= LT> ^C^ Ho "CO I U I I . g . , SCIENTIFIC PAPERS Presented by the University UNIVERSITY OF SAINT ANDREWS FIVE HUNDREDTH ANNIVERSARY MEMORIAL VOLUME OF SCIENTIFIC PAPERS Contributed by Members of the University EDITED BY WILLIAM CARMICHAEL M'INTOSH PROFESSOR OF NATURAL HISTORY JOHN EDWARD ALOYSIUS STEGGALL PROFESSOR OF MATHEMATICS JAMES COLQUHOUN IRVINE PROFESSOR OF CHEMISTRY ./ PUBLISHED BY THE UNIVERSITY MCMXI Q 4-1 63 PREFACE THIS volume of papers is published with a twofold object: that the distinguished scientific guests of the University should receive an appropriate remembrance of their sojourn amongst us during the celebration of the five hundredth anni- versary of Saint Andrews University was the first considera- tion ; the second was that the present happy occasion affords a favourable opportunity for making a record, in some measure typical, of the kind and quality of the scientific research that is being pursued at this time by her children. The editors have been more concerned with the delineation of the picture, which they hope is presented within these covers, of the real unadorned intellectual work that is being performed by the members of the University than with the introduction of highly technical or elaborate studies. They are well aware that many of her distinguished graduates have given, and are giving, to the world scientific literature of the highest standard ; and they are equally aware of the limitations imposed by time and space upon the selection that they have made for this volume. They feel that careful study of the authors' names will convince the candid reader that an attempt has been made to give a catholic representation of the present scientific life at Saint Andrews and at Dundee. They have included work from professors, from assistants, and from graduates, some of whom have but recently emerged vi SCIENTIFIC PAPERS from tutelage ; they have also included papers from members of the University who are now working elsewhere. On an occasion like the present even the strict and logical realm of Science cannot but be enveloped by the glamour inseparable from the circumstances which have called forth this book, and influenced by the thoughts that arise naturally from the consideration of the great space in time that lies between the rude beginnings of the University and the elaborate development of these later years. Amidst the reflections evoked none is stronger, and none should be more reverently valued by the disciple of Science, than that which reminds us of the debt which we owe to those who have gone before us. To the deep and difficult founda- tions that they laid, to their patient and sometimes thankless and unrewarded labour, labour too often performed with scanty or ill-adapted tools, we owe our present points of vantage, our present ambitious intellectual structures. They have laboured, and we have indeed entered into their labours. To their memory we dedicate this book, in the humble hope that in the future history of our dear University some memory may also remain of our familiar friends as helpers in carrying the torch of learning into still umllumined recesses ; in extending the empire of the human intellect ; and in giving to their fellowmen a nearer vision of the absolute yet ever unattainable truth. J. E. A. STEGGALL Chairman of the Editorial Committee PRAEFATIO Hums libri edendi ilia ratio erat, primum ut viri doctissimi rerum naturae indagandae dediti munusculo quodam accepto meminissent se apud Andreanos, dum Almae Matris natalis quingentesimus celebraretur, per aliquos dies commorari ; deinde quod hoc tarn felici tempore oblata erat occasio exemplis quodam modo idoneis demonstrandi qua ratione, quanta diligentia nostrates rerum naturam exquirere co- narentur. Quern librum cum ei quibus hoc curae fuit componere vellent, illud magis sibi proponendum censebant, ut sine fuco et fallaciis docerent qualibus in studiis Andreani versarentur, quam ut opera exquisita arte