A. The tubes K and K' slide easily through holes in rubber corks, so that after an electrolytic margin has appeared in the tubes A or A' the membrane can be withdrawn from the shoulder, and a free passage is allowed for the current. The cones C are provided with holes to allow the passage of the liquid into B and also to permit the free flow of the electric current when the membrane is resting on the projecting shoulder.

The choice of electrodes and electrode solutions needs careful consideration if accurate results are to be obtained. It is obvious that no gas evolution at the electrodes must occur if a good margin is to be maintained. The indicator solutions most frequently employed were lithium chloride and sodium acetate, and in all our experiments the cathode consisted of lead wire dipping into and completely covered by a paste of lead peroxide, which completely prevented any evolution of hydrogen. The anode consisted of a copper wire dipping into cadmium amalgam, thus con- stituting a completely unpolarisable electrode. A rough calculation showed that the total volume change at the electrodes corresponding to the changes

PbO2+H2 = Pb + 2H2O and Cd+2Cl =

was negligible. To prevent the hydroxyl ions formed at the cathode from finding their way back into the middle electrolyte, the cathode was surrounded with a solution containing acetic acid.

The cadmium chloride formed at the anode is to some extent hydrolysed, and the hydrogen ions thus formed travel towards the cathode across the cation margin and decrease the resistance of the solution there, and so also the fall of potential and the velocity of the margin. To prevent this, a little sodium or lithium hydroxide was added to the solution surrounding the cadmium electrode. If these precautions were neglected, slightly false values for the transport numbers were obtained.

The whole apparatus was held in position by means of a hinged wooden arm, and

454 DK. R. B. DENISON AND DR. B. D. STEELE ON THE

placed in a bath with plate-glass sides. The framework of the bath was of copper and provided with flanged edges, on which the plate-glass sides were clamped. Rubber tubing was placed between the copper flanges and the glass, and in this way a perfectly water-tight junction was made and the use of putty rendered unnecessary, a con- siderable advantage if the bath is to be heated. The base of the bath was provided with two holes for the insertion and support of the apparatus shown in fig. 2.

In putting together the apparatus for an experiment, the electrode vessels E are first filled and the electrodes placed in position, care being taken that none of the electrode liquid passes over into B or A.* The two halves of the apparatus are then connected by rubber tubing and the tubes A are filled just to the shoulder with the solution to be measured. The rubber cork carrying the tube K and the membrane is then placed in position! and pressed on the shoulder, care being taken not to imprison any air-bubbles below the membrane. The electrical connections having been made, the current is started by pouring the indicator through F (fig. 1) or L (fig. 2), after which the plugs P and P' are inserted. The appearance of the margins in the tubes generally occupies only a few minutes, and as soon as they have advanced about 1 centim. into the tubes the membranes are removed by lifting K or lowering K'. In this manner electric endosmose is entirely prevented. The velocities of the two margins are measured by observing the distance moved in stated intervals of time, the observations being made by means of a telescope with cross wires and a glass scale placed immediately behind the tubes A and A'.

One other point requires notice. The tubes K in apparatus fig. 1 and P' in apparatus fig. 2 must not have a greater diameter than about 1 millim. The reason for this is as follows : —Suppose that we are measuring the transport number of sodium chloride with lithium chloride and sodium acetate as indicators, both solutions being lighter than that of sodium chloride. In one measuring tube we have the solutions lithium chloride over sodium chloride, and in the other sodium acetate over sodium chloride, and the relative length of the columns of lithium chloride and sodium acetate depends on the position of the margins. Now the sodium ion travels more slowly than the chlorine ion, with the result that a longer column of sodium acetate than of lithium chloride is formed. These solutions have different densities, and so there is a difference of hydrostatic pressure tending to move the whole column of liquid up one measuring tube and down the other, and the level in the tubes K and P becomes slightly altered during the progress of the experiment. This change of level accelerates one margin and retards the other, and false values for the velocities and transport number are obtained. In order to reduce the effect of the change of level, the diameters of K and of P are made as small as possible, and hence, since the total change of level amounts to only a fraction of a millimetre, the

* In the final form of apparatus the two halves were sealed together.

t In the case of the apparatus (fig. 2) used for heavy indicators, the tube K is placed in position before putting the solution into the tube A.

ACCtTKATE MEASUEEMENT OF IONIC VELOCITIES, ETC.

455

movement of the margin in the measuring tubes A due to this cause is infinitesimal and can be neglected. In our earlier experiments, in which the apparatus was tested with potassium chloride, the wide tubes B were open to the atmosphere, but as the potassium and chlorine ions have practically the same velocity, the effect described above did not manifest itself. But with n/10 sodium chloride the introduction of the capillary tubes K changed the transport number for chlorine from 0'58(J to its correct value, 0'614.

The electrical connections and the method of illuminating the boundary are the same as those described in_ previous papers.*

Testing the Method.

The only difficulty in the present method has been found to lie in the choice of suitable indicators. The transport number of any ion in a given salt should be the same at the same concentration, provided that the indicator used fulfils certain conditions, which are briefly as follows : — •

(1) The specific velocity of the indicator ion must be less than that of the ion whose progress it indicates.

(2) It must not react with any of the ions of the -salt being measured, nor must it be hydrolysed or give rise to any other ion moving in the same direction as the indicator ion.

(3) The resistance of the indicator solution must not be too great, -i.e., the concentration of ions in the indicator must not be too small, nor must there be too great a difference between the specific ionic velocity of the indicator ion and that of the ion whose velocity is being measured.

The method was first tested by measuring the transport numbers of potassium and sodium chlorides with different indicators, and the following results (Table 1.) were

obtained : —

TABLE 1.

Salt and concentration.	Indicator ions.	Anion transport number — p.	HITTOUF'S value t
i KC, {	Li and acetate ,, formate	0-508 0-513	0-514
y Nad {	„ acetate ,, formate	0-614 0-621	0-637
ft KC1 {	,, acetate „ formate	0-508 0-508	| 0-508
• re N»C1 {	„ acetate „ formate	0-017 0-618	0-617

* STEELE, loo. dt. • DENISON, loc. at.

t From KOHLRAUSCH and HOLBOKN'S ' Leitrermogen der Electrolyte,' p. 201.

456 DR. K. B. DENISON AND DR. B. D. STEELE ON THE

These results show that the method is capable of great accuracy. There is, indeed, a slight difference between the values of the transport numbers for the normal solutions according as acetate or formate is used as anion indicator, the percentage difference being about the same with potassium chloride as with sodium chloride. Although there is only a small difference between HITTORF'S value for these solutions and that found by us with sodium acetate, it is too large to be attributed to errors of experiment. We have measured these solutions repeatedly, and with the utmost precautions, and invariably obtain results which lie between 0'508 and 0'510 for potassium chloride and between 0'613 and 0'615 for sodium chloride. It is, therefore, necessary to recognise the presence of some unknown disturbing factor in these experiments. Whatever this factor is, its effect has disappeared in the more dilute solutions, and for these accordingly we have continued to use sodium acetate as anion indicator. For stronger solutions, however, it is advisable to use sodium formate.

Sodium benzenesulphonate lias been successfully employed by us as aiiion indicator in some experiments, but when it is used for stronger solutions thann/10, irregularities occur, the cause of which we have not yet succeeded in tracing.

Experimental Result*.

We have measured the transport number and ionic velocities of those salts only which give rise to strong ions, that is, salts which undergo little or no hydrolysis in aqueous solution.

The necessity of paying attention to the possibility of hydrolysis in all electrolytic experiments cannot be too strongly emphasised. Thus, it is of no value to determine the velocity of the ions of a salt which, when dissolved, gives rise to a complicated ionic system. We cannot assume, a priori, that the transport number of any ion as determined by the present method will give us the true fraction of the total current which is carried by that ion in the presence of other ions of the same sign. In HITTORF'S analytical method the actually measured transport number gives us the fraction of the total current carried by the ion in question, whether complex ions are present or not, if we assume that the current is wholly carried by the simple ions. Whether the direct method, and the method of HITTORF, will give the same transport number for a given ion in more complex ionic systems, or whether the presence of complex ions in solutions will affect the results obtained by the two methods to a different extent, are subjects for future experimental and mathematical investigation.

By means of the present method we have been enabled to obtain for the first time an experimental determination of the transport number for such salts as potassium chlorate, bromate and perchlorate, which present considerable difficulty in the determination by the analytical method.

In the following Table II., which contains the results of our transport-number determinations, the values given in the last column are taken from KOHLRAUSCH and HOLBORN'S ' Leitvermogen der Electrolyte.' The salts used were obtained from KAHLBAUM and were not submitted to any further purification.

ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC.

TABLE II.

457

Salt.	Concentration.	Indicator.	Anion transport number. V/(U + V).	KOHLRAUSCH.
Cation.	Anion.
KC1	0-ln 0-02» 0-02»	Lithium » >> )>	Formate Acetate j> Benzenesulphonate	0-513 0-508 0-507 0-507	0-514 0-508 0-507 0-507
NaCl	0-ln 0-05« 0-04w	F| J> )» )!	Formate Acetate n 11	0-621 0-618 0-614 0-612	0-637 0-617 0-614
BaCI,	0-7n 0-ln 0-02»	1) T> »	j» » ij	0-624 0-580 0-585 0-565 0-565
SrCl2	0-ln 0-02»	J) I)	u jj	0-601 0-589	—
CaCla	0-lw 0-02»	»! ))	» ;i	0-602 0-588	0-64 0-59
KNOs	0-ln 0-02»	J) »)	j) i?	0-498 0-498	0-497
K2S04	0-lw 0-02n	» I!	j> ?)	0-515 0-512	—
KC103	O-l/i 0-02»	)) >J	)» !)	0-464 0-466	—
KBr03	0-ln 0-02»	)? II	JJ JJ	0-430 0-433	—
KC104	0-lw	)»	JJ	0-477	—
KBr	0-lw 0-02»	» J)	JJ JJ	0-519 0-518	0-507 0-507
KI	0 • In 0-02«	J) JJ	JJ JJ	0-514 0-513	0-507 0-507
KOH	1-Ow 0-ln	J) 51	Bromide jj	0-738 0-743	0-735
NaOH	1-0» 0-lM	» J»	u jj	0-839 0-842	0-82 0-81
HC1	0-lw 0-02w	Potassium i)	lodate u	0-165 0-165	0-172
HN03	0-ln 0-02n	ii »j	u jj	0-145 0-154	—
H2S04	0-lw 0-02n	jj »	u u	0-172 0-167	0-191

VOL. CCV. — A.

3 N

458 DR. R B. DENISON AND DE. B. D. STEELE ON THE

It will be seen from the table that our results agree excellently with those obtained by HITTORF'S method, the difference in most cases being only a few units in the third decimal place. One of the most marked exceptions is n/l sodium chloride, for which we find 0'621 instead of 0'637. We have repeated this experiment a number of times and always with the same result. Possibly the older value requires confirmation.

KOHLRAUSCH and HOLBORN give the same value, 0'508, for the anion transport number of potassium chloride, bromide, and iodide. On the other hand, we have found the following values :—

Potassium chloride, p = 0'508. Potassium bromide, p = 0'518. Potassium iodide, p = 0'513.

As these results were so different from what we expected, we have measured the last two salts several times and invariably with the same result, so that we were at last convinced that our numbers were correct. We have since found a paper by KOHLRAFSCH (' Zeit, fur Electrochemie,' 1902, VIII, p. 630) which had escaped our notice. In this paper KOHLRAUSCH describes experiments showing that the velocities of the ions Cl', Br', and P, as determined from conductivity measurements, are as

follows : —

Cl = 65-44. Br = 67'63. I = 60'40.

These figures give for the transport number of the three salts the following

numbers :—

Potassium chloride = 0'502, Potassium bromide = 0'511, and Potassium iodide = O'SOl,

which is practically the same ratio as that found by us. The difference between the actual values is due to KOHLRAUSCH'S figures being for infinite dilution, whereas ours are for n/50 solutions.

Further confirmation of the correctness of our results is afforded by the fact, which will be referred to again later, that the potassium ion has the same velocity in equimolecular solutions of the above three salts if our values for the transport numbers are correct, but its velocity is different if the ordinary values are correct.

Measurement of the Ionic Velocities.

The terra " ionic velocity " appears to have been used by different writers in different senses, and as a consequence there is some confusion as to its exact significance. In what follows we shall speak of: —

(1) The actual measured velocity of the ion or ionic margin, which is its velocity

under the potential gradient of the experiment.

(2) The actual mobility (U or V) or the velocity of the ion in a given solution

under a potential gradient of 1 volt per centimetre.

ACCUEATE MEASUREMENT OF IONIC VELOCITIES, ETC. 459

(3) The specific mobility (u or v) or the velocity which the ion would have under

a potential gradient of 1 volt per centimetre if the salt were completely dissociated. These are related to the actual mobilities as follows :

U = au and V = av,

a being the degree of dissociation of the salt.

(4) The " lonen Beweglichkeit " (la or lk) of KOHLRAUSCH, which is given by the

relation

p. being the molecular conductivity.

The use of the term " specific ionic velocity " for la and lk is confusing, and we suggest instead the name " specific ionic conductivity." (5) The " ionic conductivity " for a given strength of solution can then be conveniently represented by

Lre and Lk, where L,, = a/,, and L* = «//,.

The actual mobilities U and V are obtained from the measured velocities In- dividing the latter by the potential gradient in the measuring tube. This potential gradient, II, is easily calculated as follows : — According to OHM'S law the current, C, flowing through the apparatus, C = H/>: C is measured, and /•, or the resistance of 1 centim. of the liquid, is obtained from its specific conductivity, K, and the area of cross-section, A, of the tube, v = 1/*A and hence IT = C/«A.

The actual mobilities of the ions of a number of salts have been measured, and the results at 18° are contained in Columns 6 arid 9 of Table III, and, for comparison with these, two sets of figures are given in Columns 7 and 10 and 8 and 11 respectively. These figures are calculated from KoHLRAUSCH's conductivity data in the following

&> manner : —

The molecular conductivity

^ whence

K =

t) being the concentration in gramme equivalents per cubic centimetre.

Now, K is the specific conductivity, or the quantity of electricity carried in one second between two electrodes 1 centim. apart in a tube of 1 sq. ceutim. cross-section under a potential gradient of 1 volt per centimetre.

This obviously depends on (a x rj), or the number of ions present, on their specific mobility (u and v), and on the quantity of electricity carried by one gramme ion, viz., 96540 coulombs. Hence

K = «7?(«-t-v) 96540 = 17 (U + V) 96540, 3 y 2

460

DK. R. B. DENISON AND DR. B. D. STEELE ON THE

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oo oo oo o oo oo ooo oo o	0 00 OO OO 00 0
OS	c 1	COCO (MOO OO 1~- COt— (M -H* OS I-H CM CD ~i< CM too coos t^o co cooo ooco -* 100 osco t— o to o o o CD co in in -* o o o to co -* co §o oo oo o oo oo o oo oo o o oo oo o oo oo o oo oo o o oo oo o oo oo o oo oo o	-* O OS 00 CM CD i— i COOO OlO OCM OO f|S ^^ ^^ CO CO CO CO <7-l Ol <O o oo oo oo oo o o oo oo oo oo o
oo oo oo o oo oo o	0 O 0 0 0	o oo oo oo oo o
OO	Anion velocity V in centimetre/seconds.	1, "~3 ^ Tf ±3 OJ 5j •* t^ - •^ o I-H o 03 ^3 H™1 ^ w "p	OOO OC3 t— £— I-H CO OOCO OCM C3CO OS CO O to CO to O CD O O oo oo oo o o oo oo oo o o oo oo oo o o		OS 0 0 O o to 1 0 0 1 0 0 o o o o	CM O CD 00 O CM O CD CO ooo o o o 000 666	CO o o 0 6	co CO o o 6
oo oo oo o o
-	ilil ^J: c 5	ooo co I-H too ^H oo -f --< o CD -* oo coo o OOCO i— i -^ O ^ OS COOS i-HO3 1— CM CO O O CO OCO CD to CD CD 1O OO OO -* -* -f OtO O oo oo oo o oo oo o oo oo o oo oo oo o oo oo o oo oo o oo oo oo o oo oo o oo oo o	in t-cM i-Ht— t— oo ooo I-H -**<o OCM cMt— oos -* CO OCD COtO OO -^-* CO O OO OO OO OO -H o oo oo oo oo o o oo oo oo oo o
oo oo oo o oo oo o	00000	o oo oo oo oo o
CO	1	OO CM O ->* O^O C3 TO 00 r-HOS I— o to co to to to o o o o o -* oo oo oo o oo oo o oo oo oo o oo oo o oo oo oo o oo oo o	CO O -f CO t— ^H CO O O O -f -* O to O o o ooo oooo o o o o o o	o o»* oos coos -H^O o -*<as OCM CMCO CMOO t— to oo coco oo •*-* co §OO OO OO OO I-H oo oo oo oo o o oo oo oo oo o
oo oo oo o oo oo o	oooo o	o oo oo oo oo o
in'	Transport number, v/m j.v\	SiS 3 » HH "N	GO t- oo t— oo t— t— i~- OO OO OO rt 03 OO OlO OO to -* 00 66 00 6 0	1	10 m 00 CD o o 6 6	CM •**< OS !>. CO O I-H 000	I-H OS o	o co 0
J	) S o	COt- OS CX> -* CO t- OOOO -t< CM -H OO ^H I— 1 I-H I— 1 l-H OSOS l-H I— 1 CD OO OO OO CO •^•^t1 OO -*	O CO CO O O CO CO 00 CO O •^ ^ o o co	oo CMCO oo oco int- OO OOO COCO -*O t— « O COO I-H 1— 1 I-HI-H I-HI-	- o ) -* ^ t-
00 00 00 0 00 00 0	00 OOO	o oo oo oo oo o
CO	§ -o £-$	OSO OO OCCM O CO (M OSOO I-H	I-HO CM CO CM	O OO r-H
I-HO -HH OS COl-H CM -^flf I-H (M I-H l-H l-H CM OS O i-	5 O ^H OS * OS i-H OS	O •<* 1M CO O OS O OS O O3 I-H I-H	I-H t— OS l-Ht— O"*4* OCO •*+< O OOOS OCO OCD CMOO i-H i-H COCOCOCOCMCMCM
CM	Q o 't3 rd ~\~ ot	COO GOt- OO O rtto t— CO 00	-* CO t~ O CO	•^ ^ co
OOO CMOS COOS CM COC>	) O I-H 00 H OS l-H OS H I-H	CO CM I-H CO OS 03 O 03 O 00 I-H l-H	OS t-t- OOO OOS -^<O OO os ooos oco oo cooo I-H CO CO CO CO CM CM CM
r— * l-H l-H I-H l-H I-H ^H I—
r-H	Salt and concentration.	8 8 o I-H 8 8 8(M 8cMO<M 8 SCM 8 <M 8 r-H O I-HO I-H O I-H I-HO I-H O I-H	8 CM 8 CM 8 I-HO 1— 1 O I—I	8 8 8 0 rt 8 8 S CMOCM 8 CM 8cM 8cM 8 O i-HO i-HO r-HO i— 1 O i-H
oo oo oo o oo oo o CO -^ X — A O <~> ® £	o o o o o n	0 00 OO OO OO 0 eo ^

- 9

pa <! H

ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC. and from these two equations we obtain

461

= (La+L,)x

96540'

and

u + v =

96540-

We thus obtain the value of (U + V), and by multiplying this by the anion transport number U/(U+V) we obtain U and, consequently, V also.

It will be noticed that the figures given in columns 7 and 10 agree on the whole much better with the directly observed mobilities than do those given in columns 8 and 11. In calculating the former our transport number has been used, those ot KOHLRAUSCH having been used in calculating the latter. The two sets of transport numbers are given in columns 4 and 5.

The comparison can be made in the opposite manner, p. being calculated from the actual mobilities from the relation

p. =

= a(u + v) x 96540 = (U + V) x 9G540.

This calculation has also been made and the results are given in the second column of the table, KOHLRAUSCH'S values for ^ being given in the third column.

Table IV. contains a similar set of results to those given in Table III., the measurements having been made at 25° with 10"' normal solutions. As conductivity

TABLE IV.

1.	2.	3.	4.	5.	6.	7.	8.
	T = 25° C.		Anion velocity, V.	Cation velocity, U.
Salt.	Molecular	Molecular	Transport number,
	con- ductivity,	con- ductivity,	V/(U + V).	Found.	Calculated.	Found.	Calculated.
	fj., from	H, KOHL-
	U + V.	RAUSCH.
KC1	128-2	128-8	0-507	0-000674	0-000676	0-000654	0-000658
KBr	131-9	132-0	0-520	0-000710	0-000711	0-000656	0-000657
KI	130-0	130-9	0-517	0-000696	0-000701	0-000652	0-000655
KN03	119-8	119-4	0-499	0-000619	0-000618	0-000621	0-000620
KBr03	107-7	107-2	0-430	0-000480	0-000479	0-000636	0-000635
KC103	113-2	113-2	0-463	0-000544	0-000544	0-000631	0-000631
K.2S04	108-9	109-0	0-521	0-000588	0-000588	0-000540	0 • 000540
CaCl2	101-0	101-4	0-604	0-000631	0-000635	0-000412	0-000416
BaCL,	105-2	105-2	0-584	0-000638	0-000638	0-000452	0-000452
SrCl2	101-2	101-2	0-596	0-000626	0-000626	0-000424	0-000424

determinations at this temperature were not available, the calculated mobilities given in columns 6 and 8 were obtained from our own conductivity measurements, which are given in column 3 for comparison with the molecular conductivities (column 2) as

462

DE. K. B. DENISON AND DR. B. D. STEELE ON THE

calculated from our specific mobilities (columns 3 and 5), whilst the agreement between the found and calculated values is remarkably close in the majority of instances. A few cases, where the differences are larger, call for special comment. For example, the calculated and observed mobilities of the hydrogen ion in hydro- chloric and nitric acid solutions agree very well indeed, whereas the agreement is by no means good for the velocities of the anions of these acids. This is explained by the fact that the velocity of the hydrogen ion is about five times as great as that of the chlorine ion (or nitrate ion), and as a consequence the latter only moves about 10 millims., whilst the former moves through the whole length of the tube. The result of this is that there is a very much larger percentage experimental error introduced in the measurement of the mobility of the anion than in that of the cation of an acid. The same applies, but in the opposite sense, to the measurement of the mobilities of the anion and cation of an alkali.

The observed values for «/10 sulphuric acid are quite different from the calculated values. Whether this is due to the inaccuracy of the transport number, or to the occurrence of complexes in the system, it is not possible to say, but the cause of error seems to have completely disappeared in the n/50 solutions.

Interesting results are obtained when we compare the velocities of the same ion in equimolecular solutions of different salts. KOHLRAUSCH'S law of the independent wandering of the ious depends on the assumption that, in solutions sufficiently dilute for any variation in " electric friction " to be neglected, the same ion has the same velocity in whatever solution it occurs. If this is so, and remembering that it is only the specific mobility, u, that is invariable, and that the actual mobility U = au, we see at once that it is only in solutions of salts that are equally ionised that we can expect to find the same actual mobility for the same ion, and, on the other hand, where we do find this, we have strong evidence that the salts in solution are equally ionised.

On comparing the velocities of the potassium ion in equimolecular solutions of all the potassium salts that we have measured (see Table V.), we find that this velocity is identical in solutions of potassium, chloride, bromide, and iodide, but is a little smaller in the other solutions. This identity of mobility is manifested not only in

TABLE V. — Velocity of K ion in various Salt Solutions.

Salt.	»/10 at 18°.	m/50 at 18°.	n/10 at 25°.
KC1	0-00056,3	0-000606	0-000654
KBr	0-000562	0-000598	0-000656
KI	0-000564	0-000599	0 -0,00652
KC103	0-000549		0-000631
KBr03	0-000551	0-000601	0-000636
KN08	0-000536	0-000583	0-000621
K2S04	0-000510	0-000593	0-000540

ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC.

463

10th normal solutions at 18° and 25°, but also in 50th normal solutions at 18°. In the same way the mobilities of the chlorine ion in equimolecular solutions of different salts have been gathered together in Table VI. In this case, also, the same mobility

TABLE VI. — Velocity of Cl ion in various Salt Solutions.

Salt.	n/10 at 18°.	n/50 at 18°.	«/10 at 25°.
KC1	0-000582	0-000622	0-000674
NaCl	0-000585	—
BaCl2	0-000554	0-000603	0-000638
SrCl.	0-000559	0-000604	0-000626
CaCL,	0-000542	0-000596	0-000631

occurs in solutions of potassium and sodium chlorides, and similarly for the hydrogen mobilities which are given in Table VII.

These results indicate that the direct measurement of the actual ionic mobility gives us a means of comparing the degree of ionic dissociation of equimolecular

TABLE VII. — Velocity of H ion in various Acids.

Salt.	n/10 at 18°.	•«./50 at 18\	w/10 at 25°.
HC1 HNO3 H2S04	0-00304 0-00311 0-00201	0-00317 0-00315 0-00242	—

solutions of salts containing a common ion. If we assume that in dilute solutions v, or the specific mobility of a given ion, is invariable in different salt solutions, then, since U = av, a is proportional to U. Table VIII. contains a comparison on this basis of the degrees of dissociation of all the salts containing one ion in common that have been measured by us. The figures are given as the ratio of the degree of dissociation of the salt to that of potassium chloride, which has been taken as standard, and the same ratio has been calculated from the conductivity data, and the results are given for comparison in the third and fifth columns of the table.

From the very satisfactory agreement of the results obtained by the present method with the corresponding numbers given by KOHLRAUSCH, it is evident that a considerable degree of accuracy in the direct measurement of ionic velocities has been attained. It is interesting to look over HITTORF'S early transport numbers, amongst which deviations of 5 and 10 per cent, are frequently met with.

The numbers subsequently obtained were much more concordant, but the method, as already stated, is difficult and laborious, and only rarely can an experiment be

464

ON THE ACCURATE MEASUREMENT OF IONIC VELOCITIES, ETC.

completed in a day. The direct method, in its present form, gives results at least as accurate as the indirect one, and with much less trouble. In dilute solution the determination is generally complete in about half an hour, as the high voltage, which is necessary in such solutions, imparts a considerable velocity to the ions. After

TABLE VIII. — Degrees of Dissociation.

O li	w/10.	n/50.
oalt.	Found.	Calculated.	Found.	Calculated.
KBr	1-00	1-00	0-98
KI	1-00	1-01	0-98	1-01
KN03	0-95	0-97	0-96	1-00
K,S04	0-85	0-83	0-93	0-91
KC103	0-98	—	0-91	—
KBr03	0-98	—	0-90	—
NaCl	1-01	0-99	. —	—
BaCla	0-95	0-88	0-97	0-93
SrCL,	0-96	0-88	0-97	0-93
CaCL	0-94	0-87	0-96	0-91
HC1	1-01	1-07	1-02	1-04
HN03	1-02	1-01	1-01	1-00

becoming accustomed to the method, it is easy to perform five or six experiments in a day. In general the total amount of motion for anion and cation was about G centims., with a probable error in reading of about O'Ol centim. for each boundary, i.e., a total error of about 0'3 per cent. The error in measuring the conductivity might be 0-5 per cent., and the error in measuring the small amount of current flowing through the apparatus also about O'o per cent. The error in the calibration of the tube is negligible, and, therefore, in a good experiment the transport number should be correct to about 1 part in 300, and the ionic velocities to at least 1 part in 100. As a matter of fact, this degree of accuracy is easily attainable for the transport number. In the case of the separate ionic velocities the accuracy obtainable is not quite so great owing to the accumulation of errors from the various measure- ments involved.

[ 465 ]

XIV. On Mathematical Concepts of the Material World.

By A. N. WHITEHEAD, Sc.D., F.R.S., Felloiv of Trinity College, Cambridge.

Keceived September 22, — Head December 7, 1905.

PREFACE.

THE object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the material world. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the "stuff" in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows : Given a set of entities which form the field of a certain polyadic (i.e., many-termed) relation R, what "axioms" satisfied by R have as their consequence, that the theorems ot Euclidean geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean geometry. The solution of this narrower problem of the axioms of geometry is assumed (cf. Part II. , Concept I.) without proof in the form most convenient for this wider investigation. But in Concepts III., IV., and V., the entities forming the field of R are the " stuff," or part of the " stuff," constituting the moving material world. POINCARE* has used language which might imply the belief that, with the proper definitions, Euclidean geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem, here discussed, is to find various formulations of axioms concerning R, from which, with appropriate definitions, the Euclidean geometry issues as expressing properties of the field of R. In view of the existence of change in the material world, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration.

The general problem is here discussed purely for the sake of its logical (i.e., mathematical) interest. It has an indirect bearing on philosophy by disentangling the essentials of the idea of a material world from the accidents of one particular concept. The problem might, in the future, have a direct bearing upon physical science if a concept widely different from the prevailing concept could be elaborated, which * Cf. ' La Science et 1'Hypothese,' chap. III., at the end.

VOL. COV.— A 400. 3 0 16.5.06

466 DR. A. N. WHITEHEAD ON

allowed of a simpler enunciation of physical laws. But in physical research so much depends upon a trained imaginative intuition, that it seems most unlikely that existing physicists would, in general, gain any advantage from deserting familiar habits of thought.

Part I. (i) consists of general considerations upon the nature of the problem and the method of procedure. Part I. (ii) contains a short explanation of the symbols used. Part II. is devoted to the consideration of three concepts, which embody the ordinary prevailing ideas upon the subject and slight variants from them. The present investigation has, as a matter of fact, grown out of the Theory of Interpoints, which is presented in Part III. (ii), and of the Theory of Dimensions of Part IV. (i). These contain two separate answers to the question : How can a point be defined in terms of lines ? The well-known definition* of the projective point, as a bundle of lines, assumes the descriptive point. The problem is to define it without any such assumption. By the aid of these answers two concepts, IV. and V., differing very widely from the current concepts, have been elaborated. Concept V., in particular, appears to have great physical possibilities. Indeed, its chief difficulty is the bewildering variety of material which it yields for use in shaping explanations of physical laws. It requires, however, the discovery of some appropriate laws of motion before it can be applied to the ordinary service of physical science.

The Geometry throughout is taken to be three-dimensional and Euclidean. In Concept V. the definition of parallel lines and the " Euclidean " axiom receive new forms ; also the " points at infinity " are found to have an intimate connection with the theory of the order of points on any straight line. The Theory of Dimensions is based on a new definition of the dimensions of a space.

The main object of the memoir is the development of the TJieory of Interpoints, of the Theory of Dimensions, and of Concept V. The other parts are explanatory and preparatory to these, though it is hoped that they will be found to have some independent value.

PART I. — (i) GENERAL CONSIDERATIONS.

Definition. — The Material World is conceived as a set of relations and of entities which occur as forming the " fields " of these relations.

Definition. — The Fundamental Relations of the material world are those relations in it, which are not defined in terms of other entities, but are merely particularized by hypotheses that they satisfy certain propositions.

Definition. — The hypotheses, as to the propositions which the fundamental relations satisfy, are called the Axioms of that concept of the material world.

Definition. — Each complete set of axioms, together with the appropriate definitions and the resulting propositions, will be called a Concept of the Material World.

* Here in "Descriptive Geometry" straight lines are open, and three collinear points have a non- projective relation of order ; in " Projective Geometry " straight lines are closed, and four collinear points have a projective relation of separation.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 467

Definition. — The complete class of those entities, which are members of the fields of fundamental relations, is called the class of Ultimate Existents. This technical name is adopted without prejudice to any philosophic solution of the question of the true relation to existence of the material world as thus conceived.

Every concept of the material world must include the idea of time. Time must be composed of Instants (cf. BERTRAND EUSSELL, ' Mind,' N.S., vol. 10, No. 39). Thus Instants of Time will be found to be included among the ultimate existents of every concept.

Definition. — The class of ultimate existents, exclusive of the instants of time, will be called the class of Objective Reals.

The relation of a concept of the material world to some perceiving mind is not to be part of the concept. Also we have no concern with the philosophic problem ol the relation of any, or all, of these concepts to existence.

In Geometry, as derived from the Greeks, the simple elements of space are points, and the science is the study of the relations between points. Points occur as members of the fields of these relations. Then matter (the ultimate " stuff" which occupies space) in its final analysis, even if it is continuous, consists of entities, here called particles, associated with the points by relations which are expressed by saying that a particle occupies (or is at) a point. Thus matter merely occurs as one portion of the field of this relation of occupation ; the other portion consists of points of space and of instants of time. Thus " occupation " is a triadic relation holding in each specific instance between a particle of matter, a point of space, and an instant of time. According to this concept of a material world, which we will call the Classical Concept, the class of ultimate existents is composed of three mutually exclusive classes of entities, namely, points of space, particles of matter, and instants of time. Corresponding to these classes of entities there exist the sciences of Geometry, of Chronology, which may be defined as the theory of time considered as a one- dimensional series ordinally similar to the series of real numbers, and of Dynamics. There appears to be no science of matter apart from its relations to time and space.

Opposed to the classical concept stands LEIBNIZ'S theory of the Relativity of Space. This is not itself a concept of the material world, according to the narrow definition here given. It is merely an indication of a possible type of concepts alternative to the classical concept. It is not very obvious how to state this theory in the precise nomenclature here adopted. The theory at least means that the points of space, as conceived in the classical concept, are not to be taken among the objective reals. But a wider view suggests that it is a protest against dividing the class of objective reals into two parts, one part (the space of the classical concept) being the field of fundamental relations which do not include instants of tune in their fields, and the other part (the particles) only occurring in the fields of fundamental relations which do include instants of time. In this sense it is a protest against exempting any part of the universe from change. But it is not probable that this is the light in which

3 o 2

468 DR. A. N. WHITEHEAD ON

LEIBNIZ himself regarded the theory. This theory, though at present it is nominally the prevailing one, has never been worked out in the form of a precise mathematical concept. It is on this account criticized severely by BERTRAND RUSSELL (cf. loc. tit. and 'Philosophy of LEIBNIZ,' Cambridge, 1900, p. 120), who, however, has gone further than any of its upholders to give it mathematical precision. Of course, from the point of view of this paper, we are not concerned with upholding or combatting any theory of the material world. Our sole purpose is to exhibit concepts not inconsistent with some, if not all, of the limited number of propositions at present believed to be true concerning our sense-perceptions.

Definition. — Any concept of the material world which demands two classes of objective reals will be called a Dualistic concept ; whereas a concept which demands only one such class will be called a Monistic concept.

The classical concept is dualistic ; Leibnizian concepts will be, in general, monistic (cf. however Concept IV A.). OCCAM'S razor — Entia non multiplicanda prseter necessitatem — formulates an instinctive preference for a monistic as against a dualistic concept. Concept III. below is an example of a Leibnizian monistic concept. The objective reals in it may be considered to represent either the particles or the points of the classical concept. But they change their spatial relations. Perhaps LEIBNIZ was restrained from assimilating his ideas more closely to Concept III. by a prejudice against anything, so analogous to a point of space, moving — a prejudice which arises from confusing the classical dualistic concept with the monistic concepts. It is of course essential that at least some members of the class of objective reals should have different relations to each other at different instants. Otherwise we are confronted with an unchanging world. Concept V. is another Leibnizian monistic concept.

The Time- Relation, — In every concept a dyadic serial relation, having for its field the instants of time and these only, is necessary. The properties of this Time- Relation form the pure science of chronology. The time-relation is, in all concepts, a serial relation ordinally similar to the serial relation which generates the series of negative and positive real numbers.* This fact need not be further specified during the successive consideration of the various concepts, nor need any of the propositions of pure chronology be enunciated.

Definition. — The class of instants of time is always denoted by T in every concept.

The Essential Relation. — In every concept at least all the propositions of geometry will be exhibited as properties of a single polyadic relation, here called the essential relation. The field of the essential relation will consist, either of the whole class of ultimate existents (e.g., in Concepts III., IVs. and V.), or of part of the class of objective reals together with the instants of time (e.g., in Concept IVA.), or of the whole class of objective reals (e.g., in Concept II.), or of part of the class of objective

* For interesting reflections on this subject, influenced by the Kantian Philosophy and previous to the modern " Logicization of Mathematics," cf. HAMILTON, ' Lectures on Quaternions,' preface.

MATHEMATICAL CONCEPTS OP THE MATERIAL WORLD. 469

reals (e.g., in Concept I.). The essential relation ol any one concept will be a relation between a definite finite number of terms, for example, between three terms in Concepts I. and II., between four terms in Concept III., and between five terms in Concepts IV. and V.*

Definition. — In the exposition of every concept, the essential relation of that concept is denoted by R.

The Extraneous Relations. — In all the concepts here considered, other relations, here called the extraneous relations, will be required in addition to the time-relation and the essential relation. In Concepts I. and II. and IV. an indefinite (if not infinite) number of extraneous relations are required, determining the positions of particles. In Concepts III., IV. and V. one tetradic extraneous relation is required to determine the "kinetic axes" of reference for the measurement of velocity.

The time-relation, the essential relation and the extraneous relations form the fundamental relations of any concept in which they occur.

It will now be necessary to define geometry anew, since the previous definition has essential reference to the dualism of the classical concept. A proposition of geometry is any proposition (l) concerning the essential relation ; (2) involving one, and only one, instant of time ; (3) true for any instant of time.

In the classical concept everything is sacrificed to simplicity in reference to geometry, probably because it arose when geometry was the only developed science. The result is that, when the properties of matter are dealt with, an appalling number of extraneous relations are necessary.

Judged by OCCAM'S principle, this class of extraneous relations forms a defect in Concepts I. and II. and IV. Also, in both forms of the classical concept (viz., in Concepts I. and II.) geometry is segregated from the other physical sciences to a greater degree than in the other concepts.

In the study of any concept there are four logical stages of progress. The first stage consists of the definition of those entities which are capable of definition in terms of the fundamental relations. These definitions are logically independent of any axioms concerning the fundamental relations, though their convenience and importance are chiefly dependent upon such axioms. The second stage consists of the

* The idea of deriving geometry (at least protective geometry without reference to order) from a single triadic relation was (I believe) first enunciated and investigated by Mr. A. B. KEMPE, F.R.S., in 1890, cf. "On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points," ' Proc. Lond. Math. Soc.,' vol. XXI. It has since been worked out in detail for Euclidean geometry by Dr. 0. VEBLEN, cf. "A System of Axioms for Geometry," 'Trans. Amer. Math. Soc.,' vol. 5, 1904. Also cf. Professor J. ROYCE on " The Relation of the Principles of Logic to the Foundations of Geometry," 'Trans. Amer. Math. Soc.,' 1905. Professor ROYCE emphasises the importance of KEMPE'S work and considerably extends it. This memoir (which unfortunately only came into my hands after the completion of the present investigation) anticipates a general line of thought of the present paper in the emphasis laid on the derivation of geometry from a single polyadic relation ; otherwise our papers are concerned with different problems.

470 DR. A. N. WHITEHEAD ON

deduction of those properties of the defined entities which do not depend upon the axioms. The third stage is the selection of the group of axioms which determines that concept of the material world. The fourth stage is the deduction of propositions which involve among their hypotheses some or all of the axioms of the third stage. Psychologically the order of study is apt to be inverted, by first choosing propositions of the second and fourth stages because of their parallelism with the propositions of sense-perception and then by considering the first and third stages. The essential part of our task in passing concepts in review is the exhibition of the first and third stages. The second and fourth stages will only be so far touched upon as seems desirable for the purposes of elucidation.

Thus in respect to each concept considered the investigation will proceed as follows: — A certain relation R (the essential relation of the concept in question), which holds between a certain definite number of entities, is considered. The class of entities, between sets of which this relation holds, is called the "field" of R. Definitions of entities allied to R and to entities of the field of R are then given. These definitions involve no hypotheses as to the properties of R, but are of no importance unless R has as a matter of fact certain properties. For example, it may happen that the classes, thus defined, are all the null class (i.e., the class with no members) unless R has the requisite properties. Again deductions (in the second stage), made without any hypothesis as to the properties of R, may be entirely trivial unless R has certain properties. If R has not the requisite properties the deductions often sink into the assertion that a certain proposition which is false implies some other proposition. This is true* but trifling. The " axioms " respecting R are then given. These are the hypotheses as to the properties of R which are required in the concept under consideration. Finally such deductions are given as are necessary to elucidate the concept.

None of the reasoning of this memoir depends on any special logical doctrine which may appear to be assumed in the form in which it is set out. Furthermore certain contradictions recently discovered have thrown grave doubt upon the current doctrine of classes as entities. Any recasting of our logical ideas upon the subject of classes must of course simply issue in a change of our ideas as to the true logical analysis of propositions in which classes appear. The propositions themselves, except a few extreme instances which lead to contradictions, must be left intact. Accordingly the present memoir in no way depends upon any theory of classes.

The above considerations as to method must essentially hold for any investigation respecting axioms of geometry or of physics, viewed purely as deductive sciences, and apart from the question of experimental verification.

In Concepts I., II., and III. the members of the " field" of R are to be considered as points, except those members of the field which are instants of time. In these concepts the lines and planes are classes of points. In Concepts IV. and V. the

* Of. KUSSELL, ' The Principles of Mathematics,' § 16.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 471

members of the " field " of K, other than the instants ot time, are to be considered as lines taken as simple entities. Points are classes of these simple lines. But the ordinary line of geometry which has parts and segments is a class of points, and so is the ordinary plane of geometry. In Concept III., which is Leibnizian and monistic, the points (perhaps " particles " is here a better word) move, and the straight lines and planes disintegrate from instant to instant. In Concepts IV. and V. the points similarly disintegrate.

(ii) EXPLANATION OF SYMBOLISM.

This explanation is only concerned with the general logical symbolism. The special symbols which arise out of the ideas of the paper are defined in their proper places. PEANO'S* chief symbols are used. The changes and developments from PEANO, which will be found here, are due to RUSSELL and myself working in collaboration for another purpose. It would be impossible to disentangle our various contributions, t

None of the reasoning of the paper is based upon any peculiarity of the symbolism. It is used here only as an alternative form for enunciations, for the sake of its conciseness and (above all) its precision. In the verbal enunciations precision has been to some extent sacrificed to lucidity ; and the exact statement of what is meant is always to be sought in the symbolic alternative form. The proofs have been translated into words out of the symbolic form in which they were mostly elaborated.

On D, =, c, e, =, = Df

There are five copulas, namely, D, =, c, e, =. Here x^y means x implies y ; and x == y means x implies y and y implies x; and xcy means all x's are y's; and xey means x is a member of y ; and x, yen means x and y are members of u. Note that xcy implies that x is a subclass of y ; a class will be said to contain a subclass and to possess a member. Lastly, x = y means x is identical with y. Note that, if Df, short for Definition, is placed at the end of the line, thus,

x = y Df

the symbols mean that x is defined to stand for y. In such a case y is some complex of symbols, and x will be an abbreviated symbol standing for y.

On (f>lx, (x).<f>!x, fax) . <l>\x, (x, y), fax, y), D,

Propositioned Functions. — <f>lx means x has the property <j>, where <j> is given different forms corresponding to different properties; ftlx means x has the property

* Cf. 'Notations de Logique Mathematique,' Turin, 1894; .and ' Formulaire Mathematique,' Turin, 1903.

t See, however, RUSSELL'S articles, " Sur la Logique des Relations," ' Revue de Mathematiques,' vol. VII., 1900-1901, Turin.

472 I>B- A- N. WHITEHEAD ON

of being a class possessing at least one member; and (x) . $lx means $\x is true for every value of x; and fax) . <j>\x means there exists a value of x for which <f>lx is true. Note that (x) and fax), written before any proposition involving x, give the above meanings, even if the proposition is not in the symbolic form <f>\x. If the proposition involve both x and y, then (x, y) prefixed means that the proposition is true for all values of x and y ; and similarly for (x, y, z), and so on. Also (gee, y) prefixed means that there exist values of x and y, such that the proposition is true ; and similarly for fax, y, z), and so on. Furthermore <j>\ x -DX \jj\x stands for (x) . {(f)lx 3 \)flx}, and similarly for two and three variables.

On the Use of Dots, viz., ., :, .'., : :

p . q or p : q or p . '. q or p : : q all mean p and q are both true propositions. As an example, x, y e u, which has been defined above, is really the proposition xeu.yeii', and x, y, z e u is the proposition xeu . y eu . zeu.

Dots as Brackets. — The different symbolic forms for the joint assertion of pro- positions arise from the fact that dots are also used as bracket forms for propositions according to the following rules : —

(i) The larger aggregation of dots represents the exterior bracket, (ii) The dots at the end of a complete sequence of symbols are omitted, (iii) The dots immediately preceding or succeeding the implication sign, viz., D, are exterior brackets to any equal number of dots occurring in other capacities (e.g., as above in the joint assertion of propositions), (iv) The dots which also serve to indicate the joint assertion of propositions are interior brackets to any equal number of dots occurring in other capacities, (v) The dots after (x) and fax) are increased in number according to the necessity for their use as brackets.

In reading a symbolic proposition it is best to begin by searching for that implication sign, viz., 3, which is preceded or succeeded by the greatest number of dots. This splits up the proposition into hypothesis and consequent ; and so on with these subsidiary propositions, if necessary.