elaborata in medium proferrent. Sciebant sane complures nostrorum libellos in quibus de rerum natura quaereretur et scripsisse optimos et scribere ; prae- terea se multa, dum hunc librum intra breve temporis spatium edere conantur, invitos omisisse. Illud vero persuasum sibi habebant, si quis scriptorum nomina diligenter perlegisset, eum affirmare non dubitaturum id saltern temptatum esse, ut demonstraretur quo modo cum Andreapoli turn Taoduni haec studia vigerent. Etenim alia professores scripserunt, alia lectores, alia alumni quorum nonnulli modo e statu pupillari excesserunt, alia qui antea apud nos versati, nunc alibi in his studiis versantur. Atque hoc tali tempore eis quoque qui omnia diligenter perpendere et ad certam rationis normam dirigere solent vii viii SCIENTIFIC PAPERS necesse est profecto ut mentis aciem praestringat illius diei species, dum secum reputant quanta interfuerit annorum series inter initia ilia Academiae humillima atque hunc florentissimum eiusdem statum. Illud vero summa diligentia, summa reverentia recordari decet, illis qui ante nos operam in rerum natura indaganda collocaverunt maximam a nobis deberi gratiam. Nam quod illi semper summa patientia, interdum nulla gratia nullo munere, saepe vel paucis vel male aptis instrumentis quasi fundamenta iacere moliti sunt, idcirco nos iam haec excelsa aedificia tenemus, in hoc tarn sublimi fastigio stamus. Illi laboraverunt, et nos in labores eorum introivimus. Itaque illis hunc librum ita dicatum volumus ut simul sperare audeamus nonnullam apud posteros servatum iri memoriam amicorum quoque et familiarium nostrorum qui doctrinae facem in penetralia etiam nunc luce carentia inferre, ut latius mens humana dominetur efficere, denique ad scientiam illam perfectam atque absolutam, quam tamen nemo umquam assequi possit, propius accedere pro virili parte contendunt. T. R. M. CONTENTS MATHEMATICS AND PHYSICS PAOE PREFACE v IDEM LATINE REDDITUM vii THOMAS ROSS MILLS Lecturer in Latin at University College Dundee CONCRETE REPRESENTATIONS OF NON - EUCLIDEAN GEOMETRY 3 DUNCAN M'LAREN YOUNG SOMMERVILLE Lecturer in Applied Mathematics in the United College St Andrews ON THE ALGEBRAICAL SOLUTION OF INDETERMINATE CUBIC AND QUARTIC EQUATIONS .... 47 ROBERT NORRIE Assistant Lecturer in Mathematics at University College Dundee THE PROBLEM OF PARTITION OF ENERGY, ESPECIALLY IN RADIATION 93 WILLIAM PEDDIE Professor of Physics at University College Dundee ON THE ACCURACY ATTAINABLE WITH A MODIFIED FORM OF ATWOOD'S MACHINE . 99 JOHN PATRICK DALTON Carnegie Research Fellow at University College Dundee b x SCIENTIFIC PAPERS PiOK THE DEVIATION OF THE OSCILLATIONS OF A VISCOUS SOLID FKOM THE ISOCHKONOUS LAW . . .109 WILLIAM PEDDIE Professor of Physics at University College Dundee THE DISSIPATION OF ENERGY IN TORSIONAL OSCILLA- TION 113 JAMES BONN YM AN RITCHIE Carnegie Research Scholar at University College Dundee WAVE IMPACT ON ENGINEERING STRUCTURES . .129 ARNOLD HARTLEY GIBSON Professor of Engineering in the University and WILLIAM NELSON ELGOOD CHEMISTRY THE PREPARATION OF PARTIALLY METHYLATED SUGARS AND POLYHYDRIC ALCOHOLS . . 155 JAMES COLQUHOUN IRVINE Professor of Chemistry in the United College St Andrews A GENERAL REVIEW OF PURDIE'S REACTION . .177 CHARLES ROBERT YOUNG Formerly Assistant to the Professor of Chemistry in the United College St Andrews THE PREPARATION OF ANHYDRIDES OF ORGANIC ACIDS 225 WILLIAM SMITH DEN HAM Assistant in the Department of Chemistry at the United College St Andrews INDIUM AND THALLIUM IN CRYSTALLOGRAPHICAL RE- LATIONSHIP .241 ROBERT CHARLES WALLACE Lecturer on Geology and Mineralogy in the University of Msmitoba ' CONTENTS ri NATURAL HISTORY AND MEDICINE MM A BRIEF HISTORY OF THE CHAIR OF NATURAL HISTORY AT ST ANDREWS 273 WILLIAM OARMICHABL M'INTOSH Professor of Natural History in the United College St Andrews MAGNALIA NATURAE ... . 