On V, •*•, ^e, ?£, ^(x).4>lx, -^fax).<t>\x

Again £>V? means one or other or each of p and q is a true proposition; and -^-p means p is not true. Thus ~- <j> ! x means x has not the property <j> ; also x -~ e u stands for — (x eu); and x ^ y stands for ^-(x = y); and -^(x) . <f>lx stands for ^ {(x) . <j)lx} ; and ^fax) . <f)\x stands for ^ {fax) .

Onx(<f>!x), (ix)(<l>\x), i', 7', u, n, u', n'

Non-Propositional Functions.— x(<j>\x) denotes the class of terms which have the property (ft, and (ix) (<f>\x) denotes the single entity (if there is such) which, when

MATHEMATICAL CONCEPTS OF THE MATEKIAL WOELD. 473

substituted for x, makes (j>lx to be a true proposition. It is not necessary for the above symbolism that the proposition involving x should be in the symbolic form <j>\x. Again, L'X denotes the class possessing x as its sole member, and i'x denotes the sole member of the class x, and uuv denotes the logical sum of u and v, that is, the class possessing all members of u and all members of v and no other members. Thus, i'aui'& denotes the class whose sole members are a and b. Again, unv denotes the logical product of u and v, that is, the complete common subclass of u and v; and u'w denotes the class which is the logical sum of all members of u, that is, the class which has as members all members of members of u; and n'w denotes the class which is the logical product of all members of u. The exact symbolic definition of n'n is

n'w = x {v f. u . ov . x e v} Df

It follows from this definition that, if u possess no members, r\'u is the class of all entities.

On A, els', -, Nc'

Again, A denotes the null class, that is, the class with no members ; els'", denotes the class ivhose members are the subclasses contained in u, including u itself and the null class. It follows that the propositions, vecls'u and vcu, have practically identical meanings. Again, u-v denotes the class u with the exception of those members which it possesses in common ivith v.

The cardinal numbers* are themselves classes. Thus, 1 is the class whose members are the unit classes, 2 is the class whose members are couples. Accordingly, xc2 means x is a class with two members ; Nc'it denotes the cardinal number of the class u.

On fi, fi", i,", i", u", and so on.

The general form for a non-propositional function whose value depends on x is fix, where fi receives different forms for different functions, as has been illustrated by the particular cases considered above. The apostrophe may be read as "of"; it is the general symbol for the connection of the preceding functional sign with the succeeding argument. According to this rule we should write sin'x for sin x and log'x for log ,r. Again, fi"u denotes the class of values of fix, ivhen the various members of u are substituted for x ; it may be read " the class of fi's of u's." Thus, if fix is " the head of x," and u is "the class of horses," then fi'u is " the class of heads of horses." The exact symbolic definition of fi'u is as follows :

fi'u = z {(333) . xeu . z = fix} Df

It follows from the definition, by substituting for fi, that i"«, i"u, u"«, n"«, cls'X Nc"w are now defined.

* Cf. RUSSELL, 'Principles of Mathematics,' chap. XL, and FKEGE, 'Grundlagen der Arithmetik,' Breslau, 1884, pp. 79, 85.

VOL. CCV, — A, 3 P

474 DR- A- N- WHITEHEAD ON

. . •' . . On(ExWy).. [' ".• ^ '^Jmv^^

Again, (Ex£<l>'y) means there exists an entity ivhich is denoted by the non- propositional function <f>'z, when z has the particular value y. For example, if u is a class, there is such an entity as its cardinal number, denoted by Nc'w ; but if u is not a class, there is no such entity as its cardinal number.*

On Symbols of the Type R;( ).

Relations. — R'(xyz) means x, y, z form an instance in which the triadic relation R holds, the special "positions" of x, y, z in this instance of that relationship being indicated by their order of occurrence in the symbol ^'(xyz). Again, R;( ' yz) means there exists an entity, x say, such that ~R'(xyz). The symbolic definitions of R;( ' and of analogous symbols, are

Df

z) . = . (W) . R(xyz) Df

Df Df 2) . E'(a^) Df

and so on. Again, R; ( ; yz) denotes the class of terms, such as x (say), which satisfy K,-(xyz), and ~R'(-;z) denotes the class of terms, such as y (say), such that there exists a term, x say, such that ~R'(xyz) holds. The symbolic definitions of R:(;yz) and of analogous entities, and of R;(';«) and of analogous entities, are

Df Df Df Df Dt

Df

and so on :

Df

Df Df

The difficult question of the import of a proposition, which contains a non-propositional function (with some particular entity as argument) to which no entity corresponds, has recently been elucidated by RUSSELL, cf. 'Mind,' October, 1905. All propositions containing such a function are untrue, unless the function is merely a constituent of a subsidiary proposition whose truth is not implied by the proposition in question.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 475

Thus, R:(;>-) denotes the class of terms, such as x (say), such that there exist terms, y and z say, such that ~R-(xyz) holds.

Again, R:(;;z) denotes the class which is the logical sum o/R;(;-z) and R:(-;z); and R;(;;-) denotes the class which is the logical sum q/"R;(;") and R;(-;-); and R:(;;;) denotes the class which is the logical sum of R;(;") and R;(v) and R;(--;), that is, the "field" of the relation R. The symbolic definitions of the above, and of similar entities, are :

R'(;;z) = R'(;-z)uR'(-;z) Df

»(;y;) = R'(;jr)uR;(-y;) Df

R'(z;;) = R'(* ;>&(*•;) Df

R'(;;-) = Ri("')uRi(-;-) Df

arid so on, and

R'(;;;) = R'(;--)uR'(-;-)uR'(--;) Df

This notation, which has been explained for triadic relations, can obviously be extended to any polyadic relations. Thus, ~R-(abcd) and ~R'(abcdt) are defined in a similar manner, and so are the symbols for the allied propositions and classes.

On 1 — 1.

A dyadic relation, S say, is called one-one, when each referent has only one relatum, and each relatum has only one referent. The class of one-one relations is denoted by 1 — *•!. The symbolic definition is

Df

On I-

The Assertion Sign. — A proposition, which is stated in symbols as being true, i.e., which is asserted as distinct from being considered, has the symbol " prefixed to it, with as many dots following as will serve to bracket off the proposition. This symbol I" is called the assertion sign*

PART II.- — THE PUNCTUAL CONCEPTS.

Those concepts of the material world in which the class of objective reals is composed of points, or particles, or of both, will be called the punctual concepts. The classical concept is a punctual concept, and will be considered first. The other punctual concepts can be explained briefly by reference to the classical concept.

Concept I. — (The Classical Concept). — This is dualistic, the class of objective reals

* This symbol is due to FREGE, who first drew attention to the necessity of the idea which it symbolizes; cf. his ' Begriffschrift,' HALLE, 1879, and RUSSELL, ' Principles of Mathematics,' p. 35.

3 P 2

470 DR. A. N. WHITEHEAD ON

being subdivided into points of space and particles of matter. The essential relation has for its field the points of space only. Slight variants (not considered here) can be given to the concept by varying the properties of the essential relation, so as to make the geometry non-Euclidean, or, retaining Euclidean geometry, so as to give various forms to the essential relation and the resulting axioms. In the exposition of a system of geometrical axioms for Concept I., VEBLEN'S memoir (cf. loc. cit.), to which I am largely indebted, will be followed. The changes which are made from VEBLEN'S treatment are (i) in the addition of the symbolism which emphasizes the idea of the essential relation, and (ii) in the fact that the question of the independence of axioms is here ignored, through a desire not to overload this memoir with difficulties (both for the author and reader) belonging to another part of the subject. As the result of (ii), some of VEBLEN'S definitions and axioms have been simplified (and, in a sense, spoiled). The axioms thus obtained for Concept I. will shorten our investigations of other concepts by serving as a standard of comparison to determine whether the axioms of the other concepts are sufficient to yield three-dimensional Euclidean geometry.*

The essential relation (called R) is triadic. R;(«6c) means the points a, b, c are in the linear order (or the Tel-order) abc. The relation R, when R'(afrc) holds, is not symmetrical as between the three points a, b, and c ; namely, it will be found that a and b (or b and c) cannot be interchanged.

Definitions of Concept I.

Definition. — The class R;(a;6) is the segment between a and b ; and the class R;(«6;) is the segmental prolongation of ab beyond b ; and the class R:(;a6) is the segmental prolongation of ab beyond a. It follows from the subsequent axioms that R;(a6;) is identical with R;(;6a).

Definition. — The straight line ab is the logical sum of R;(a;i) and R;(;afe) and R;(a6;) together with a and b themselves. Its symbol is R;a6. The definition in symbols is

R;;a6 u R;a;& u R;afc; u i'a u i'& Df

* On the philosophical questions connected with the mathematical analysis of geometry cf. ' A Critical Exposition of the Philosophy of LEIBNIZ,' Cambridge, 1900, and 'The Principles of Mathematics,' Cambridge, 1903, both by BERTRAND KUSSELL ; and also two articles by L. COUTURAT in the 'Revue de Metaphysique et de Morale' (Paris) for May and September, 1904, one entitled "La philosophic des Mathematiques de KANT " and the other " Les principes des Mathe"matiques — VI. La geometric"; also POINCARE'S ' Science and Hypothesis,' Part II., English translation, London, 1905.

For expositions of exact systems of axioms cf. 'Vorlesungen iiber neuere Geometric,' Leipzig, 1882, by PASCH ; also ' I Principii di Geometria,' Turin, 1889, by PEANO; also " I Principii della Geometria di Posizione," 'Trans. Acad. of Turin,' 1898, by PIERI; also HILBERT'S 'Foundations of Geometry,' Engl. Transl., Chicago, 1902 ; also Professor E. H. MOORE, " On the Projective Axioms of Geometry," ' Transact. of the Amer. Math. Soc.,' 1903; also Dr. 0. VEBLEN (loc. cit.); also Professor J. ROYCE (loc. cit.).

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 477

Definition. — The class whose members are the various straight lines is denoted by linR. The definition in symbols is

linR = v {($a, b) . a, b e K;( ; ; ; ) . a ^ b . v = R:"o&} Df

Definition. — Any class of points [i.e., members of R;(;;;)] is called a figure. Definition. — The class of lines defined by a figure u is the class of lines defined by any two distinct points of u. Its symbol is lnR'?<. The definition in symbols is

lnR'M = v {(RX, y) . x, y e u n R;( ; ; ; ) . x ^ y . v = R'iw/} Df

Definition. — The linear figure defined by a figure u is the logical sum of all the lines defined by u (i.e., is all the points on such lines). Thus its symbol is u'lnR'w.

Definition. — Three points form a triangle, if there is no line which possesses them all. The symbol expressing that a, b, c are points forming a triangle is AH'(abc). The definition in symbols is

A R'(abc) . = . a, b, c e R:( ; ; ; ) . -»- (QV) . v e lin,{ . a, b, cer Df

Definition.— The space defined by the triangle abc is the linear figure defined by the linear figure defined by the three points a, b, c. Its symbol is UH(abc). The definition in symbols is

UR(abc) = u'lnH'uclnB'(i'a u t'b u t'c) Df

Definition. — The class of planes is the class of spaces defined by any three points a, b, c when they form a triangle. Its symbol is pleR. The definition in symbols is

pleR = v{(^a, b, e) . AR'(abc) . v = UR(abc)} Df

Definition. — The space defined by the four points a, b, c, d is the linear figure defined by the figure which is the logical sum of UR(bcd) and UR(acd) and UR(abd) and HR(abc). Its symbol is UR(abcd). The symbolic definition is

TIR(abcd) = u'lnn' {UR(bcd) u HR(acd) u HK(abd) u TlR(abc)} Df

The above definitions are sufficient to exhibit the dependence of the various geometrical entities on the essential relation, and also to enable us, as far as geometry is concerned, to pass on to the third stage. Owing to the simplicity of the definitions, the second stage for this concept is of very small importance.

It will be noticed that none of the definitions contain any reference to length, distance, area, or volume. This is because none of these ideas appear in the axioms, and only such definitions are given here as are necessary for the enunciation of the axioms. According to the well-known* method of projective metrics, the ideas are introduced by definition and require no special axiom.

* Cf. VEBLEN, loc. at. ; also ' Vorlesungen iiber Geometric,' by CLEBSCH, third part ; also ' The Principles of Mathematics,' by RUSSELL, chap. XLVIII.

478 DR. A. N. WHITEHEAD ON

Axioms of Concept I.

The axioms, it must be remembered, are merely an enumeration of various propositions concerning the properties of the fundamental relations, which will occur as hypotheses in the propositions of the fourth stage. In this instance we are merely considering the axioms of geometry, and these concern the essential relation (R) only. The axioms will be named systematically thus, I Up R, II Hp R, III Hp R, and so on. Their enumeration will take the form of defining these names as abbreviations standing for the various statements, which will be used subsequently as hypotheses.

I Hp R is the statement that there is at least one set of entities, a, b, c, such that R;(«fec) is true. The definition in symbols is

. = .a!R;(;;;) Df

II Hp R is the statement that R;(«fcc) implies R;(c&«). The definition in symbols is

II Up R . = : (a, b, c) : R(abc) . 3 . R:(c&«) Df

III Hp R is the statement that R:(«6c) is inconsistent with R;(&ra). The definition

in symbols is

IIIHpR . = : (a, 6, c) : R'(afcc) . 3 . — R'(&ca) Df

IVHpR is the statement that R;(«6c) implies that « is distinct from c. The definition in symbols is

IV Hp K . = : (a, b, c) : R;(a6c) . 3 . a y* c Df

V Hp R is the statement that, if « and b are distinct points, the segmental prolongation of ab beyond b possesses at least one member. The definition in symbols is

VH^R. = :(a, &):»,& eR'(;;;). a* 6.3.3! {R'(o&;)} Df

VI Hp R is the statement that, if c and d are distinct points, possessed by the line defined by the points a and b, then a is possessed by the line defined by c and d. The definition in symbols is

VIHpR. = : («, b, c, d): c, d tftab . c * d . 3 . aeR'ceT Df

VII Hp R is the statement that there exist at least three points forming a triangle. The definition in symbols is

VII Hp R . = . (g a, b, c) . A K-(abc) Df

VIII Hp R is the statement that, if a, b, c be three points forming a triangle, and R'(&cd) and R;(cea) hold, then there exists a point possessed both by the segment ab, and by the line defined by d and e. The definition in symbols is

VIII HpR . = :<«, 6, c, d, e): AH:(afec) . ft(bcd) . R:(cea) . = . a!{R;denR!(a;fe)} Df

MATHEMATICAL CONCEPTS OF THE MATERIAL WOULD. 479

IXHpR is the statement that there exists a point and a plane, such that the plane does not possess the point. The definition in symbols is

IXHpR. = :(ap, d) . pep\eR . dift(;;i)-p Df

X H^> R is the statement that there exist four points a, b, c, d, such that the three- dimensional space UR(abcd) contains the whole class of points. The definition in symbols is,

XHpR. =.(aa,6, c, d). ~R-(;;;) cUH(abcd) Df

XI Hp R is some statement which secures the continuity (in CANTOR'S sense) of the points on a line. The axiom need not be given here, since there will be no reasoning in this memoir connected with it.

XII Up R is the statement that, if a be any plane and a a line contained in it, then there exists a point c in a, such that there is not more than one line, possessing c and contained in the plane a and not intersecting a. The definition in symbols is

XII Hp R . =.'.«.€ pleH . a e linK n cls'a . DO a : (g;c) : c e a :

I, I' e linK n cls'a . c e I n V . I n a = A . V n « = A . =>,, v . I = I' Df

Of these axioms, IX Hp R secures that space is of three dimensions at least, and X Hp R secures that it is of three dimensions at most, and XII Hp R is the "Euclidean" axiom. From these twelve axioms the whole of geometry* can be deduced. The well-known parabolic (i.e., Euclidean) definition of distance (not given here) assumes an important significance, and all the usual metrical properties follow.

The Extraneous Relations. — Nothing could be more beautiful than the above issue of the classical concept, if only we limit ourselves to the consideration of an unchanging world of space. Unfortunately, it is a changing world to which the complete concept must apply, and the intrusion at this stage into the classical concept of the necessity of providing for change can only spoil a harmonious and complete whole. Owing to the fact that the instants of time are not members of the field of the essential relation, the time relation and the essential relation have (so to speak) no point of contact. To remedy this, another subdivision of the class of objective reals is conceived, namely, the class of particles (where the particles are the ultimate entities composing the fundamental " stuff" which moves in space). These particles must form part of the fields of a class of extraneous relations. Each such extraneous relation is conceived as a triadic relation, which in any particular instance holds between a particle, a point of space, and an instant of time. Also the field of each such extraneous relation only possesses one particle, and no particle belongs to the field of two such relations. Thus each extraneous relation is peculiar to one particle. Also, as has been pointed out by RUSSELL, t to whom the above analysis of these extraneous relations of the classical concept is in substance due, the impenetrability

* Of. VEBLEN, loc. at.

t Of. 'Principles of Mathematics,' vol. I., § 440.

480 DE. A. N. WHITEHEAD ON

of matter is secured by the axiom that two different extraneous relations cannot both relate the same instant of time to the same point. The general laws of dynamics, and all the special physical laws, are axioms concerning the properties of this class of extraneous relations.

Thus the classical concept is not only dualistic, but has to admit a class of as many extraneous relations as there are members of the class of particles.

Instead of the specific relations of occupation for the various particles, one general triadic relation of occupation can be considered. Thus, O'(pAt) may be considered as the statement the particle p occupies the point A at the instant t. Then for any given A and t there is either one only or no particle p for which O'(pAt) is true. Then the laws of physics are the properties of this single extraneous relation O. But the use of this single relation apparently introduces no real simplification, differing in this respect from the use of the essential relation which so simplifies the statement of the axioms of geometry. The general relation 0 remains a mere alter- native statement of the facts respecting the various specific relations of occupation.

Concept II. — This concept is a monistic variant of the classical concept suggested by RUSSELL.* In the classical concept the particles only occur as terms in the triadic extraneous relations. If we abolish the particles (in the " classical " sense), and transform the extraneous relations into dyadic relations between points of space arid instants of time, everything will proceed exactly as in the classical concept. The reason for the original introduction of " matter" was, without doubt, to give the senses something to perceive. If a relation can be perceived, this Concept II. has every advantage over the classical concept. Otherwise the material world, as thus conceived, would appear to labour under the defect that it can never be perceived. But this is a philosophic question with which we have no concern.

Concept III. — This is a Leibnizian concept, and also a monistic variant of the classical concept, obtained by abandoning the prejudice against points moving.

This concept can be otherwise considered, as obtained from the modern (and Cartesian) point of view of the ether, as filling all space. The particles of ether (or moving points) compose the whole class of objective reals. The essential relation (R) is a tetradic relation, and, in each specific instance of the relation holding, three of the terms are objective reals and the remaining term is an instant of time. R;(a&c£) may be read as stating the objective, reals a, b, c are in the R-order abc at the instant t. Instead of ~R'(abct), it will be convenient to write R(;(«6c). Then the geometrical definitions are exactly those of Concept I., replacing R everywhere by R«. Also the geometrical axioms are those of Concept I. ; except (i) that R is replaced by R(, (ii) that IHpR, and VII HpR, and IXHpR, are further modified by the introduction of the hypothesis t e T — thus I Hp R of Concept I. becomes

;;:) Df

Cf. ' Principles of Mathematics,' vol. I., § 441.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 481

and similarly for the other two axioms, and (iii) that one additional axiom (the axiom of persistence) must be introduced, namely,

XIIIHpR. = :«eT.D(.R;(;;r)cR;(;;;0 Df

This axiom of persistence is unnecessary for the geometrical reasoning, but is an integral part of the " physical" side of the concept. Also the hypothesis £eT, which is introduced in the three axioms (I, VII, IX) Hp R, is unnecessary in the other axioms, since it is implied by the hypotheses already existing. The same explanation holds of the absence of the hypothesis, t e T, from many axioms and propositions of subsequent concepts.

Thus at each instant the objective reals may be considered as the points of the classical concept, and the whole of Euclidean geometry holds concerning them. But at another instant the points will not have preserved the same geometrical relations as held between them at the previous instant. Thus, in the comparison of the states of the objective reals at different instants, the objective reals assume the character of particles.

Tke Extraneous Relation. — A single extraneous relation is necessary to obviate the difficulty of comparing straight lines and planes at one instant with similar entities at another instant. In what sense can a point at one instant be said to have the same position as a point at another instant ? This definition can be effected by introducing into the concept a single tetradic extraneous relation S, so that, when S'(uvwt) holds, t is an instant of time, and u, v, w are intersecting straight lines mutually at right angles. Also corresponding to any instant t in the fourth term, there is one and only one line for each of the other terms respectively. This last condition, expressed in symbols, is

<£T.D(.S;(;--£)el • S;( •;-t)fl . &(--;t) el

The straight lines indicated at each instant by this relation are to be taken as the " kinetic axes."* Velocity and acceleration can now be defined, and a general continuity of motion (in some sense) must be included among the axioms.

This concept has the advantage over Concepts I. and II. that it has reduced the class of extraneous relations to one member only, in the place of the innumerable and perhaps infinite number of extraneous relations in the other two concepts. The concept pledges itself to explain the physical world by the aid of motion only. It was indeed a dictum with some eminent physicists of the nineteenth century that " motion is of the essence of matter." But this concept takes them rather sharply at their word. There is absolutely nothing to distinguish one part of the objective reals from another part except differences of motion. The " corpuscle" will be a volume in which some peculiarity of the motion of the objective reals exists and persists. Two

* Cf. W. H. MACAULAY, ' Bulletin of the Amer. Math. Soc.,' 1897. VOL. CCV. — A. 3 Q

482 DR- A. N. WHITEHEAD ON

different developments, viz., Concept III A. and Concept Hie., are now possible, according as the persistence is taken to be of one or other of two possible types.

Concept IIlA. — Here the persistence is that of the same objective reals in the same special type of motion. KELVIN'S vortex ring theory of matter can be adapted to such a concept.

Concept IIIn.— Here the persistence is that of the type of motion in some volume, but not necessarily of the identity of the objective reals in the volume. The continuity of motion of a corpuscle as a whole becomes then the definition of the identity of a corpuscle at one instant with a corpuscle at another instant.

PART III. — (i) GENERAL EXPLANATIONS OF LINEAR CONCEPTS.

These concepts depart widely from the classical concept. The objective reals (at least those which, with the instants of time, form the field of the essential relation) have properties which we associate with straight lines, considered throughout their whole extent as single indivisible entities. These objective reals, which in Concept V. are all the objective reals, will be called linear objective reals. Perhaps, however, a closer specification of the linear objective reals of these concepts is to say that they are the lines of force of the modern physicist, here taken to be ultimate unanalysable entities which compose the material universe, and that geometry is the study of a certain limited set of their properties. But this mode of realizing the nature of the linear objective reals has also its pitfalls, for a line of force suggests ends, while these linear objective reals have no properties analogous to the properties of the ends of lines of force. The whole of a straight line, viewed as a point-locus, will be found to be associated with a linear objective real. The " linear " concepts here considered are all Leibnizian.

Concept IVA. is dualistic, and requires among the objective reals a class of "particles" in addition to the linear objective reals. Concept IVB. is the monistic variant of Concept IVA., obtained exactly as Concept II. is derived from Concept I. Both of the Concepts IVA. and IVfi. labour under the same defect as Concepts I. and II. in requiring an indefinitely large class of extraneous relations. Concept V. is monistic, and is by far the most interesting of the set of linear concepts. It requires only one extraneous relation to perform a similar office to that of the extraneous relation in Concept III.

Points are now defined complex entities, being certain classes of linear objective reals. Geometers are already used to the idea of the point as complex. In projective geometry, as derived from descriptive geometry, the projective point is nothing but a class of straight lines.* This idea will now be extended to all

* Cf. PASCH, loc. cit., and SCHUR, " Ueber die Einfiihrung der sogennanten idealen Elemente in die projective Geometrie," 'Math. Annal.,' vol. XXXIX., and BONOLA, "Sulla Introduzione degli Enti improprii in Geometria projettiva," ' Giorn. di Mat.,' vol. XXXVIII.

MATHEMATICAL CONCEPTS OF THE MATEKIAL WOULD. 483

points ; and the descriptive point, from which in the current theory the projective point is ultimately derived, is here abolished. The "Theory of Interpoints" [cf. Part III. (ii)] and the "Theory of Dimensions" [cf. Part IV. (i)] represent two distinct methods of overcoming the following initial and obvious difficulty of these "linear" concepts: — A point is to be defined as the class of objective reals " concurrent" at a point. But this definition is circular. How can this circularity be removed ? The Theory of Interpoints and the Theory of Dimensions give two separate answers to this question. The points in the linear concepts, being only classes of objective reals, are capable of disintegration. In fact, when motion is considered, it will be found that the points of one instant are, in general, different from the points of another instant, not in the sense of Concept III. that they are the same entities with different relations, but in the sense that they are different entities. More difficulty will probably be felt in conceiving anything analogous to a line as a simple unity. Here it is to be observed that a linear objective real does not replace a line of points of ordinary geometry. On the contrary, the class of those points (here called a, punctual line), which have a given linear objective real as a common member, is this ordinary geometrical line. A punctual line has parts and segments in the ordinary way. The idea of a single unity underlying a straight line is not wholly alien to ordinary language. The idea of a direction, as it could also be used in non-Euclidian geometries where each line will have its own peculiar direction, may be conceived as being that of a line taken as a unit. But it is unnecessary to elaborate these considerations, as they have no relation to the logic of the subject.

In the dualistic Concept IVA. the particles form another class of objective reals in addition to the linear objective reals. Each particle is associated at each instant with some one point, that is, with some class of linear objective reals. Thus the two points, respectively associated at any instant with two particles, have in common one linear objective real. Thus, when mutually determined motions are considered, these linear objective reals assume the aspect of lines of force. In the monistic Concept V. the analogy of objective reals to lines of force arises in a similar way. In this case particles, in the sense used above, do not exist. Corpuscles, to use another term, are defined entities, analogous to the corpuscles of Concept III. ; any general consideration of them is best deferred till the definitions can be understood.

In Concepts IV. and V. the conception of an ether is (in a sense) rendered unnecessary, or (in another sense) is largely modified. The collection of linear objective reals (i.e., in Concepts IVs. and V., of all objective reals) now forms the entity (the ether) which " lies between " the corpuscles of gross matter. These corpuscles must be conceived as volumes with some peculiarity either of motion or of structure. Of course it might be found useful, for the explanation of physical phenomena, to assume that corpuscles of some sort are generally distributed between bodies of gross matter, thus forming an ether in a secondary sense. The ancient controversy concerning action at a distance becomes irrelevant in these concepts. In

3 Q 2

484 DR A. N. WHITEHEAD ON

one sense there is something, not mere space, between two distant -corpuscles, namely, the objective reals possessed in common ; in another sense there is a direct action between two distant corpuscles not depending on intervening corpuscles. In fact, the premises common to both bands of disputants are swept away.

The Essential Relation.— In both of the Concepts IV. and V. the essential relation (R) is a pentadic relation, and has for its field both the class of instants of time and that of linear objective reals, that is, in Concept V. the field is the complete class of ultimate existents. The proposition ~R-(abcdt) can be read as the statement that the objective real a intersects the objective reals b, c, d in the order bed at the instant t. This conception of " the intersection in order of three linear objective reals by a fourth at an instant of time " must be taken as a fundamental relation between the five entities. But the properties of the relation are not to be limited by the suggestion of the technical name " intersection." The axioms will be so assumed that ~R'-(abcdt) implies that a, b, c, and d are distinct. Also, when points are defined, it will be found that the axioms secure that a intersects b, c, and d in distinct points. Furthermore, in general, b, c, and d are not co-punctual ; so that the case when a is a transversal of the pencil b, c, d of co-punctual lines is only a particular case of the satisfaction of T\,'(abcdt).

Definitions. — The notation of the general symbolism provides us with the symbol E/(;;;;-) for the class of linear objective reals, and with R;( — ;) for the class of instants. But these symbols are long. Accordingly 0 will be defined to stand for the class of linear objective reals, and T for the class of instants. Thus, in symbols,

0 = B/(;;;;-) Df

T = R;(----;) Df

When "particles" (in Concept IV.) are not being directly considered, the term " objective real " will be used instead of " linear objective real," or " member of 0."

(ii) THE THEORY OF INTERPOINTS.*

*1. The theory of intersection-points (shortened into interpoints) is required in both of the Concepts IV. and V. Accordingly, it is convenient to investigate it before the special consideration of either concept. In Concept IV. the interpoints are the points, and there are no other points. In Concept V. the interpoints are, in general, only portions of points, and a point may contain no interpoint or many interpoints. Thus

* From this point a continuous argument commences, and the sections and included propositions are numbered by a combined integral and decimal system, the whole number for the section and the decimal part for the proposition, also the symbol (*) is placed before an integral number marking a section. All the easier proofs of propositions are omitted, those proofs remaining being retained either as specimens, or as containing some point of difficulty. The omitted proofs are often replaced by references to the preceding propositions used in them, as a guide to their reconstruction. Note that "cf. *2'31 -41 '5 " is used as a shortened form of "r/ *2'31 and *2'41 and *2'5."

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 485

the axioms of Concept IV. (cf. *2) and those of Concept V. (cf. *22) are two alternative sets of hypotheses as to the properties of R in connection with which the theory of interpoints, as given in the present *1, assumes importance. Some axioms, involving interpoints in their statements, are identical in Concept IV. and Concept V. These axioms are stated now in *1, and their simple consequences are deduced. The theory of interpoints depends on that of " similarity of position " in a relation. This general idea will only be explained in the special form in which it is here required in respect to the essential relation R.

*ril. Definition. — An entity, y, will be said to have a position in the pentadic relation R, similar to that of the entity x, with a as first term and t as last (fifth) term, if, whenever the relation holds between five terms, a being the first term and t the last term, and either x or y or both occurring among the other terms, the relation also holds when x is substituted for y (whenever y occurs), and also holds when y is

/a??? A substituted for x (whenever x occurs). The symbol R;( " ' j denotes the class of

entities with positions similar to that of x in the relation R, a being first term and t last term. The definition in symbols is

. = . R1 (a&yt)} Df

/a??? A /a??? A

*1'12. Proposition. — If y is a member of RM ' \' j, then R;f ' ' ' j is identical

/a??? A with R;( "x I In symbols,

,

\ / \ v /

fa ? ? ? A *1'13. Proposition. — x is a member of R'(

V x

*T21. Definition. — A class P of objective reals is called an intersection-point on a (shortened into interpoint on a), when there exists an objective real x, which is a member of R: (a ;;;t), and P is the class whose members are a together with all the

members of the class R;( ' ' V The symbol R; (a???«) stands for the class of inter -

\ x j

points on a at the instant t. The definition in symbols is

R'a???* = P x . xeR;a;;;0 . P = i'auB»~ Df

*1'22. Definition. — P is called an interpoint of the relation R at the instant t, if there exists an objective real a, such that P is a member of R;(a ???«). The symbol intpntR( stands for the class of interpoints o/R at the instant t. In symbols,

intputB( = P{(3«) . PcR:(a???OI Df

486 OR A. N. WHITEHEAD ON

*T23. Proposition. — If P and Q are distinct members of R!(a???£), then a is the sole member common to P and Q. In symbols,

h : P, Q €»'(«???«) .P^Q.3.i'a = PnQ

proof—.Cf. *1'1 1-12-21.

*1'31. Definition. — The interpoints B, C, D will be said to be in the interpoint- order BCD at the instant t with respect to the relation R, when there exist objective reals a, x, y, z, such that (1) B, C, D are members of R! («???£); (2) a; is a member of B, y of C, z of D; (3) U'(axyzt) holds. The symbol Rin;(BCD«) stands for the statement that the interpoints B, C, D are in the interpoint-order BCD at the instant t with respect to the relation R. In symbols,

Rin-(BCD«) . = . (fta, x, y,z) . B, C, DeR'(a???0 .xeB.yeC.2eD. ll^axyzt) Df

*l-32. |-.intpntK( = Rin:(;;;0

Proof. — The class Rj,,^;;;^) is part (or all) of the class intpntH( (cf. *l'3l). Again (cf. *1'22), if B is a member of intpntR(, objective reals a and x exist, such that x is a member of R: («;;;<), and B is the iuterpoint possessing a and x. Hence there are objective reals y and z, such that either ~Rl(axyzt) or R,'(ayxzt) or ^(ayzxt). Hence (cf. *1'31), there are interpoints C and D, such that either Rin;(BCD«) or R;(CBD«) or R' (CDBi). Hence B is a member of Rjn: ( ; ; ; t ).

The theory of interpoints has its chief interest when the following axiom is satisfied. It is named intpnt Hp R.

*1'41. Intpnt Hp R is the statement that if A be an interpoint at the instant t, and a be any member of A., then A is a member o/"R: (a ???<). In symbols,

Intpnt Hp R . = : Aeintpntm . aeA . D,(,A,t . AeR;(a???£) Df

*T42. Proposition. — Assuming intpnt HpR, then if A and B are distinct members of intpntHj, A and B have either no members in common or one only. In symbols,

I" .'. intpnt Hp R . D : A, B e intpntR( .A^B.D.AnBeOul.

Proof— Cf. * 1-23 -41.

The interest of the relation of interpoint-order (Rhl) arises when the relation R satisfies four axioms specifying the idea that ~R-(abcdt) expresses that a intersects b, c, d in the order bed. These axioms (together with intpnt Hp R) will be employed both in Concept IV. and in Concept V. They will be named aHpR, £H|?R, yHpR, SHpR.

*1-51. allpR is the statement that a is not a member of R;(a;;;£). In symbols,

«HpR. = . (a, t) .a^e~R<(a;;;t) Df

*1'52. ^HpR is the statement that R'(abcdt) implies ^'-(adcbt). In symbols,

R . = : (a, b, c, d, t) : ~R''(abcdt) . o . R(adcbt) Df

MATHEMATICAL CONCEPTS otf THE MATERIAL WORLD. 437

*1'53. yHpR is the statement that T&-(abcdt) and R-(acdbt) are inconsistent. In

symbols,

y Hp R . = :(«, b, c, d, t) : U-(abcdt) . D . ^ E'(acdbt) Df

*1'54. 8 Hp R is the statement that R;(«6ccfa) implies that b and d are distinct.

In symbols,

8 Hp R . = : (a, b, c, d, t) : H'(abcdt) .o.b^d Df

*1'61. Proposition. — Assuming (a, ft, y, 8) Hp R, then R,-(abcdt) implies that a, b, c, d are all distinct. In symbols,

t" . '. (a, ft, y, 8) Hp R . D : ~R-(abcdt) .^.a^b.a^c.a^d.b^c.b^d.c^d

*T62. Proposition. — Assuming (a, ft, y, 8) Up R, then Rin;(BCD<) implies that B, C, D are all distinct. In symbols,

\- . : (a, ft, y, 8) Up R . D : Rin;(BCD«) .D.B^C.B^D.C^D

Proof. — By definition (cf. *l'3l) RIn:(BCD() implies that a, x, y, z exist such that B, C, D are members of R;(a???£), a; is a member of.B, y of C, z of D, and K,'(axyzt). Hence (cf. *1'61) a, x, y, z are distinct. Now if any two of B, C, D are identical, e.g., B and C, then x and y are both members of B. Hence (cf. *1 -12-21), since a is distinct from x and y, x can be substituted for y in Hi'(axyzt). Hence ~R;(axxzt), which contradicts *1'61.

*1'63. Proposition. — Assuming ftEpR, then Rin:(BCDO implies Rin;(DCBf). In

symbols,

I- /. ft Up R . D : Rin'(BCD<) . 3 . Rin'(DCB<)

Proof.— Cf. *1-31'52.

*l-64. Proposition.— Assuming (a, ft, y, 8) Hp R, then RIn;(BCD«) and Bln;(CDB«) are inconsistent. In symbols,

I- .-. («, ft, y, 8) Hp R . D : BIn'(BCD«) . D . - Rin;(CDBO

Proof. — Rin;(BCD£) implies (cf. *1'31) that a, x, y, z exist such that a is a common member of B, C, D, x is a member of B, y of C, z of D, and ~R'-(axyzt), and B, C, D are members of R: («???«). Hence (cf. *1'61) a, x, y, z are all distinct. Similarly also if Rin;(CDBi), then a', *', yf, z' exist with similar properties, viz., x' a member of B, &c., except that T\,-(a'y'z'x't). Hence (cf. *r23'62) a and a' are identical. Thus R:(axyzt) and ^(ai/z'x't). But (cf. *1'21) x can be substituted for x', y for yf, and z for z'. Hence R-(axyzt) and ~R'(ayzxt). But this contradicts yHpR.

*1'65. Proposition. — Assuming (intpnt, a, ft, y, 8)HpR, the classes R;(;-"«) and R:( ;;;;£) are identical. In symbols,

|- : (intpnt, a, ft, y, 8) Hp R . D . R!( ;•• • •*) = R:( ; ; ; ;«)

488 DR. A. N. WHITEHEAD ON

Proof. — If a; is a member of R;( •;;;t), y exists such that a? is a member of R;(2/;;;<) ; also (cf. *P61) x and y are distinct. Hence (cf. *1'21) P exists such that it is a member of R;(?/???£), and a; is a member of it. Hence (cf. *1'41) P is a member of E: («???<), and hence (cf. *l'2l) y is a member of R: (x ;;;t). Hence x is a member of E'( ;*"*).

*171. Proposition. — Assuming aHpR, every interpoint possesses at least two members. In symbols,

(-.-. «HpR. => : A e intpntR( . D . Nc'A^ 2

Proof.— Cf. *1-13-21-22'51.

*172. Proposition. — Assuming (intpnt, a, /3, y, S) Up R, then on every objective real there exist at least three interpoints. In symbols,

|- .'. (intpnt, a, ft, y, 8) Rpll . D : a eR;( ;;;;<) . D . Nc'R!(a???«) ^ 3

Proof.— Cf. *l-2r31'62-65.

*173. Proposition. — Assuming (intpnt, a, /?, y, 8) Hp R, then, if there are any objective reals, the interpoints are not all on any one objective real. In symbols,

. I".', (intpnt, «, J3, y, 8) Hp R . a : a!R;(;;;;i) . a . 3! {intpntH, - R;(a???*)} Proof.— Cf. *1 -4271 72.

(iii) CONCEPT IV.

*2. This concept bifurcates into two alternate forms, namely IVA. and IVB. Concept IVs. is related to IVA. just as Concept II. is related to the classical concept. Thus Concept IVA. is dualistic, and Concept IVs. is the monistic variant of it. Both concepts can initially be considered together as Concept IV. In Concept IV. the essential relation (R) is pentadic, one of the terms being an instant of time. ~R-(abcdt) can be read as a intersects b, c and d, in the order bed at the instant t. The class of those entities, appearing among the first four terms in any instance of the relation holding, is called the class (0) of " linear objective reals." The remaining class of objective reals, required for Concept IVA., is called the class of " particles."

The geometrical points of this concept are simply the interpoints of R, as defined above (cf. *l). During the consideration of this concept they will be called points. The further definitions, beyond those of *1, required for a concise statement of the geometrical axioms are almost exactly those of Concept I., with the Rjn of this Concept IV. written for the R of Concept I., and modified by the mention oft, as in Concept III. This mention of t can be managed in a similar (though not identical) way to that in Concept III. by writing

*2'0l. R<-(ABC = Rln=ABC Df

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 489

Then the definitions of Concept I. will be assumed to apply to K<. For example, the punctual line joining the points A and B is the class of points which is the logical sum of Rt;(;AB) and R,;(A;B) and Rt;(AB;) together with A and B themselves. Its symbol is R(AB. The definition in symbols is

KvAB = By ( ; AB) u BY ( A ; B) u BY ( AB ; ) u i' A u I'B Df

It will follow (cf. *1'23'31) from the axioms that a punctual line is the class of those points with some member of 0 as sole common member. The other definitions can be managed in like manner, only in the symbolism a suffix to a suffix will be avoided by writing AR(;(ABC), and so on, instead of AR;(ABC), and so on.

The Axioms. — The earlier axioms have to be modified from those of Concept I., but the later axioms are simply those of Concept I. with the R of that concept replaced by the R( of Concept IV.

IHpE. = :*eT.Dt.OcR;(;;;;0 Df

IIHpE. = .a!R:( — 0 [Y.e., 3 !T] Df

IIIHpR. = .aHpR Df (cf. *1-51)

IVHpR. = .£HpR Df (c/*l-52)

VHpR. = .yHpR Df (cf. *l-53)

VIHpR.=.SHpR Df (cf. *1'54)

VII Hp R . = . intpnt Hp R Df (cf. *1'41)

VIII Hp E. = .-. (A, B, C).-. A, B, CeR,-(;;;). A^B. A^C.B^C.

3 ! ( A n B n C) . D : R(; ( ABC) . V . R<:(BCA) . V . IV (CAB) Df IXHpR.= :(A, B): A, BeE,'(;;;). A ^ B . D . gr !R;(AB;) Df

XHpR . = : (A, B, C, D, E) : A,«;(ABC) . Re;(BCD) . R(;(CEA) . => .

a!{R('DEnRe;(A;B)} Df

XIHpR . = : JtT .=H . ta D) .^epleR( . DeE,'(;;;) -p Df

XII Hp R . = . (3 A, B, C, D) . Rt;( ; ; ; ) c Uut (ABCD) Df

XIII Hp R . = . the axiom of continuity, cf. XII Hp R of Concept I. Df

XIV Hp R . =.'.«£ pleRi . a e linKi n cls'a . Da> . : (gC) : C e a :

I, V e linR( n cls'a . C e I n V . I n a = A . I' n a = A . ot.v • I = V

Note that only I Hp R and XI Hp R require the hypothesis t e T ; in all the other axioms there is a hypothesis which can only be true when t e T. For the purpose of comparison with the axioms of Concept I., the following propositions are required :—

VOL. ccv. — A. 3 R

490 DE. A. N. WHITEHEAD ON

*2-ll. I-.'. (Ill, IV, V, VI, VII) Hp-R . D : t £T . 3 . Nc<Rt>(;;;) 5 2 Proof.— Cf. *172.

*2-21. I" .-. IV Hp R . 3 : R(:(ABC) . o . BV(CBA) Proof— Cf. *1'63.

*2-22. |- .-. (Ill, IV, V, VI) HpR . D : R(;(ABC) . D . - R,:(BCA) Proof— Cf. *1'G4.

*2-23. |- .-. (Ill, IV, V, VI) Hj?R . D : BY (ABC) . D . A * C Proof— Cf.*V&2.

*2'31. Proposition. — Assuming (VII, IX)HpR, if A and B are two distinct points at the time t, they possess one, and only one, common member. In symbols,

i- .-. (VII, IX)HpR . D : A, BeR(;(;;;) . A ^ B . D . AnBel

Proof— Cf. *1-31'42 and (VII, IX)HpR.

*2'32. Proposition. — Assuming (VII, VIII, IX) Hp R, a line at any instant t (i.e., a member of linH() is the complete class of points (Intel-points) possessing some linear objective real. In symbols,

L:(VII,Vin,IX)HpR.D.linlu=^[(aa).«eR;(;;;;0.p = A {AeR,'(;;;) . ae A}]

Proof.— Cf. VIII Hp R and *2'31.

Propositions *2'31'32 effect the identification of the punctual line, as defined above, and the class of points on some linear objective real. Thus a straight line considered as an entity with parts is a punctual line, and considered as a simple unit is a linear objective real.

*2'33. Proposition. — Assuming (VII, VIII, IX) HpR, if C and D are two points in the punctual line R(;AB, then A is a point in the punctual line R(;CD. In symbols,

t- .-. (VII, VIII, IX) Hp R . 3 : C, D e R(:AB . C ^ D . D . A eR,!CD

Proof— Cf. *2'32.

*2-41. I- .-. (Ill -IX) Hp R . 3 : * e T . 3 . (a A, B, C) . A ,i(;(ABC)

Proof— Cf. *17273.*2-32.

*2'5. Proposition. — Assuming (III-XIV) Hp R of Concept IV., then all the axioms of Concept I. hold when the Rt of Concept IV. is substituted for the R of Concept I., and t is a member of T.

Proof.— Cf *2-ll-21-22-23'33-41 and (IX-XIV)HpR (of Concept IV.) and (I-XII) Hp R of Concept I.

It will be noticed that IHpR (of Concept IV.) is not required in the above

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 491

comparison. It does not belong to the purely geometrical side of the concept, but is a necessary part of the " physical" ideas. IIHpE, (of Concept IV.), though it does not occur explicitly in the above comparison, is required to give the geometry " existence." Thus the geometry of Concept IV. requires thirteen axioms.

For the purpose of the transition to projective geometry (cf. VEBLEN, loc. cit.), it is now unnecessary to conceive a new class of " projective points." The points already on hand are exactly the entities required. All that is necessary is to define the class of those linear objective reals (cf. XIV Hp R), coplanar with any given linear objective real and not intersecting it, as the point at infinity on that objective real. Then with these new points at infinity, and the old points, the complete set of " projective points" is obtained.

The Extraneous Relation. — For the purpose of the definition of motion, one extraneous tetradic relation is required, exactly as in Concept III. Also the same hypotheses must hold respecting it. The three mutually rectangular and intersecting punctual lines, thus indicated at each instant, are to be taken as the " kinetic axes," and all motion measured by reference to them. A given set of kinetic axes does not, in general, correspond to the same three linear objective reals at different instants of time.

Matter. — It is necessary to assume that the points in this concept disintegrate, and do not, in general, persist from instant to instant. For otherwise the only continuous motion possible would be representable by linear transformations of coordinates ; and it seems unlikely that sense-perceptions could be explained by such a restricted type of motions. We have therefore to consider what, in this concept, can represent the permanence of matter. A " corpuscle," as we may call it, may be conceived to be a volume with some special property in respect to the linear objective reals " passing through " it. This is the procedure adopted in Concept V. ; and the methods of overcoming the obvious difficulties which suggest themselves will be considered in detail there. It is sufficient here to notice that, in this Concept IV., the special property of the volume must relate merely to the motion of the objective reals. For the only alternative is to make the property consist of the permanence of the points within the volume. But then the difficulty of permanent coll ineat ions, mentioned above, recurs. To find a special property of motion, we require a kinematical science for linear objective reals in this concept analogous to the kinematical parts of hydrodynamics. In the absence at the present time of such a science, we proceed to other alternatives.

Concept IVA. — Conceive a class of particles, each particle being associated at each instant with some point, but not necessarily each point with some particle. Then the particles represent the " matter" which " occupies" space. Laws of motion must then be stated (i) for the particles and (ii) for the linear objective reals. Also the motion of the particles may be conceived to be influenced by that of the linear objective reals, and vice versa. The endeavour to state such laws appears to reduce

3 R 2

492 DR. A. N. WHITEHEAD ON

itself to rewriting with appropriate changes a chapter of any modern treatise of electricity and magnetism. It would seem necessary to subdivide the class of particles into " positive " and " negative " particles, a charged volume containing an excess of one type. The conception of an ether conveying lines of force is replaced by the class of the linear objective reals. The details can be managed much as in the analogous case of Concept V., considered later. An indefinite number of extraneous relations are required to " locate " the particles, just as in Concept I. This concept (as thus developed with "particles") is not completely a "linear" concept. It is a hybrid between the "linear" and "punctual" concepts. In its dualism it is not superior to the classical concept. But, in possessing moving linear objective reals as well as moving particles, it is richer in physical ideas.

Concept TVs. — In this concept, just as in Concept II., each triadic extraneous relation of Concept IVA. between an instant of time, a particle, and a point is replaced by a dyadic extraneous relation between a point and an instant of time.

PART IV. — (i) THE THEORY OF DIMENSIONS.

*3. Concept V. depends upon a treatment of the theory of dimensions different from -that which at present obtains. The theory here developed is relevant to any definite property which (1) is a property of classes only, and (2) is only a property of some classes. It will be clearer, and no longer, to explain the theory in its full generality, and in Concept V. to make the special application required.

This general theory of dimensions may, perhaps, have a range of importance greater than that which is assigned to it in the sequel. In *10 a set of hypotheses are given respecting the property ^> ; and when these are true of <£, the propositions and definitions of *3 to *8 acquire importance and emerge from triviality, also in this case further deductions of propositions can be made. The Concept V. to which this theory is applied is explained in the definitions of *20 and the axioms of *22. In this Concept V. a special property <j> is taken, which is termed "Homaloty" (cf. *20'11'12), and (cf. *22) in the axioms a relation R is considered such that " homaloty," defined in respect to R, has the properties of the axioms in *10.

*3'01. Definition. — If <^\x is some proposition involving the entity x, which may be varied, so that (j)lx and <j>ly make the same statement (<£) about x and y respectively, then any entity z, for which (f> I z is true, is said to possess the property <j).

*3'02. Definition. — A $-class is a class with the property <£, that is to say, if u is a <£-class, then <j)lu is true.

*3'11. Definition. — The ^-region is the logical sum of all classes which possess the property <£. The symbol O^ will denote the (^-region. The symbolic definition is

Df

*3'12. The common (f>-subregion for u is that class which is the common subclass

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 493

of all (^-classes with the class u as a subclass. The symbol cm^'u will denote the common <£-subregioii for u. The symbolic definition is

cm/ it = n'v{(f>\v . uecls'v} Df

Note. — If no class v, with the property <f> and containing u as a subclass, exists, then cm/tt will be the class of all entities. But if a class v exists which has the property <j) and contains u, then cm^'u is a subclass of O^. In the sequel it will be found that this latter is the only relevant case for our purposes.

Elucidatory Note. — -Assuming our ordinary geometrical ideas, let the property of the " flatness" of a class of straight lines be defined thus : A class of straight lines is flat, either, when it is a necessary and sufficient condition for membership that a straight line meets two members of the class, not at their point of meeting, or, when the class is a unit class with one line as its sole member. Thus a plane (as a line-locus) is flat, a three-dimensional space (as a line-locus) is flat, and so on. Now let the property <j) in the above definition be the property of flatness. If then u is a class consisting only of two straight lines, the common (^-subregion for u is either a three-dimensional space or a plane, according as the two lines are not, or are, coplanar. Also in a space of higher dimensions than three, if u be a class consisting of three straight lines, the common <£-subregion for u may be either (l) a plane, or (2) a three- dimensional space, or (3) a four-dimensional space, or (4) a five-dimensional space, according to the circumstances of the lines. It will be noticed that, in the application of this theory of the common ^-subregion to the particular case of geometrical flatness, the common <£-subregion of any class of lines is itself flat. But this is not, in general, the case when any property not flatness is considered. It is this peculiar property of flatness which has masked the importance in geometry of the theory of common (^-subregions.

*3'121. Definition. — Two classes u and v have ^-equivalence if cm/w = cm/v. The class of those classes (including u itself as a member), which have (^-equivalence with u, is denoted by equiv/w. The symbolic definition is

equiv/ u = v (cm/ v = cm/ u) Df

*3'13. Definition. — A class u (not the null class) is fy-pvime, when, if v be any proper part (part, not the whole) of u, v is not (^-equivalent to u. The class of those classes which are ^>-prime will be denoted by the symbol prm^,. The symbolic definition is

prm^ = u{^\u :vcu . ^.(u- v) . r>0 . cm/v ^ cm/?«} Df

Elucidatory Note. — With the assumptions of the elucidatory note on cm/w, it is at once obvious that two straight lines form a ^-prime (where <j> is flatness) class, whether they are or are not coplanar. But if u consist of three straight lines, (1) u is not (£-prime if cm/M is a plane, (2) u is not, in general, ^-prime if cm/w is a space of three dimensions, but u is (in this case) ^>-prime if the three lines are concurrent,

494 DR- A- N- WHITEHEAD ON

(3) u is (£-prime if cm+'u is of four dimensions, (4) u is <£-prime if cm^'u is of five dimensions.

*3'21. The ^-dimension number (or the <f>-dimension») of a class u is the greatest of the cardinal numbers of ah1 classes (including possibly u itself) which are both ^-equivalent to u and <£-prime. The ^-dimension number of u will be denoted by dimwit. The symbolic definition is

dim/M . = : (ia) : a eNc"(prm^, n equiv/w) : p e Nc" (prrn^ n equiv/w) . DP . p ± a Df

Elucidatory Note. — With the assumptions of the previous elucidatory notes (where (f> is flatness), we see that those c^-prime classes, the common ^-subregions for which are spaces of three dimensions (as ordinarily understood), are all pairs of non-inter- secting lines and all trios of concurrent non-coplanar lines ; also no class of four lines in such a space can be prime. Thus three is the greatest cardinal number of any (£-prime class of lines for which the common </>-subregion is such a space. Hence, according to the above definition, three is the ^-dimension number of the space.

*3-22. Definition. — A class n is <j>-ctxial when (l) it is 0-prime and (2) its cardinal number is equal to its ^-dimensions. The class of ^-axial classes is denoted by the symbol ax^. The symbolic definition is

ax^, = u{ue prm^, n dim^' u} Df

Elucidatory Note. — With the assumptions of the previous elucidatory notes (where <f> is flatness), we see that two coplanar lines form a ^-axial class, and so also do three concurrent non-coplanar lines.

*3'23. Definition. — A class u is (^-maximal when (l) all those of its subclasses (possibly including u itself), which are both <^>-prime and (^-equivalent to u, are <£-axial, and (2) there are such subclasses. The class of ^-maximal classes will be denoted by mx^. The symbolic definition is

= u { 3 ! (prm^, n equiv^,' u n els ' u) . prm^, n equiv^,' u n els ' u c ax0 } Df

Elucidatory Note. — Referring to the previous elucidatory notes (where <f> is flatness), we see that any set of coplanar lines form a (^-maximal class ; similarly any set of concurrent lines form a (^-maximal class.

*3'31. Definition. — The ^-concurrence, of u and v, where u and v are classes, is that subclass of u (possibly u itself), such that any couple, formed by any member of it and any member of v, is <£-axial. The (^-concurrence of u with v is denoted by the symbol u^v. The definition in symbols is

uj v = x {x e u : y e v . uy . L'X u I'y e ax,,,} Df

The (^-concurrence of the (^-region (O^) with any class v will be written O^'v instead of 0$ ' v.

Elucidatory Note. — Referring to the previous elucidatory notes (when (f> is flatness),

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. :495

we see that, when u and v are classes of straight lines, u+v (i.e., the ^-concurrence of u with v) is that complete set of lines of u which is such that any member of it is coplanar with every member of v.

*3'32. Definition. — A class u is a self-<j>-concurrence if the ^-concurrence of u with itself is the whole class u. The class of those classes which are self-^>-concurrences will be denoted by conc^,. The symbolic definition is

conc^, = u {u = iifi'u} Df

*3'33. Definition. — x will be said to be ^-concurrent with y, if the class composed of a; and y only (i.e., the class I'x u <,'?/) is <£-axial.

*3'41. Definition. — A <f>-plane is a class u such that there exists a class v, which (1) is <£-axial, (2) is composed of two members only, and (3) is such that u is the class cm^v. The class of those classes which are <£-planes is denoted by pie,,,. The symbolic definition is

le, = u V .ve2na,x.u = cm'v Df

Note. — It requires an axiom to establish that a ^-plane is a self-^-concurrence (cf. *16'11).

*3'42. Definition. — A class u is a <j>-point, if there exists a class v, which (1) is <£-axial, (2) is composed of three members only, and (3) is such that u is the (^-con- currence of the (^-region with v. The class of those classes which are ^-points is denoted by the symbol pnt^. The symbolic definition is

. v e 3 n ax, . w = Qv Df

Note. — It requires axioms to establish that a <£-point is (^-maximal and is a self-<£- concurrence (cf. *14'11'12). Also note that this definition does not apply unless the number of dimensions of O^ is at least three, but then applies unchanged however great this number may be.

Elucidatory Note. — Referring to the previous elucidatory notes (where (j> is flatness), we see that a ^-point now becomes simply that class of straight lines concurrent at a point. The analogy with KLEIN'S " ideal," or " protective," points is obvious. Only when the present theory is applied, it will be found that the original " descriptive " point has entirely vanished.

*3'43. Definition. — A class is <j>-coplanar if there exists a <£-plane of which it is a subclass. The symbol cople^.u denotes that the class u is ^-coplanar. The definition

in symbols is

cople^,! u . = . (ftp) . p e ple^ . u e cls'j? Df

*3'44. Definition. — A class is <$>-copunctual if there exists a (£-point of which it is a subclass. The symbol copnt^,!w denotes that the class u is <£-copunctual. The definition in symbols is

copnt ! u . = . (3?) . P e pnt^, . u e cls'P Df

496 DR. A. N. WHITEHEAD ON

General Deductions Concerning Dimensions.

A large chapter of interesting propositions concerning the entities defined above in *3 can be compiled. The following are chosen as being directly wanted in the subsequent investigations : — •

*4. On Common ^-subregions.

*4'21. Proposition. — If v is a subclass of u, then cm/?; is a subclass of cm/ it. In

symbols,

h : v c u . D . cm/i; c cm/ u

Proof.— Of. *3'12.

*4'25. Proposition. — A class u is itself a subclass of cm/M. In symbols

I" . u c cm/ u Proof.— Cf. *3'12. *4-27. Proposition. — If u is a class with the property (f>, then u is identical with

cm/ if. In symbols,

1" : <f> ! u . D . u = cm/ u

Proof. — Cf. *3'12.

*4'28. Proposition. — If there exist two classes, both with the property <£, which possess no common member, then cm/ A is itself the null class (A). In symbols,

l~ : (g; u, v) . u n v = A . <£ ! u . <f) ! v . D . cm/ A = A

Proof. — Note that cm/ A is the common part of all (^-classes.

Corollary. — If x and y exist such that they are distinct, and the two unit classes with them as members respectively each have the property <£, then cm/ A is A.

Note that when *4-28 is appealed to, it will be this corollary which is directly used.

*4'31. Proposition. — The common c^-subregion for the common ^-subregion for u is the common <£-subregion for u. In symbols,

I" . cm/cm/w = cm/w

Proof. — For cm/M is contained in every <£-class containing u. Hence (of. *3'12) cm/cm/M is contained in cm/w. Also (cf. *4'25'2l) cm/w is contained in cm/ cm/ u.

*4'32. Proposition. — If u and v are (^-equivalent, and w is any class, then the common ^-subregion for the logical sum of u and w is identical with the common <£-subregion for the logical sum of v and w. In symbols,

I" : cm/tt = cm/v . => . cm/ (it u w) = cm/(v u w)

Proof. — For (cf. *4P21) cm/v is contained in cm/(i?uw), and hence (hypothesis) cm/w is contained in cm/(vuw), and hence (c/ *4'25) MUW is contained in cm/(vui0), and hence (c/ *4'21) cm/(wuw) is contained in cm/ cm/ (v u w), and

MATHEMATICAL CONCEPTS OF THE MATEKIAL WOKLD. 497

hence (cf. *4'31) cm^(u U w) is contained in cm^'(v u w). Then interchanging u and v, and combining the two results, the proposition follows.

The following propositions are not cited subsequently, so their verbal enunciations are omitted : —

*4'41. |" . cm^'(cm^'?< n cm^'v) = cm^'u n cm^v

*4'42. |" . n 'cm^'p = cm,/ n 'om^'p

*4'43. h . cm^'(cm^M u cm^'v) = cm^'(u u v)

*4'44. |" . cm,,,' u 'cm^'p = cm,/ u 'p

*5. On ^-Primes.

*5'23. Proposition. — If u is a ^-prime, and v is a subclass of u, and is not the null class, then v is a <£-prime. In symbols,

t" : u e rm . v e cls'u . \v . D . we

Proof. — For if w be any subclass (not the null class) of v, then (cf. *3'13) cm^(tt-w) is not cm,/w. But (u-w) can be written {(v-w) u (u-v)}, and u can be written {vu(u-v)}. Hence cm^' { (v - iv] u (u - v) } is not cm,,,' {v u (u-v)}. Hence (cf. *4'32) cm^'(v-w) is not cm^'v. Hence (cf. *3'13) v is a ^-prime.

Note. — This theorem, together with *4'31'32, is the foundation of the whole theory. It is remarkable that it requires no axiom concerning <f>. The companion theorem (cf. *12-42), with ax,j, substituted for prm^, requires axioms respecting <f>.

*5'231. Proposition. — Necessary and sufficient conditions, that a class u may be </>-prime, are : (1) u is not the null class, and (2) if x be any member of u, then cm^'(M-t'a;) is not cm^u. In symbols,

I" .'. g;!w : xeu . DZ . cm,/ (M-I' a;) ^ cm^'u : = . ?ieprm^

Proof— Cf. *3'13 and *4'21.

*5'233. Proposition. — If cm,/ A = A, then every unit class is a <£-prime. In

symbols,

I" : cm^'A = A . D . 1 c prm^,

Proof— Cf. *3'13 and *4'25.

*5'235. Proposition. — If x and y are distinct, and the unit classes i'x and i'y have the property <j), then the class, which is the couple composed of x and y, is a ^>-prime. In symbols,

|" : x ?± y . (j)\t.'x . <j>h'y . D . i'x u i'y eprm^,

Proof.— Cf. *3'13 and *4'25'27.

*6. On (j>- Dimensions and <j>-Axial Classes.

*6'23. Proposition. — The (^-dimension of u, if there is such an entity, is a cardinal number not zero. In symbols,

|- : (ExYdim/tt) . D . dim,' u eNc-i'O proof. _Cf. *3'21.

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498 DE. A. N. WHITEHEAD ON

*6'25. Proposition. — If v is a (£-prime and has a ^-dimension number, then the cardinal number of v is less than, or equal to, dim+'v. In symbols,

h : (ExXdim/w) . v e prm^, . D . Nc'v = dim/ v

Proof.— Of. *3'21.

*6'26. Proposition. — If v is <£-axial and is ^-equivalent to u, then the cardinal number of v is equal to dim,,,' u. In symbols,

i" : v € ax^, n equiv/ u . D . Nc' v = dim/ u Proof.— Cf. *3'21'22.

*8. On (^-Concurrences.

*8'21. Proposition. — If u is contained in w, then the ^-concurrence of u with v is contained in the ^-concurrence of w with v. In symbols,

i~ : u c.w . D . ftj,' v c. Wj v Proof.— Cf. *3'31.

*8'22. Proposition. — If t' is contained in w, then the ^-concurrence of u with w is contained in the (^-concurrence of u with v. In symbols,

h : v c w . D . ?«/ w c ?7/ v

Proof— Cf. *3'31.

*10. Geometrical Properties. — A property <£ is called geometrical if it satisfies the five axioms (X, yu, i>, TT, />) H^; $ stated below. The axiom v Up <$> takes the special form for three dimensions. It is to be noticed that three dimensions is the lowest number for which a <£-point (cf. *3'42) can be defined. The reasoning can be applied to higher dimensions, only more elaborate inductions and an extra axiom are required. Other axioms and definitions are wanted to enable all the propositions of projective geometry to be proved. These will not be considered here as such an investigation would involve some repetition when we come to Concept V. The class O^, is the class of straight lines of the geometry, conceived as simple unities. The class put^, is the class of points, each point being a class of lines. The class ple^ is the class of planes, each plane being a class of lines.

*10'1. XH/» (£ is the statement that 0^, has the properly (f>. In symbols,

XHp^. = .^!O+ Df

*10'2. p. Hp (f> is the statement that, ifx is any member of Of, the unit clans L'X has the property <$>. In symbols,

p.Kp<l>. = :ajeOt.3s.^!icoj Df

*10'3. vHp<f> is the statement that the ^-dimension number of 0^ is three. In symbols,

O, = 3 Df

MATHEMATICAL CONCEPTS OF THE MATERIAL WOELD. 499

*10'4. TT Hp (j> is the statement that, if u is a subclass ofO^,, and v is <f>-axial and contained in cm^'u, then there exists a class w (possibly the null class) such that the logical sum ofv and w is (j>-axial and ^-equivalent to u. In symbols,

it Hp <fr . = : u e cls'O0 . v e ax,,, n cls'cm^'tt .=„,„. (g;w) . v u weax^, n equiv^'w Df

*10'5. />Hp<£ is the statement that ifu and v are both (f>-axial, and if they possess at least two members in common, then their logical sum is <f>-maximal, In symbols,

p Tip <ft . = : u, v e ax0 . NC'(M n v) = 2 . DBit) . u u v e mx0 Df

Elucidatory Note. — Referring to the previous elucidatory notes (where <£ is flatness), we see that *10'4 in effect assumes that a line can always be added (1) to two con- current lines to form a set of three concurrent non-coplanar lines, and (2) to one line in a plane to form a set of two concurrent lines in that plane. Also *10'5 assumes that, if two sets of three concurrent lines have two members in common, the four lines are concurrent.

Deductions from the Axioms. *11. Preliminary Propositions.

*11'11. Proposition.- — Assuming (X, i>)Hp<£, O^, has at least three members. In symbols,

I- : (X, v) Hp (/>.=>. Nc'O^ 3

Proof— Of. *4-27 and *10'1'3.

*11'12. Proposition. — Assuming (X, /i, v)Hp<f), cm^'A is the null class (A). In symbols,

|- : (X, JLI, v) Up <£.=>. cm/A = A

Proof— Cf *4'28 and *10'2 and *1M1.

*1T21. Proposition. — Assuming (X, /i) Hp <f>, all ^-prime classes with more tlian one member are contained in O^,. In symbols,

I" . '. (X, p.) Hp <f> . D : v e prm^, . Nc't' > 1 . ID . v e cls'O^,

Proof. — For if a; is not a member of O^,, then cm^Ya; is the class of all entities. Hence (cf. *4'21'27 and *10'1'2) the conclusion follows.

*12. On $- Axial Classes and $- Dimensions.

*12'11. Proposition. — Assuming (X, p., v) Hp <j>, every unit class whose single member belongs to O^, is ^-axial. In symbols,

|~ . '. (X, /A, v) Hp (j> . => : x e O^, . DT . L'X e ax^

Proof.— Cf. *3'21-22 and *4'27 and *5'233 and *10'2.

*12'12. Proposition. — Assuming (X-Tr)Hp^, every subclass of O^,, not the null

3 s 2

500 DK. A. N. WHITEHEAD ON

class, has a set of <£-axes. (Note.— A class which is (/.-axial and (/.-equivalent to a class tt is said to be a set of $-axes ofu.) In symbols,

|- /. (X-ir) Hp </> . r> : u ecls'O^, . R\u . D . g!(ax^ n equiv^w)

Proof.— Since there is at least one member of u, there is (cf. *12'll) a (/.-axial class contained in «. Hence (c/ *10'4) this class can be augmented so as to become a set

of «£-axes of u.

*12'13. Proposition. — Assuming (X-7r)Hp(£, if u and v are subclasses of 0^, and v is not the null class, and cm^'-u is contained in cm/w, then there exist two sub- classes of 0^, w and w', say, such that w is a set of (/.-axes of v, and wuw? is a set of (/.-axes of u. In symbols,

I".'. (X-7r) Hp(/> . D : M, -wccls'O^ . g!w . cm/vccm^w . => .

(3*0, w') . weaXf n equiv^'v . wuw/e ax^ n equiv^'w.

.— Cf. *10'4 and *12'12.

*12'21. Proposition. — Assuming (X - TT) Hp (/>, if w and v are subclasses of O^, and v is not the null class, and cm^'v is contained in cm^'u, then the (/.-dimension number of v is less than, or equal to, the (/.-dimension number of u. In symbols,

I:.'. (X-7r) Hp (/>.=>: u, -uecls'O^ . Q\v . cm+'v c cm^'u . => . dim,,,' v = dim^,' M.

prooy; — From *6'26 and *12'13, w and w' exist (assuming wnu/ = A) such that Nc'?0 = dim^'v and Nc'w + Nc'w' = dim^,'?<. Hence dim^,'v < dim^'w, unless w1 is the null class, or unless the numbers are not finite, in which cases dim/w = dim^'tt is possible.

*12'23. Proposition. — Assuming (X-7r)Hp(/>, if u and v are subclasses of 0,,,, and v is not the null class, and cm^'v is contained in cm^'w, then, if dim^'-y = dim^'n, we have cm^'v = cm^'u, and conversely. In symbols,

I" : : (X - TT) Hp <j> . D . '. v, u e cls'O^, . 3 ! v . cm,,,' v c cm^,' u . D :

cm^'-w = cm^'tt . = . dim^'v = dim/u

Proof. — Assuming dim^,' v = dim^' u, and also assuming the notation of the proof of *12'21, then w and w' are such that (l) w is (/.-equivalent to v and wuw' to w, (2) Nc'w = dim^'v and Nc'ty + Nc'w/ = dim^'u. Hence, by hypothesis and (2), Nc'w + NcV = Nc(w. Also (cf. *10'3 and *12'2l) Nc'w + NcV = 3. Hence NcV = 0, that is, W = A. Hence from (1), cm^'w = cm^'v. The converse is obvious.

*12'33. Proposition. — Assuming (X-Tr)Hp^., if u and v are subclasses of O^, and v is not the null class, and cm^'v is contained in, but is not identical with, cm^'w, then dim^'v is less than dim^w. In symbols,

1" . '. (X - TT) Hp <£ . D : u, v f. cls'O^, . 3 ! v . cm^,' v c. cm,,,' u .

cm6'v T± cm/M . D . dim/!' < dim/« Proof.-Cf. *12'21-23.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 501

The following proposition should be compared with *12'12 : —

*12'37. Proposition. — Assuming (X - IT) Hp <f>, if u is a subclass of 0^, and is not the null class, there exists a subclass of u which is <£-prime and ^-equivalent to u. In

symbols,

h . •. (X - TT) Hp (f> . D : u e cls'O^ . g ! M . D . g ! (prm^, n els' u n equiv,,,' u)

Proof. — From *12'12'21, u is either of one, or of two, or of three ^-dimensions. If u is of one (^-dimension, the conclusion follows from *5'233 and *10'2 and *11'12. If u is of two (^-dimensions, then (of. *5'235 and *10'2) any two members of it form a <£-prime class, and (cf. *12'21) the ^-dimension number of this class is not greater than two, and hence (cf. *6'25) it is two, and hence (cf. *12'23) this subclass is (^-equivalent to u. If u is of three ^-dimensions, it must contain at least one subclass v consisting of two members, and, as before, v must be </>-prime. If v is of three ^•-dimensions, then (cf. *12'23) u arid v are ^-equivalent. Iff is of two ^-dimensions, then there is a member of u, x say, which is not a member of cm^'v. Then, either v u L'X is (^-prime and (cf. *12-23) ^-equivalent to u, or the class composed ofx and some one (not necessarily any one) of the members off is <£-prime and ^-equivalent to u. The following proposition should be compared to *3'13 and *12'33 :— *12'41. Proposition. — Assuming (X- 77) ~Rp<f>, if u is <£-axial, and v is a subclass of u, and both v and (u-v) are not the null class, then dim^r is less than dim^K In symbols,

|~ . '. (X - TT) Hp (£ . D : u e ax^, . v e els ' u . g ! v . 3 ! (u - r) . D . dim^' v < dim,/ u

Proof.— Cf. *3'13 and *4'21 and *11'21 and *12-21'23. The following proposition should be compared to *5'23 : —

*12'42. Proposition. — Assuming (X — ir)Hp$, then, if u is a subclass of O(/) and is (^-axial, any subclass of u, not the null class, is <^-axial. In symbols,

1" . '. (X - TT) Up <f> . D : u f. ax^, n els ' O^, . v e els' u . g ! v . D . v e ax^,

Proof. — From *6'26 we have Nc'tt = dim^'w ; from *5'23 and *6'25 we have Nc'v = dim^'n Hence (cf. *12'41), if v is not identical with v, we have

Nc' v = dim^,' v < Nc' u ......... (1)

Firstly, assume that v omits one member of u only. Then Nc'-j'+l = Nc'?*. Hence, from (l), Nc'w = dini^'w, and hence (cf. *3'21) v is ^-axial.

Secondly, if v omits two members of u, then it is a unit class, and (cf. *12'11) is

It is convenient to conclude this section (*12) with three theorems which are fundamental to the theory of (/>-points and of (^-planes.

*12'51. Proposition. — Assuming (X - TT) H_p <£, if u is of two ^-dimensions, and x and y are members of cm^'w, then the class composed of x and y is ^>-axial and is a subclass of O.J,. In symbols,

I" .•. (X-7r)Hp(£ . D : dim^'M = 2 . x, yfcm+u . D. L'X u t'«/eax^ n cls'O^,

502 DE. A. N. WHITEHEAD ON

Proof.— cnV« is contained in O* (cf. *4'21'27 and *10'1 and *11"21). If x and y are identical, cf. *12'11. If* is distinct from y, then c/ *3'21 and *12-ll'21-23. ^

*12'52. Proposition.— Assuming (X - TT) Hj9 (/., if u has three members only, and is a self-(/>-concurrence, and its (/.-dimension number is three, then u is <£-axial and a subclass of O^. In symbols,

h.'. (\-ir)Hjpc£ . ID : «e3 n conc^ . dim^, = 3 . => . u e ax^, n els' O^

Proof.— I? v is contained in u and possesses two members, then (cf. *3'31'32) v is f-axial, and (cf. *12'23) is not (/.-equivalent to u. Hence (cf. *5'231) M is 0-prime, and hence (cf. *3'22) is (/.-axial, and also (cf. *11'21) is a subclass of O+.

*12-53. Proposition.— Assuming (X - TT) Hp (/>, if x, y, and z are three distinct entities forming a (/.-axial class, then the common subclass of cm/(»'xUt'y) and cm^'(i'a; u i'z) is the unit class L'X. In symbols,

|-/. (\-ir)H2>^ .3: *'*U t'7/u t'2e3 nax^, .D. cm^'^'aju icy) ncm^(i'aju t'z) = i'a; Proof— Cf. *4-21'27 and *10'2 and *12'23-42.

*13. On ^-Maximal Classes and Self -^-Concurrences.

*13'11. Proposition. — Assuming (X - TT) lip </>, if p is a ^-maximal class and a subclass of O^, and q is a subclass of p, not the null class, then q is a (/.-maximal class. In symbols,

|~ . •. (X. - TT) Hp <j) .mpe mx^, n els' O^, . 7 e cls'j) .^Ip.^.qe mx^,

Proof. — The class q must (cf. *10'3 and *12'21) be of one, or two, or three ^-dimensions. If the ^-dimension number of q is one or two, then (cf. *10'2 and *12'lt'51) q is a ^-maximal class. If the ^-dimension number of q is three, then (cf. *12'23) q is (^-equivalent to p. Hence if v be a subclass of q, which is ^-prime and ^-equivalent to q, it is ^-equivalent to p, and hence (cf. *3'23) it is <^-axial. Hence (cf. *3'23 and *12'37) q is ^.-maximal.

*13'31. Proposition. — Assuming (X — ir)Hp^, if u is a self-(^-concurrence and a subclass of O^, then u is a ^-maximal class. In symbols,

I" . '. (X - TT) lip <f) . D : u e cone,,, n els' O^ . D . u e mx^,

Proof. — There exists (cf. *12'37) a subclass (v) of u, which is ^-prime and ^-equivalent to u. If v is a unit class, then (cf. *10'2 and *1211) v and u are identical, and u is of one (/(-dimension and ^-maximal. If v is a couple, then (cf. *3'31-32) v is ^-axial, and (cf. *3'22'23) u is of two ^-dimensions and is (/.-maximal. If v is composed of three members, then u is of three (/.-dimensions, and neither of the previous cases can hold. Hence again u is (/.-maximal.

MATHEMATICAL CONCEPTS OF THE MATERIAL WOELD. 503

*13'32. Proposition. — Assuming (\ - TT) Up (j>, all subclasses of O^ which are ^-maximal are self-<£-concurrent, and conversely. In symbols,

I" : (X - TT) Hp (f> . D . mx^, n els' O0 = conc^ n els' O0 Proof.— Cf. * 13-1 1-31.

*14. On Points.

*14'11. Assuming (X-/3)Hjp<£, every ^-point is a self-(£-concurrence and a subclass of O^. In symbols,

h : (X - p) Up (j> . => . pnt^, c confy n els' O^

Proof. — Every <£-point (cf. *ll*2l) is a subclass of O^. Again let x and y be two distinct members of a <£-point P. Then (cf. *3'42) a, b, c exist such that i'rt u i'6 u I'c is <£-axial and of three dimensions, and x and y are each ^-concurrent with each of a, b, and c. Hence (cf. *12'53) at least one pair of a, b, and c exist (say a and b) such that i' x u i' a u i' b and t' y u <,' a u <.' b are both three ^-dimensional and self-(^-concurrences. Hence (cf. * 1 2 -52) t' x u t' a u i' b and i' x u t' « u t' b are both ^>-axial. Hence (cf. *10'5 and *13'32) i'.x u i'y is </>-axial. Hence P is a self-(/>-concurrence.

*14'12. Proposition. — Assuming (A.-p)Hp<£, every ^-point is ^-maximal. In symbols,

h : (X -

Proof— Cf. *13'32 and *14'11.

*14'13. Proposition. — Assuming (X'-p)Hp^, if P is a ^-point, then P is the ^-concurrence of O^, with P. In symbols,

I- .'. (X-p) % </, . D : P epnt, . D . P = (\'P

Proof.— Cf. *3'42 and *8'21'22 and *14'11.

*14-14. Proposition. — Assuming (X-TT) Hp ^>, if P is a ^>-point, it possesses at least three members. In symbols,

|-.\ (\-7r)Hpc/> . = : Pepnt* . D.Nc'P^ 3

Proof. — P possesses (cf. *3'42) every member of O^ which is ^-concurrent with each of a certain <£-axial set of three members. Hence (cf. *12'42) P possesses this set of three members.

*14'21. Proposition. — Assuming (\-p)Hp<£, ^-points with more than one member in common are identical. In symbols,

I- .'. (X-p) Hp 0 . D : P, Q epnt^ . Nc'(P n Q) > 1 . o . P = Q

Proof. — Let a and 6 be two distinct members of P n Q. Then (cf. *3'42 and *12'52 and *14'11) c and d exist, such that c is a member of P and d of Q, and

504 DK. A. N. WHITEHEAD ON

t'aui'&uc'c and i'a u i'6 u iVZ are both of them three ^-dimensional and <£-axial. Hence (cf. *10'5 and *13'32) cHs a member of O/(i'a u I'b u i'c), and hence (cf. *3'42 and * H'll '13) d is a member of P. Thus P and Q are identical.

*16. On <f>-Planes.

*16'11. Proposition. — Assuming (X-7r)Hp<£, every ^-plane is (^-maximal, self-<£- concurrent, and a subclass of 0+. In symbols,

i" : (X - TT) Hp <f) . z> . ple^, c mx,,, n conc^, n cls'O^

Proof.— Cf. *12-21-23'51.

*16'21. Proposition. — Assuming (X - TT) Hp <£, <£ -planes with more than one member in common are identical. In symbols,

tv. (\-7r)Hp<f> . D \p, Q-eple^ . Nc'(^ n </) > 1 . D . p = q

Proof. — Cf. *12'23 and *16'11.

*1G'31. Proposition. — Assuming (X - •n-) H/> <£, every self-<£-coucurrence is either 0-copunctual or ^-coplanar. In symbols,

\ . : (X - TT} Hp 4> • ^ : u e conc^, . => . copnt^, ! u . V . cople^, ! u

Proof. — A proof is only required when u is of three ^-dimensions. Then «, b, c exist, such that they are three distinct members of u and are not a <£-coplanar class. Hence (cf. *12'52) they form a <£-axial class of three members. Hence (cf. *3'42) u is, in this case, <£-copunctual.

*16'32. Proposition. — Assuming (\-p)H^>(£, if p is a <£-plane, and P and Q are distinct (^-points, and p and P have common members, and also p and Q, then the member (if any) common to P and Q is a member of p. In symbols,

i- .-. (X -p) Hp<£ . D : p eple* . P, Q epnt, . P * Q . R\(p n P) . ^\(p n Q) . D . P n Q cp

Proof. — If P n Q is the null class, then P n Q is contained in p. If P n Q is not null, let c be a member ; also let a and b be, respectively, members of p n P and of p n Q, which exist by hypothesis, (i) If c is identical with a or b, then (cf. *14'2l) P n Q is contained in p. Again (ii) if c is not identical with a or 6, then (cf. *14'11 and *16'll)a, b and c form a self-<£-concurrence. Hence (cf. *16'3l) this class is either <£-copunctual or ^-coplanar. If the class is <£-copunctual, then (cf. *14'21) P and Q are identical. Hence it is ^-coplanar, and hence (cf. *16'2l) c is a member of p.

*16'33. Proposition. — Assuming (X-/3)H^<^, if P is a <£-point and p and q are distinct ^-planes, and P and p have common members, and so, also, have P and q, then the member (if any) common to p and q is a member of P. In symbols,

.p j± q . an^ . a Proof. — The proof is in all respects similar to that of *16'32.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 505

*16'42. Proposition. — Assuming (\-/>)H/><£, if p is a <£-plane and is not ^-co- punctual, then p is the ^-concurrence of O^ with p. In symbols,

h . '. (X - p) Up (f> . D : p c ple^ . •> copnt^ ! p . => . p = O^'p Proof.— From *8'21 and *16'11 we have

Let x be any member of the ^-concurrence of O^ with p. Hence (cf. *16'll) p u L'X is a self-(£-concurrence. Hence (cf. 16 '31) p u i'x is either ^-copunctual or t^-coplanar. But on the first alternative p is </>-copunctual. Hence p u i'x is <£-coplanar. Hence x is a member of p. Hence from (l) the proposition follows.

Note. — In Concept V. the hypothesis of *16'42, that a <£-plane is not copunctual, is verified (cf. *28'll), where <j> represents " homaloty," and the axioms of that concept are assumed.

Summary of the Complete Development of this Subject. — By the use of further axioms the whole theory of projective geometry, apart from " order " and apart from FANG'S axiom t respecting the distinction of harmonic conjugates, can be proved for (/•-points and the associated geometrical entities. Then FANG'S axiom can be added, and the theory of order and continuity can be introduced, as in PIEPII'S memoir (loc. cit.). In the sequel a somewhat different line of development is adopted, suitable for the special ideas of Concept V.

(ii) CONCEPT V.

This concept is linear and monistic. It makes use both of the theory of interpoints and of the theory of dimensions. The points are classes of objective reals, and disintegrate from instant to instant. The corpuscles are capable of various and complicated structures, and are thus well fitted to bear the weight of modern physical ideas. The concept is Leibnizian, and only requires one extraneous relation for the same purposes as that of Concept III.

The essential relation is the pentadic relation T&-(abcdt), as explained at the commencement of Part III. The four first terms, namely, a, b, c, d, are objective reals and are mutually distinct, the fifth term is an instant of time.

The relation R,-(abcdt) can be read, a intersects b, c, d in the order bed at the instant t. In this concept copunctual objective reals do not necessarily intersect, though two intersecting objective reals are necessarily copunctual. The relation of intersection is not to be limited in properties by the mere geometrical suggestion of its technical name.

Since points are defined by the aid of the theory of dimensions, it follows (cf. note

t Cf. FIERI, loc, tit. VOL. CCV. — A. 3 T

506 DR. A. N. WHITEHEAD ON

to *3'42) that the geometry cannot be of less than three dimensions. Hence in this concept geometry of three dimensions occupies a position of unique simplicity.

The points at infinity, here called cogredie.nl points, are points in exactly the same sense as the other points. They are defined by a property not hitherto taken as fundamental. The properties of cogredient points play an essential part in the construction of a relation which assigns an order to the points on any straight line.

*20. Definitions.

*20'11. Definition. — An objective real p is doubly secant with a class u at an instant t if there exist two objective reals, members of u (x and y, say), which are both intersected by p at the instant t, and are such that there exists no interpoint on p of which x and y are both members. The symbol (uu)M \p will denote that p is doubly secant with u at the instant t. The symbolic definition is

(uu)w\p . = : fax, y) . x ^ y . x, yen n K;(p;;;<) . - far) . v eR;(p???£) . x, yev Df

*20'12. Definition. — A class u is homalous at an instant t, either when a necessary and sufficient condition, that x should be a member of u, is that x should be doubly secant with u, or when u is a unit class contained in R;( ;;;;£). The symbol /AR(!M will denote that u lias the property of homaloty at the instant t. The symbolic definition is

p.]ulu . = .'. xeu . =x . (uu)Ht\x : V : ue 1 n cls'B/( ;;;;*) Df

This property (p.Rt) of homaloty will now be taken as the special value of <f>, to which the theory of dimensions will be applied. The common /iRrsubregion for u is denoted, according to the definition of *3'12, by cmM '«. But a suifix to a suffix will be avoided by using the simpler symbol cmlu'u, and similarly for the other entities defined in *3. Thus the following symbols are also defined, namely,

OHt, equivH(X prmR(, dimlt(, ax,u, mxR(, uRt'v, concH(, pleH,, pntu(, copleH(h«, copntH,!w.

With regard to the nomenclature, the term " ^-equivalence" should be particularized into " homaloty-equivalence," and "<£-prime" into " homaloty-prime," and so on. But, except where confusion is likely to occur, the term " homaloty " will be dropped ; and the terms " equivalence," " prime," " dimensions," " axial," " maximal," " concurrence of u with v," " self-concurrence," " plane," " point," " coplanar," " copunctual" will be used in the senses defined in *3, with <£ particu- larized into homality.

Elucidatory Note. — This definition of homaloty should be compared with the definition of the flatness of a class of punctual lines which has been used in the elucidatory notes of *3. Thus a class of punctual lines is flat, either when it is a unit class whose single member is a straight line, or when it is a necessary and

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 507

sufficient condition of a straight line x being a member of it, that x should meet two members of the class in points which are not their point of meeting (if they have a point of meeting). Owing to the fact that " intersection " (as used here) is wider in intension and narrower in extension than the idea of the " meeting " of two punctual lines, two punctual lines may " meet " without the corresponding objective reals " intersecting." The result is that homaloty and flatness have some different properties, for example, cf. *21'21.

*20'21. Definition. — The punctual associate of a class u is the class of those points which have a member in common with u. The punctual associate of u is denoted by assKj'tt. The definition in symbols is

assltj< u = P {P e pntB( . 3 ! (P n u)} Df

Note. — The punctual associate of the class i'a, ivhere a is an objective real, will be called the punctual associate of a. Its symbol is assR(' t' a.

*20'22. Definition. — A punctual line is a class of points such that there exist two planes, p and q, which are distinct and are such that the class of points is the common subclass of the punctual associates of p and q. The class of punctual lines at any instant t is denoted by linju. The symbolic definition is

linK( = m{(Kp, q) . p, qepleRt . p ^ q . m = &ss!U'p n ass,,/?} Df

Note. — Those punctual lines which are not " lines at infinity " (to be explained later) will be proved as the result of the axioms to be the punctual associates of the various objective reals.

*20'23. Dejinition. — The point, if there is one and one only, which contains a class u is called the dominant point of u. The dominant point of u is denoted by «K(. The

symbolic definition is

uKt = (tp) {p e pntH, . u e cls'p} Df

Note. — The idea of a dominant point obtains its importance from the fact that, according to the axioms given below, each interpoint is contained in one and only one point.

*20'231. Dejinition. — The nonsecant part of u is that subclass of u of which no member is a member of any interpoint which is a subclass of u. The nonsecant part of u is denoted by nscK(' u. The symbolic definition is

nscK(' u = x {x e u . -*- (3^) • v e intpntK( n els' u . xev} Df

Note. — This definition takes its importance from the fact that (assuming the subsequent axioms) a point in general consists of a nonsecant part and of a part made up of interpoints contained in it. Either the interpoints or the nonsecant part may be wholly absent.

*20<232. Definition. — A class of points is called a Figure.

3 T 2

508 DR. A. N. WHITEHEAD ON

*20'233. Definition. — A point, which is a member of a figure, will be said to lie in

that figure.

*20'234. Definition. — A point, which lies in the punctual associate of a class of objective reals, will be said to be on, or upon, that class.

*20'235. Definition. — A punctual line is said to join two points if both the points lie in it.

*20'236. Definition. — -Two punctual lines, which possess a common point, will be said to meet at that point. Similarly, any two classes of points will be said to meet in their common subclass, and this subclass will be called their meeting.

*20'24. Definition. — A class of points is called collinear if there exist a punctual line in which they all lie. The symbol collRt!w will denote that u is a class of collinear points at the instant t. The symbolic definition is

. melinR( . wecls'm Df

*20'31. Definition. — Two figures are in perspective if (i) they have a one-one correspondence to each other, (ii) the joint figure formed by the two figures combined is not collinear, and (iii) there exists a point (the centre of perspective) which lies in every punctual line joining two distinct corresponding points. The statement that u and v are in perspective with each other at the instant t, and that S is the requisite one-one correspondence, will be denoted by u (S persp)R( v. The symbolic definition is

w(Spersp)u, v . = : u, vecls'pntR( . — co\\ml(u u v) . S e 1 — *• 1 . u — S'(;-) . v = S;(-;) :

(gV) : melinR, . S;(AA') . A ^ A'. A, A'em . D,,,,A,A,. V em Df

*20'32. Definition.— The symbol [AB] persp,« [A'B'] denotes that A, B, A', B' are points, and that the figure formed by A and B is in perspective with the figure formed by A' and B', and that the one-one correspondence of the perspective is of A to A' and of B to B'. Also [ABC] perspR( [A'B'C'] has a similar meaning, and so on. In symbols,

[AB] PerspK( [A'B'] . = . (gS) . (I'A u I'B) (S persp)Ht (i'A' u i'B') . S;(AA') . S;(BB') Df

[ABC] perepH, [A'B'C'] . = . (gS) . (i'A u i'B u i'C) (S persp)H( (i'A' u i'B' u i'C') .

S;(AA').S;(BB').S:(CC') Df

*20'33. Definition. — The symbol MperspR(v denotes that there exists a one-one relation S such that, at the instant t, u is in perspective with v and S is the requisite one-one correspondence. In symbols,

u perspRt v . = . (gS) . u (S persp)H( v Df

*20'41. Definition.— Two objective reals, a and c, are called cogredient at an instant t when (1) if u, v, iv are three interpoints on a, and u', v', it/ are three interpoints on c, and the dominant points ust, vw, wRt are a trio of points in

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 509

perspective with the trio of dominant points u'Rt, t/H<, u/Kt, then the interpoint relation (Rin), if it arranges either trio of interpoints in an interpoint order, arranges both trios of interpoiuts in the same interpoint order (i.e., either uvw and u'v'w', or vwu and t/u/u', or so on), and (2) there exist three interpoints u, v, w on a in the interpoint order uvw, and three interpoints u', v', w' on c such that «R(, vut, wm are in perspective with «'„„ 1/R(, w'R(. The symbol cogrdR('a denotes the class of objective reals cogredient with a. The symbolic definition is

cogrdR<'a = x {(u, v, w, u', v', iv') : u, v, w c R:(a ? ? ? t) . u', v', w' e R;(« ? ? ? t) .

perspR( OVRXR J . => . Rhl:

perspR( [u'^v'^w'^} Df

. — The class cogrdR('a does not include a itself (of. *27'43). It will be noticed that universal preservation of order by ranges in perspective on a pair of lines is a characteristic of a pair of parallel lines in Euclidean space, and of nonsecant lines in hyperbolic space. The choice of this property for the definition of parallelism (or nonsecancy) arises from the facts that (l) any two coplanar objective reals are copunctual (according to the subsequent axioms), so that the property of nonsecancy (in its ordinary acceptation) is not available, (2) we do not wish to make "cogredience" synonymous with " nonintersection " (using " intersection " in the special sense here defined), as this would impose an unnecessary limitation on the concept. The idea of cogredience is an essential element in the definition of a relation which, with the aid of axioms, distributes the points in any punctual line into an order.

*20'42. Definition. — A Cogredient Point is the class of objective reals cogredient with some objective real a, together with a itself. The symbol QOIM denotes the class of cogredient points at the instant t. The definition in symbols is

OOK( = it {(3«) . a e OR4 . u = I'a u cogrdK('ct} Df

Note. — In the case of Euclidean geometry, which is the only case considered here, each cogredient point is a point according to the definition of *3'42. The present definition would be very inconvenient, unless this were the case. The symbol o°K( is reminiscent of the fact that the cogredient points are the points at infinity.

*20;51. Definition. — The Point-Ordering Relation is a tetradic relation holding between three points and an instant of time. Its symbol is Rp,,, and R|m:(ABC£) is defined to mean that, at the instant t, (1) A, B, C are non-cogredient points upon the same objective real, a say, and (2) there exist an objective real x and three interpoints u, v, w on x such that (i) x is cogredient with a, and (ii) u, v, w are in the interpoint order uvw, and (iii) A, B, C are in perspective with the dominant points uRt> vxt> wm- The definition in symbols is

510 DR. A. N. WHITEHEAD ON

B|in»( ABC«) . = . A, B, C e pntK( - OOR( . (ga, x, u, v, w) . a e A n B n C . x e cogrdR(' a .

u, v, t(;eR:(a;???«) . ~Rin-(ui>wi) . [ ABC] perspR( [>R(t>R(i{>m] Df

Since (cf. *27'43) x in the above definition is distinct from a, three collinear (i.e., on a) points, A, B, C, cannot directly take their point-order from three inter- points which they themselves may severally contain (cf. however *2T5l). The point-order of A, B, and C must arise from the order communicated (in a sense) to a copunctual pencil of three punctual lines by three interpoints contained respectively in points in these lines, and all three interpoints possessing an objective real (x) in common. The punctual lines of this pencil must possess A, B, and C respectively. This intervention of a pencil for the communication of point-order is necessary for the comparison of the orders of different ranges. If the apparently simpler plan is adopted, inextricable difficulties seem to arise. Also it will be remembered that not every point will necessarily contain an interpoint.

*20'61. Definition. — A Punctual Plane is a figure which is either the punctual associate of some plane, or is the class OOR(. The class of punctual planes is denoted by pple,«. The definition in symbols is

pplem = assu,"pleE( u i' ooRt Df

Note. — This definition is only convenient for Euclidean geometry.

*2072. Definition. — A figure is called Punctually Coplanar if there is a punctual plane containing it. The symbol coppleu(!w will denote that u is a punctually coplanar figure at the instant t. In symbols,

coppleK(!w . = . (3^) . p e ppleR( . u e cls'p Df

Note. — This definition should be compared with that of copleR(!?< in *3'43.

*21. General Deductions.

*21'01. Proposition. — All the general deductions in the theory of dimensions, namely, *4 to *8, hold.

The following propositions, dependent on the special definition of homaloty, also hold :-

*2ril. Proposition. — OR< is the class R; (;;;;t). In symbols,

Proof.— Cf. *20'12.

Note.— If t is not an instant of time, the classes ORi and R;(;;;;£) are both the null class, and are thus identical. Accordingly the hypothesis, <eT, is not required in this proposition. A similar explanation of the absence of the hypothesis, t e T, holds for many other propositions.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 511

*21'21. Proposition. — An objective real, which is doubly secant with the common subregion for u, is a member of the common subregion for u. In symbols,

I" : w = cmR('w . (ww)Rt\x . z> . xecmRt'u

Proof.— Cf. *3'12 and *20'11.

Note. — The converse is not in general true, namely, that, if u is a class of objective reals, and x is a member of cmBt'u, then a; is doubly secant with cmR('w. Nor does this converse follow from subsequent axioms. In the absence of this converse proposition the properties of homaloty differ from those of " flatness" for classes of punctual lines. For if u is a class of punctual lines, and (f> stands for the property of flatness, then cm^'u is flat.

*21'31. Proposition. — The proposition p. Hp<£ is true when pRt is substituted for <f>. In symbols,

Proof.— Cf. *10-2 and *20'12.

*21'41. Proposition. — If a is an objective real cogredient with c, then c is cogredient with a. In symbols,

I" : a e cogrdR(' a . = . c e cogrdm' a Proof.— Cf. *20'41.

*21'51. Proposition. — If u, v, w be three interpoints, possessing the same objective real, and with dominant points um, vRt, wRt, then Rpn; (uRtvRtivmt) implies ^-^(uviut).

In symbols,

|- .'. u, v, w e K; (a ???«). D : R,,n; (uRtvRtwRtt) . D . ~R-m; (uv-wt)

Proof. — -By definition (cf. *20'5l) ^v^(uKtvmwRtt) implies (l) the existence of an objective real x, cogredient with a, and also of three interpoints, u', v', w', all possessing x, and (2) that ~R,in-(u'v'iv't}, and (3) that u', v', w' are contained in dominant points u'Rt, t/Rt, i»'Rt in perspective with uRt, vRi, ivm. Hence by the definition of cogredience (cf. *20'41) also Rin;(uvwt) holds.

*22. The Axioms. — Just as in Concept III., the axiom of persistence (cf. *22'l) does not enter into the geometrical reasoning, but it is essential to the physical side of the concept.

*22'1. I Hp II is the statement that, if t be an instant of time, O is contained in OR(. In symbols,

= :«eT.D(.OcOR< Df

The next four axioms, viz. (II.-V.)HpTl, are the axioms of order. They have already been explained in *l'5r52'53'54.

*22'21. IIHpR=aHpIl Df

*22-22. IIIHpR = £H2>K Df

*22'23. IVHpR = yHpR Df

*22-24. VHpR=8HpR Df

512 DR. A. N. WHITEHEAD ON

The next three axioms, viz. (VI.-VIlI.)HpR, are the axioms establishing the relation of interpoints to points. Intpnt Rp R has been denned in * 1 "4 1 .

*22'31. VIHpR = intpntHpR Df

*22'32. VII Hp R is the statement that, if u is an interpoint, there exists a point containing u. In symbols,

VII Hj> R . = : (t, u):we intpnt» . a . fop) . p e pnt» . « cp Df

*22-33. VIII Hp R is the statement that, if p be a point, and u and v be two distinct interpoints contained in p, then u and v possess no common member. In symbols,

VIII Hp R . = : (p, u,v,t):pe pntK1 .u,ve intpntR( n cls'p . u ^ v . => . uav = A Df

The next set of three axioms, viz. (IX.-XI.)HpR, supplies the missing hypotheses requisite to make homaloty a "geometrical property," as defined in *10.

*22'41. IX HpR is the statement that, if t is an instant of time, j/Hp/*K( is true.

In symbols,

IX Hp R . = : tt T . 3( . v Hp ft™ Df (cf. *10'3)

*22'42. XHpR is the statement that, if t is an instant of time, irHp/iHt is Zrwe.

In symbols,

. = :*eT.=>« . irH^/t,,, Df (c/ *10-4)

*22'43. XI HpR is the statement that, if t is an instant of time, p Hp/*Ki is true.

In symbols,

XIHpR. = :«eT.Df ./oHp/iH, Df (cf. *10'5)

*22'51. XII HpR is the statement that, ifp and q are distinct planes, and there exists a point, not a cogredient point, ivhich is a member of the punctual associates of both planes, then p and q possess a common member. In symbols,

= : p,

Df

The next axiom, XIII HpR, is the " Euclidean" axiom.

*22'61. XIII HpR is the statement that the cogredient points are points. In symbols,

XIII Hp R . = . oo R, c pntHt Df

The next three axioms, namely (XIV.-XVI.) Hp R, establish the theory of the order of points as determined by the point-ordering relation (cf. *20'5l). Incidentally some existence theorems can be deduced from them, which would else have to be provided for elsewhere.

*2271. XIV HpR is the statement that, if A and B are two distinct non-

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 513

cogredient points, then there exists at least one point C such that A, B, C are in the point-order ABC at the instant considered. In symbols,

XIV Up R . = : (A, B, «) : A, B e pnt* - <*„, . D . g ! Rpn<(AB ; t) Df

*2272. XV Hp R is the statement that, if A, B, C are three distinct non-cogredient points, on the same objective real, then at the instant considered one of the point- orders ABC, or BCA, or CAB holds. In symbols,

XVH^R . = .-. (A, B, C, t) .: A, B, C epntK. ooR< . a!(A n B n C) .

A^B.B^C.C^A.D: RI>U:(ABCO . v. R,,U;(BCAO . v. R^CAE*) Df

The next axiom, XVIHpR, is the well-known "transversal" axiom.

*2273. XVI HpR is the statement that, if at the instant t the points B, C, D are in the point-order BCD, and the points C, E, A are in the point-order CEA, and the points A, B, C are not collinear, and F lies in the punctual associates botJt of A n B and o/D n E, then the points A, F, B are in the point-order AFB. In symbols,

XVI HpR . = : (A, B, C, D, E, F, t) : R^BCD*) . R

AnBnC = A.F€ assH('(A n B) n assR('(D n E) . D . Rim; (AFB«) Df

*22'74. As XVII Hp R, an axiom of continuity will be wanted.

Note. — The above axioms are all axioms of geometry, in the sense of "geometry" as defined in the sense definition of it given in "Part I. (i.). But geometry in this Concept V. includes more than does geometry in Concept I. For in Concept I. geometry has only to do with points, punctual lines, and punctual planes ; but in Concept V. geometry has, in addition, to consider the relation of the objective reals (which are all "linear") and of interpoints to the above entities. In this respect, geometry in Concept V. merges into physics more than does geometry in Concept I. Thus the excess of the number of axioms in Concept V. over the number in Concept I. arises from the fact that there is a larger field to be covered. Also, I Hp R is not required in the geometrical reasoning.

*25. Preliminary Propositions.

*25'11. Proposition. — Assuming (II. -VI) Hp R, all the propositions of the theory of interpoints (cf. *l) hold of the interpoints of this Concept.

*25'12. Proposition. — Assuming (II. -VI.) Hp R, if t be an instant of time, then 0R, possesses at least four members. In symbols,

tv. IXHpR . D : £eT . D . Nc'OKa 4

Proof.— Cf. *1-61717273 and *2M1 and *25'11. VOL, ccv. — A. 3 u

514 DR. A. N. WHITEHEAD ON

*25'13. Proposition. — Assuming (II.-VL) HpR, if t be an instant of time, then 0R( is homalous. In symbols,

Proof— From *T65 and *21'11, OK( is identical with R;(;---£). Hence (cf. *l-3r61'62) every member of Om is doubly secant with OR(. Again (cf. *2M1), every objective real which is doubly secant with 0R( is a member of it.

*25'14. Proposition. — Assuming (II.-VL, IX-XI.)HpR, if t be an instant of time, then all the special deductions of the theory of dimensions, namely *11 to *16, hold respecting homaloty, that is, with p.}U substituted for <£.

Proof— Cf. *21"31 and *22'4r42'43 and *25'13.

*25'21. Proposition. — Assuming I HpR, if I is an instant of time, then O = OR(. In symbols,

Proof.— Cf. *2fll and *22'1.

Note. — The above theorem is not used in any geometrical reasoning. *25'31. Proposition. — Assuming (II. -VIII.) HpR, if a be a member of OR(, then the number of points on a is at least three. In symbols,

|- . '. (II. VIII.) Hp R . D : « e Ol{( . D . Nc'assK(Va ^ 3

Proof.~Cf. *172 and *2M1 and *22'32'33 and *25'1.

*25'32. Proposition. — Assuming, (II. -VII., IX.-XI.)HpR, if u be an interpoint, there is one and only one point containing it. In symbols,

h .'. (IT. -VII., IX.-XF.) Hp R . D : u t intpntB, . D . P {P e pntR( . u c P} e 1 Proof— Cf. *171 and *14'21 and *22'32.

*2G. On Cogredient Points.

*26'11. Proposition. — Assuming (II.-VL, IX-XL, XIII.) HpR, if a point possesses two members which are cogredient to each other, it is a cogredient point. In symbols,

h.'. (II.-VL, IX-XL, XIII.) HpR . D :

A e pntK( . a, b e A . b e cogrdK('a . « ^ b . D . A e OOR(

Proof— Cf. *14'21 and *20'42 and *22'61.

*26'22. Proposition.— Assuming (II.-VL, IX. -XL, XIII.) Hp R, if A is a cogredient point and a is a member of A, then A is identical with «,' a u cogrdlu' a. In symbols,

|- . •• (II.-VI. , IX. -XL , XIII. ) Hp R . : A e OOR( . a e A . D . A = t' a U cogrdR(' a Proof— Cf. *14'21 and *20'42 and *21'41 and *22'61.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 515

*26'23. Proposition. — Assuming (II. -VI, IX -XL, XIII. )HpR, there is one and only one cogredient point lying in the punctual associate of an objective real. In symbols,

|- . '. (II. -VI, IX.-XI., XIII.) Hp R . s : a e OH, . 3 . «>R( n assKYa £ 1

Proof.— Cf. *26'22.

*26'24. Proposition. — Assuming (II. -XL, XIII.) HpR, there are at least two points, not cogredient points, lying in the punctual associate of an objective real. In symbols,

h .-. (II.-XL, XIII.)HpR.D: «eOR( . D . Nc'{assR(Va- <»„,} S 2 Proof.— Cf. *25'31 and *26'23.

*27. Ow Punctual L/ines.

*27'11. Proposition. — Assuming (II. -VI, IX.-XI.) Hj? R, if p and q are distinct planes, and p n q possesses a member, then assKt'p n assKt'q is identical with &ssKt'(p n q). In symbols,

|- .". (II.-VI, IX.-X

p, gepleR( .p ?± q . R\(p n q) . z> . &ssKt'p n &ssRtfq = assR('(p n q)

Proof.— Cf. *16-33 and *20'21.

*27'12. Proposition. — Assuming (II.-VI, IX.-XII)HpR, if p and q are distinct planes, and a point, not a cogredient point, lies in the punctual associates both of p and also of q, then p n q possesses one and only one member. In symbols,

tV. (II.-VI, IX. -XII.) HpR . ^ :

p, q e pleu, . p ^ q . 3 ! {(assK(^p n assK(' q) - OOE(} . D . p n q e 1

Proof.— Cf. *1 6 -21 and *22'51.

*27'13. Proposition. — Assuming (II-XIII.) HpR, if p and q are distinct planes, then if p n q possesses one member, there are non-cogredient points lying in the punctual associates both of p and of q ; and also conversely. In symbols,

I- : : (II-XIII) Up R . D .'. p, q e pleR( . p * q . D :

3 ! {(assR('jp n assK('g) - a>R<} . = . p n q e 1 Proof.— Cf. *26'24 and *27'12.

*27'21. Proposition. — Assuming (II.-VI, IX.-XLT.) Hp R, if m is a punctual line possessing a non-cogredient point, then there exists an objective real such that m is its punctual associate. In symbols,

I .'. (II -VI, IX.-XII.) HpR . D :

melinR( . ft!(m- COR() . D. (ga) . aeOR( . m =

Proof.— Cf. *27'11-12.

3 u 2

516 DR. A. N. WHITEHEAD ON

*27'22. Proposition.— Assuming (II.-XIII.) HpR, a punctual line possesses either more than one non-cogredient point or no such point. In symbols,

I-.-. (IL-XIIL)HpR . D : me ling, . g!{ro- »»} . D . Nc'(m- OOB) > 1

Proof.— Cf. *20'22 and *26'24 and *27'21.

*27'23. Proposition.— Assuming (II.-VI., IX.-XIII.)HpR, a punctual line, possessing a non-cogredient point, possesses one and only one cogredient point. In

symbols,

.D :melinR( . a!(m-ooK) .^.mn ooR(el

Proof.— Cf. *26'23 and *27'21.

*27-31. Proposition.— Assuming (II.-VL, IX.-XII.)HpR, if m is a punctual line possessing a non-cogredient point, and A and B are two distinct points lying in it, then m is the punctual associate of A n B. In symbols,

I- /. (TI.-VL, IX.-XIL) HpK . D :

melinB, . g;!(m- oo){() . A, Be m . A ^ B . D . meassK('(A n B)

Proof.— Cf. *14'21 and *27'11'12.

*27'41. Proposition. — Assuming (II. -XI.) Hp 11, if a is any objective real, then there exist two planes p and q such that « is the sole member of p n q. In symbols,

|- .'. (II. -XL) Hp R . r> : a e OR( . D . (ftp, q) . p, q e pleR( . p n q = t'a

Proof.— Cf. *12'21 and *14'12'14 and *22'41 and *25'31.

*27'42. Proposition. — Assuming (II.-XIII.) HpR, if a be an objective real, then assK('i'a is a punctual line with a non-cogredient point, and conversely, if m is a punctual line with a non-cogredient point, there exists an objective real a such that m = assju't'a. In symbols,

I" .'. (II.-XIII.) HpR . D : melinjj; . g!(m- COK() . = . (get) . a e. ORt . m = assfi,Ya

Proof.— Cf. *27-ll'13-21-41.

Note. — This proposition, *27'42, establishes the connection between the objective reals and the punctual lines.

*27'43. Proposition. — Assuming (II.-XIII.)HpR, if a and c are cogredient, they are distinct. In symbols,

I" . '. (II.-XIII. ) Hp R . D : c € cogrdHt' a . D . a ^ c

Proof.— Cf. *20'31-41 and *27'42.

*27'51. Proposition. — Assuming (IL-XIV.)HjpR, if A is any non-cogredient point

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 517

and B is any other point, then A n B possesses one and only one member. In symbols,

|- .-. (II.-XIV.) Hp R . D : A e pntR( - OOR( . B e pntR( .A^B.o.AnBel

Proof.— (i) If B is non-cogredient, cf. *14'21 and *20'51 and *2271. (ii) If B is cogredient, then (cf. *14'14) let b be a member of B. Then (cf. *26'24) there is on b a non-cogredient point D. Hence by (i) A n D possesses a single objective real, d say. Hence (cf. *14'12) b and d are coplanar and cm,w'(i'6 u i'd) is a plane whose punctual associate possesses both A and B. Also, since a point is three-dimensional, B possesses another objective real, c say, not coplanar with b and d. Hence by similar reasoning cmR('(i'6 u i'c) is a plane, not identical with cmR('(i'6 u i'd), whose punctual associate also possesses A and B. Hence (cf. *27'13) these two planes have one objective real in common, and hence (cf. *16'33) this objective real is a member of A n B, and hence (cf. *14'21) A n B possesses one and only one member.

*27'52. Proposition. — Assuming (II.-XIV.) HpR, if A be a non-cogredient point and B be any other point, then assR(' (A n B) is a punctual line with a non-cogredient point, and conversely, if m be a punctual line with a non-cogredient point, then there exist two points A and B, such that A is not cogredient and m is identical with assR('(A n B). In symbols,

(gA, B) . A e pntR( - OOR( . B e pntK( . A j± B . m = assR('(A n B) Proof.— Cf. *27'22-31-42-51.

*28. On Figures.

*28'01. Proposition. — Assuming (II.- VI., IX.-XI.)HpR, if t be an instant of time, there exists at least one punctual plane, not the plane OOR(. In symbols,

[- . '. (II.-VL, IX -XL) HpR.3:*eT.D.a! (pplew - t'

Proof.— Cf. *12'42 and *22'41.

*28'11. Proposition. — Assuming (II.-XIII.) HpR, if p be any punctual plane, not the plane ooR(. it possesses at least three non-cogredient points, which are not collinear. In symbols,

| . '. (II.-XIII.) Hp R . D : p e ppleR( . i'

(gw) . u f. (3 n cls'p). u n °OR« = A . -~ collHt!w

Proof.— Cf. *14'2l and *26'24 and *27'21'31.

Note.—Cf. *16'42 and the note on it.

*28'12. Proposition. — Assuming (II.-XIII.)HpR, if p be any punctual plane, not

518 DK. A. N. WHITEHEAD ON

the plane »„,, there exists at least one non-cogredient point not lying in it. In symbols,

|- . '. (II.-XIII.) Hp R . D : p € ppleE< - t < OCR, . a . (3 A) . A e (pntB( - COR,) -p

Proof— Cf. *12'21 and *16'32 and *22'41 and *26"24 and *28'11.

*28'21. Proposition— Assuming (II.-XIV.) HpR, if a: be any objective real and p be any plane, then either the punctual associates of x and p have one and only one common member, or x is a member of p. In symbols,

h : : (II. -XI V.) Hp R . D . '. x e OK( . p e pleK( . ^ : assR(Ya; n assR(> c 1 . V . x ep

Proof. _ Take (cf. *26'24 and*28'll)two non-cogredient points A and B upon x, and a non-cogredient point C in assR(> but not on x. If either A or B lie in uss^'p, then cf. *16'32. If neither A nor B lie in ass,t/_p, then (cf. *16'32 and *27'5l) cm,,/ {(B n 0) u (A n C)} is a plane possessing x, and its punctual associate possesses C. Hence (cf. *22'51) there is a common member of p and this plane, y say, and x and y are coplanar, hence (cf. *10'4 and *1G'11) the punctual associates of a; and y possess a common point. Hence assR,'i'ic and assKt'p possess a point in common, and then cf. *16'32.

*28'22. Proposition. — Assuming (II. -XIV.) Hp II, if p and q are punctual planes, and p is not identical with QOH(, and p n q is contained in QOH<, then p n q and p n oo u( are identical. In symbols,

I".-. (II. -XIV.) HpR .=>:p, tfepplejj, .p ^ com . jongcooR( . z> .pr\q =pn<x>Rt

Proof — If p and q are identical, then pcooK(, but (cf. *28'll) this is impossible. Hence p ^ q. If q = OOH(, then p n q = p n oolt(. Assume q ^ OOR(. Then (cf. *20'6l) there exist planes, p' and q' say, such that p is assm'p' and q = assR,'(/. Since pru/cooK!, there is (cf. *26'24) no objective real common to p' and q'. Hence (cf. *28-21) upon every objective real possessed by p' there is one and only one point lying in </. Hence (cf. *26'23) p n q = p n ooK(.

*28'31. Proposition. — Assuming (II. -XIV.) Hp R, in every punctual line there lie at least three points. In symbols,

[ . : (II.-XIV.) Rp R . a : m e linH( . 3 . Nc'm ^ 3

Proof — If m possesses non-cogredient points, then cf. *27'22'23. If m is contained in ooRj, then (cf *28'22) m is identical with pr\a°RI, where p is a punctual plane. But (cf *27'51'52 and *28'll) there are three distinct punctual lines contained in p, meeting two by two in three non-cogredient points ; then cf. *27'23.

*28'32. Proposition. — Assuming (II.-XIV.) HpR, a punctual line is the common meeting of two punctual planes, and conversely. In symbols,

h . '. (II.-XIV.) Hp R . D : linB( = m {(ftp, q) . p, q e ppleB« .p^q.m=pnq}

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 519

Proof. — The direct proposition follows from *20'22'61. For the converse, let p and q be a pair of punctual planes. If neither p nor q be oo8f, then cf. *20'22'61. Consider now p n ooK(, where, p is distinct from OOK(. Take (cf. *28'12) a non- cogredient point A, not in p. Also (cf. proof of *28'3l) there are two distinct points B and C in p n OOK(. Hence (cf. *27'5l) cmR<'{(A n B) u (A n C)} is a plane, and its punctual associate, q say, possesses A and B and is distinct from p. Hence (cf. *27'23 and *28'22) p n q is identical with p n OOK(. But (cf. *28'22) p n q is a punctual line, and hence p n oou( is a punctual line.

*28'33. Proposition. — Assuming (II.-XIV.) HpR, two distinct points lie in one and only one punctual line. In symbols,

|- .'. (II. -XIV.) Hp R . D : A, BepntK, . A * B . D . m{meliriJM . A, Bern} e 1

Proof. — Firstly, only one punctual line (if any) possesses both A and B (cf. *27'31 and *28'22-32). Secondly, to prove that a punctual line exists possessing both A and B. If either point is non-cogredient, cf. *27'52. If both points are cogredient, then (if. *28'11'12) two non-cogredient points C and D exist such that the four points A, B, C, D are not punctually coplanar. Hence (cf. *27'5l) the meeting of the punctual associates of cmm' {(A n C) u (B n C)} and of cmK(' {(A n D) u (B n D)} is a punctual line possessing A and B.

*28'41. Proposition. — Assuming (II.-XIV.) HpR, three points, which are not collinear, lie in one and only one plane. In symbols,

h . '. (II.-XIV.) Hp II . D : u e 3 n pntK( . -*~ collK( ! u . D . p {p e ppleIH . u c p} e 1

Proof. — If the three points are all cogredient, then (cf. *28'22'32) (»R4 is the only punctual plane which possesses them all. If the three points are A, B, C, and A be non-cogredient, then (cf. *27'51 and *28'32) the punctual associate of cmn('{(A n B) u (A n C)} is a punctual plane, possessing A, B, and C, and is the only one.

*28'42. Proposition. — Assuming (II.-XIV.) HpR, three punctual planes, which do not meet in a punctual line, meet in one point. In symbols,

h . •. (II.-XIV.) Hp R . D : u e 3 n els' pplem . ^ . n ' u e linK« u 1

Proof. — Let p, q, r be the three punctual planes. Assume that p and q are neither the punctual plane ooBi. If q n r is contained in OOH(, then cf. *20'22'61 and *27'23 and *28'22'32. If q n r is not contained in o>R(, then cf. *27'12 and *28'21.

*30. Perspective.

A few propositions on perspective (cf. *20'3t'32'33) are required as a preliminary to the discussion of the point-ordering relation (cf. *20'5l).

520 DR. A. N. WHITEHEAD ON

*30'1. Proposition. — Assuming (II.-XIV.)H/)R, if two figures are in perspective, their cardinal numbers are equal and each greater than one. In symbols,

h .-. (II. -XIV.) HpR . = : uperspRtv . D . Nc'w = Nc'v . Nc'w > 1

Proof. The equality of the cardinal numbers follows from the definition ; also if

both figures were unit classes, then (cf. *28'33) they would be collinear.

*30'3. Proposition. — Assuming (IL-XIV.)HpK, if the figure u is in perspective with the figure v, and also with the figure w, and if u, v, w are respectively collinear, and the punctual lines respectively containing u, v, w possess a common meeting, then either v is in perspective with w, or the joint class of v and w (i.e., v u w) is collinear. In symbols,

I" : : (II. -XIV.) Hp R . D . '. u persp,i( v . u perspB( v . m, m', m" e linm .

u c m . v c m' . w c m" . g; ! (m n mf n m") . D : v perspB, w . V . m' = m"

Proof. — DES ARGUES' well-known propositions respecting triangles in perspective being coaxial, and its converse, can now (cf. *28"11"12"31132'33"41'42) be proved. Then by drawing a figure for the present proposition the conclusion easily follows from some pure geometrical reasoning.

*31. Tfie Point-Ordering Relation.

It will be proved in this section that the point-ordering relation (Rpn) has at any instant the same properties as the essential relation of Concept I. (cf. *31'3). It follows that the ordinary Euclidean geometry holds of the figures of Concept V., the points at infinity being the points of the punctual plane ooR(, and the metrical ideas being introduced by appropriate definitions.

*31-11. Proposition.— Assuming (II. -XL, XIII, XIV.) HpR, the class Rpll;(;;;£) is identical with the class of non-cogredient points. In symbols,

I- .'. (II.-XL, XIII, XIV.)H^R . =,. Rpn;(;;;0 = pntM- oof

'St

Proof.—Cf. *3'42 and *20'51 and *22'7l and *26'24.

*31'12. Proposition. — Assuming (II. -XV.) HpR, if a is a punctual line, and A and B are two non-cogredient points on it, then a, without its cogredient point, is identical with the whole class formed by Rim;(;AB«) and Rpn:(A;B£) and R;(AB;«) together with A and B added as members. In symbols,

I-.-. (II-XV.)HpR.D: aelinK. A, Bea-oo^. A ^ B .3.

a- QOB( = Rpn;(;AB*) u Rpn;(A;BO u Rpn>(AB;«) u I'A u i'B

Proof— The identity is to be proved by showing that each class contains the other. For one half of the proof, cf. *2272 and *27'2l. For the other half, cf. *20-5l and*27-21'31.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 521

*31'21. Proposition. — Assuming (II. -VI. )HpR, the point-order ABC implies the point-order CBA. In symbols,

Proof.— Cf. *1'63 and *20'51.

*31'22. Proposition.— Assuming (II. -VII, IX.-XI.) HpR, if A, B, 0 are in the point-order ABC, then A, B, and C are distinct. In symbols,

I- .-. (II.-VIL, IX. -XI.) Hp R . a : R,,ni(ABCO . D . A ^ B . A ^ C . B ^ C

Proof. — There are (cf. *20'51) interpoints u, v, w on a common objective real a, such that ~R-m'(uvwt) and [ABC] perspE( [MK(vKiwm], where (cf. *20'23 and *25'32) um, vRt, wRt are the dominant points of u, v, iv. But (cf. *1'62 and *22'33) uRl, vRt, w}it are distinct points. Hence (cf. *20'32) A, B, C are distinct points.

*31'23. Proposition. — Assuming (IL— XV.)HpR, the point-order ABC is incon- sistent with the point-order BCA. In symbols,

|- .'. (II. -XV.) Hp R . : Rim;(ABa) . D . - R;(BCAO

Proof. — Since this proof is long, the paragraphs will be numbered for reference by (i), (ii), &c., prefixed.

(i) If u, v, w are interpoints on the objective real a, and u', v', w' interpoints on the objective real a', and a and a' are cogredient, and [ihuvntWui] perspR( [u'Rtv'ntw'ni], then (cf. *20'23'41 and *25'32) u, v, iv and u', v', iv' must agree in interpoint order, and (cf. *1'64) each set of interpoints has only one (if any) interpoint order (counting uvw and ivvu as the same order), and (cf. *27'43) a and a' are distinct.

(ii) From *20-51, R11U;(ABC<) implies (a) that A, B, C are on an object real d; (ft) that there are interpoints u, v, w on an objective real a, cogredient with d ; (y) that l^'(uvwt) ; (8) that [ABC] perspRj [uEtvRtwRt~] ; and (e) (cf. *27'43) that a and d are distinct.

(iii) Assume that Rpn:(BCA^) also holds. Then, in addition to the entities of (ii), there exist interpoints u', 1/, iv' on an objective real a', satisfying all the conditions of (ii) without changes, except that (ii, y) becomes Rin;(i/wV£). This assumption (iii) will now be proved to be absurd.

(iv) From XIIIHpR(c/: *22'61) and *26'11'22, d, a, a' are copunctual and a and a' are either cogredient or identical.

(v) Hence (cf. *30'3) either (Case I.) [uRtvRtwRt~] perspB( [u'Rtv'Rtw'Rt'] or (Case II. ), a and a' are identical.

(vi). Case I. — We have [_uRtvRtwRt']perBpRt[u'Rtv'RtU''Rt~]. Hence (cf. *20'41) the interpoint orders of u, v, w and u', v', v/ must agree. Hence, from (ii, y), Rin;(u'v'iv't) holds. But (cf. *1'64) this is inconsistent with ~Rin-(v'iv'u't). Hence Case I. cannot hold.

VOL. CCV.— A. 3 X

522

DK. A. N. WHITEHEAD ON

(vii). Case II. —We have a and a' identical. Now (cf. *31'22) A, B, C are distinct points. Hence [cf. (ii, 8) and *30'1] uRt, vRt, wRt are distinct points, and [cf. (ii, £)] they are collinear. Hence, by XVHpE (cf. *2272), they are in some point-order. Hence [cf. (ii)] three interpoints u", v", w" exist on a common objective real x, which is distinct from a and cogredient with it, and also [u"mv"mw"Rt~\ perspR( [uRtvRtivm~].

(viii). Hence Case II. divides into two subclasses, either (Case II., «) x is not identical with d, or (Case II., ft) x is identical with d.

(ix). Case II., a. — a and a' are identical and distinct from both d and x, and d and x are distinct; also a, d, x are cogredient and therefore (cf. *26'11'22) copunctual. Hence [cf. (ii, 8) and (vii) and *30'3] we have [ABC] perspK( \u" *$' *&/' TU\- Hence [cf. (iii, 8) and *30'3] [u'Rtv'Rtu>'nt~\ perspR{ [u\tv"Jttw"Rt~]. Hence [cf. *20'41 and (ii, y) and (vii)] Ilin; (u'v'iv't}. Hence (cf. *1'64) Case II., a, cannot hold.

(x). Case II., ft. — a and a' are identical, x and d are identical, and a and d are cogredient and distinct. Since (cf. *20-5l) A, B, C are not cogredient points, they are distinct from u'K, v'K, w/K, since the only point common to the punctual associates of a and d is a cogredient point. Thus none of u'm, v'm, it/Rt can be cogredient points. Hence (cf. *28-33) there is a punctual line joining B and w'm which does not possess A or u'Rt or V (cf. figure annexed). Hence (cf. *28'42) there is at least one other

V

If!

B"

point, A" say, lying in the punctual line joining A and u'Rt in addition to V and u' and A. Hence (cf. *28'33) there is a punctual line, z say, joining A" to the cogredient point common to the punctual associates of a and d. This punctual line must meet (cf. *28'42) the punctual lines VB and VC (cf. figure annexed) in the points B" and C". Hence

[A"B"C"]perspm[ABC], and hence (cf. *30'3) we have

and hence (cf. *30'8) we have

[A"B"C>erspm [WV>

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 523

But also

[A"

and hence (cf. *30'3) we have

Thence, by the same reasoning as that in (ix) for Case II., «, it follows that Case II., ft, cannot hold.

(xi). Hence neither Case I. nor Case II. can hold ; and therefore the proposition follows.

*31'3. Proposition. — -The point-ordering relation (R,,n) satisfies all the axioms satisfied by the essential relation of Concept III., except the axiom of persistence for that concept.

Proof. — In order to prove this, we make a comparison, as in Concept III., with the axioms of Concept I.

For I Hp R of Concept I., cf. *25'12 and *2G'24 and *3ril.

For II Hp R of Concept I. , cf. *3 1 '2 1 .

For III Hp R of Concept I. , cf. *3 1 '23.

For IV Up R of Concept I, cf. *31 "22.

For V Hp R of Concept I. , cf. XIV Hp R (*22 7 1 ).

For VI Hp R of Concept L, cf. *28'33 and *31'12.

For VII Hp R of Concept I, cf. *28-01'll.

For VIII Hp R of Concept L, cf. XVI HpR (*2273).

For IX Hp R of Concept L, cf. *28'12.

For X Hp R of Concept I. , cf. *28 -42.

For XI Hp R of Concept I, cf. *XVII Hp R (*2274).

For XII Hp R of Concept L, cf. *26'23 and *28'33.

Note. — -In order to complete this comparison, it must be noticed that it follows from *31'12 that the punctual line, with its cogredient point excepted, is the line as defined on the analogy of Concept I. Also, it follows, from *28'32 and *28'42 and the propositions of *31, that the punctual plane, with its cogredient points excepted, is the plane as defined on the analogy of Concept I. Then the transition to projective geometry is made, not by constructing a fresh type of points (the projective points), but simply by putting back the class ( <»R() of cogredient points. Metrical geometry is then constructed in the well-known way,t making the plane ( COE() of cogredient points to be the plane at infinity.

The Extraneous Relation. — For the purpose of enabling velocity and acceleration to be measured, an extraneous relation is required, in all respects similar to those required in Concepts III. and IV., and the description already given need not be repeated.

* Cf. VEBLEN, loc. tit., for a sketch of this method ; also CLEBSCH and LINDEMANN, loc. at.

3x2

524 DE. A. N. WHITEHEAD ON

The Corpuscles.— We may distinguish five types of points. A point of Type (1) contains no interpoints, and consists only of its nonsecant part (of. *20'231). A point of Type (2) contains a single interpoint and no nonsecant part. Such a point is a single interpoint. A point of Type (3) contains a single interpoint together with a nonsecant part. A point of Type (4) contains many interpoints with no nonsecant part. A point of Type (5) contains many interpoints together with a nonsecant point.

We seem to be precluded from considering the " particles" to be stable points by the same difficulty as to the resulting permanence of collinearity, which was explained in considering the corpuscles of Concept IV. It is evident that at this stage many subdivisions of Concept V. are possible, in respect to the ideas which may be formed of the nature of the corpuscle. The following sketch of a possible development is o-iven because of its superior simplicity, and also because of a certain consonance which it possesses with some modern physical ideas.

It is evident that volumes, in which, in some sense, there is an excess or a defect of interpoints, can be conceived as being charged with one or other of the two sorts of electricity. This idea is taken as the basis of the following brief outline of a possible development of the concept. Let the interpoints be identified with negative electricity and the nonsecant parts of points with positive electricity. A point of type (1) is a negative electron; a point of type (2) is a positive electron. The persistence of existence of an isolated electron of either type is to be defined by persistence of type and continuity of motion. If the electron is not isolated, consider, for example, a volume in which electrons of type (2) either compose all the points, or, at least, are everywhere dense. Then the persistence of such a collection of electrons must be considered as a whole, and is defined, as in the simpler case, by persistence of type and continuity of motion.

Three methods of procedure now suggest themselves, either (Case I.) to assume that the electrons consist of single points, so that a corpuscle is a volume containing a large finite number of points of type (2), and a small finite number of points of type (l), or (Case II.) to assume that a corpuscle is a volume in which points of type (2) are (at least) everywhere dense, and which contains a finite number of points of type (1), or (Case III.) to assume that an electron of either type is essentially a volume (possibly with internal boundaries) in which points of the appropriate type are at least everywhere dense. In Case III. a corpuscle will be a relatively large electron of type (2) containing within it a finite number of relatively small electrons of type (1). Case III. has the merit, such as it is, of making the " inverse square" law of electricity appear somewhat natural. The field of force " at a point" produced by an electron may be conceived as proportional to the number of objective reals shared in common by the point and the " electric points" in the electron, and also to the number of these electric points. The number of electric points would be measured by the mass of the electron, the number of objective reals by the solid angle subtended at the point by the electron.

MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 525

What is wanted at this stage is some simple hypothesis concerning the motion of objective reals and correlating it with the motion of electric points and electrons. From such a hypothesis the whole electromagnetic and gravitational laws might follow with the utmost simplicity. The complete concept involves the assumption of only one class of entities as forming the universe. Properties of " space" and of the physical phenomena " in space " become simply the properties of this single class of entities. In regard to the simplification of the preceding axioms, viz., of (I. -XVI.) Hp R, the ideal to be aimed at would be to deduce some or all of them from more general axioms which would also embrace the laws of physics. Thus these laws should not presuppose geometry, but create it.

t 527 J

INDEX

PHILOSOPHICAL TRANSACTIONS.

SERIES A, VOL. 205.

A.

ABNEY (Sir W. DE W.). Modified Apparatus for the Measurement of Colour' and its Application to the Determination of

the Colour Sensations, 333.

ARCHIBALD (E. H.). See STBELE, MCINTOSH, and ARCHIBALD. Atmospheric electricity in high latitudes (SIMPSON), 61.

B. BURRARD (Lieut.-Col. S. G-.). On the Intensity and Direction of the Force of Gravity in India, 289.

C.

Chlorine, atomic weight of, and action on glass (DlxON and EDGAR), 169.

Colour, modified apparatus for measurement of, and application to determination of colour sensations (ABNBY), 333.

Colours in metal glasses, metallic films, &c. (GARNETT), 237.

Conducting solvents, halogen hydrides as (STEELE, ic.), 99.

Cordite, nitro-cellulose, &c., explosive decomposition of (NOBLE), 201 ; rate of combustion and explosive pressure of

(PBTAVEL), 357.

CUNNINGHAM (E.). On the Normal Series satisfying Linear Differential Equations, 1. C0THBBHTSON (C.) and PRIDEAUX (E. B. E.). On the Refractive Index of Gaseous Fluorine, 319.

D.

DBNISON (R. B.) and STEELS (B. D.). On the Accurate Measurement of Ionic Velocities, with Applications to Various Ions,

449.

Differential equations, linear, normal series satisfying (CUNNINGHAM), 1. DIXON (H. B.) and EDGAR (E. C.). The Atomic "Weight of Chlorine : an Attempt to determine the Equivalent of Chlorine

by Direct Burning with Hydrogen, 169.

B.

EDGAR (E. C.). See DIXON and EDGAK. Explosives, researches on, Part III. (NOBLE), 201 ; solid and gaseous, experiments on pressure of explosions (PETAVEL), 357.

F.

Fluorine, refractive index of gaseous (CUTHBBBTSON and PRIDEAUX), 319.

VOL. CCV. — A 401. 3 Y 22.5.06

528 INDEX-

Or.

GfABNETT (J. C. MAXWELL). Colours in Metal Glasses, in Metallic Films, and in Metallic Solutions.-!!., 237. Geometry, on the axioms of (WHITEHEAD), 465. GHass, coloration by gold, silver, &c. (&ABNBTT), 237. Gravity, intensity and direction in India (B0RBAED), 289.

H.

Halogen hydrides as conducting solvents (STEELE, &c.), 99.

HBKSCHKL (Sir W.). Fifth and Sixth Catalogues of the Comparative Brightness of the Stars, in continuation of those printed in the ' Phil. Trans.' for 1790-99, prepared from the MS. Records by Col. J. HEBSCHEL, 399.

I.

Iodine and sodium thiosulphate, reaction between (UixoN and EDGAR), 169. Ionic velocities, accurate measurement of (DENISON and STEBLE), 449.

M.

MclNTOSH (D.). See STBELB, MC!NTOSH, and ARCHIBALD.

MAoMAHos (P. A.). Memoir on the Theory of the Partitions of Numbers.— Part III., 37.

Magic squares, construction of general (MAcMAHON), 37.

Mathematical concepts of material world (WHITEHEAD), 465.

Metals, colouring properties of, in glasses, films. &c. (Gt-ARNETi), 237.

N. NOBLE (Sir ANDREW). Researches on Explosives. — Part III., 201.

P.

Partitions of numbers, theory of (MAcMAHON), 37.

PETAVEL (J. E.). The Pressure of Explosions. — Experiments on Solid and Gaseous Explosives. Parts I. and II., 357.

Point, on the projective (WHITEHEAD), 465.

PRIDKAUX (E. B. R). See CITTHBEHTSON ami PRIDEAUX.

R.

Radio-activity, atmospheric, in high latitudes (SIMPSON), 61.

Refractive index of gaseous fluorine (CUTHBERTSON and PEIDEAI'X), 319.

S.

SIMPSON (Q-. C.). Atmospheric Electricity in High Latitudes, 61. Stars, liftii and sixth catalogues of the comparative brightness of the (HiiESCHEL), 399.

STEELE (B. ]).), MC!NTOSH (D.), and ARCHIBALD (E. IT.). The Halogen Hydrides as Conducting Solvents. — Parts I. to IV., 99. (See also DENISON and STEELS.)

W.

WHITEHEAD (A. N.). On Mathematical Concepts of the Material World, 465.

HABBISON AND SONS, PRINTERS IN OBDINABT TO HIS MAJESTY, ST. MABTIN's LANB, LONDON, W.C.

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Cordite 0-035 inch diameter. Gravimetric density = 0'0744. Charge uniformly distributed in spherical enclosure.

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N.B. — The above figures are reproduced full size from the original negatives. The series of lines which cross the records in a direction nearly parallel to the axis of time correspond to successive portions of the cooling curve. They represent the pressure after 1, 2, 3, &c., complete revolutions of the chronograph drum. In most cases the light was cut off and the record stopped less than a second after the explosion, thus leaving the lower part of the diagram clear.

0'

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