305 D'ARCY WENTWORTH THOMPSON Professor of Natural History at Uniyersity College Dundee ST ANDREWS AND SCIENTIFIC FISHERY INVESTIGA- TIONS 327 EDWARD ERNEST PRINCE Dominion Commissioner of Fisheries in Canada ON THE TOXICITY OF LOCAL ANAESTHETICS . . 343 CHARLES ROBERTSHAW MARSHALL Professor of Materia Medica in the UniTersity MATHEMATICS AND PHYSICS CONCRETE REPRESENTATIONS OF NON- EUCLIDEAN GEOMETRY INTRODUCTORY NOTE WHEN Euclid composed his logical system of the Elements of Geometry he was no doubt aware that it was based upon many unproved assumptions. Some of these assumptions are explicitly stated, either as postulates or as axioms (or common notions). The fifth postulate, often given as the eleventh or the twelfth axiom, is a lengthy statement relating to parallel straight lines, and is conspicuous by its want of any intuitive character: 'If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.' The universal converse of this state- ment is proved (with the help of another assumption, that the straight line is of unlimited extent) in Prop. 17, while its contrapositive is proved (again with the same assumption) in Prop. 28 of the First Book. Such considerations induced geometers and others to attempt its demonstration. Hundreds of such attempts have been made, with a display of great ingenuity. All these attempts, however, if they do not actually involve fallacious reasoning, are based upon some equivalent assumption either tacit or expressed. An entirely different mode of attack was devised by a Jesuit, Gerolamo Saccheri.1 He attempted to discover con- 1 G. Saccheri, Eudides ab omni naevo vindicalus, Milan, 1733. This work was for a long time forgotten. It was brought to the notice of Beltrami in 1889, who published an account of it in the Rendiconti of the lancei Academy. It has been translated into English by G. B. Halsted, Amer. Math. Mmi., 1-5 (1894-98), German by Stackel and Engel in Theorie der Paralldlinien, 1895, and Italian (Manuali Hoepli, 1904). 4 CONCRETE REPRESENTATIONS OF tradictions in the systems of geometry which would be evolved from a contrary assumption. The two geometrical systems which he develops at some length, which are characterised by the sum of the angles of a triangle being greater or less than two right angles, are the well-known non-Euclidean geometries, called by Klein Elliptic and Hyperbolic respec- tively. Saccheri himself, as also Lambert,1 who struck out the same line independently, believed that the geometry of Euclid was the only logical system, and it was not till Lobach- evsky 2 published the first of his epoch-making works in 1829 that non-Euclidean geometry emerged as a system ranking with Euclid's. With the name of Lobachevsky must always be associated that of Bolyai Janos, who arrived independently at the same results by similar methods. His work 3 was published as an appendix to a book of his father, Bolyai Farkas, in 1832. While Saccheri and Lambert both develop the two non-Euclidean geometries, neither Lobachevsky nor Bolyai admitted the possibility of Elliptic geometry, which requires that a straight line be of finite extent. To Riemann 4 is due the conception of finite space, but in his Spherical geometry two straight lines intersect twice like two great circles on a sphere. The conception of Elliptic geometry, in which the 1 J. H. Lambert, ' Theorie der Parallellinien,' Leipziger Mag. r. ang. Math., 1786. Reprinted in Stackel and Engel's Theorie der Parallellinien. 2 N. I. Lobachevsky, [On the Foundations of Geometry] (In Russian. German translation by Engel, Leipzig, 1898). Oeometrische Untersuchungen zur Theorie der Parallettinien, Berlin, 1840 (2nd ed., 1887), has been translated into English by Halsted (Austin, Texas, 1891 ). One of the most accessible of his papers is ' Geometric imaginaire,' J. Math., Berlin, 17 (1837). Several other papers, originally composed in Russian, have been translated into French, German, or Italian. Lobachevsky's researches first became generally known by means of the translations of Hoiiel in 1866-67. 3 J. Bolyai, ' Appendix, Scientiam spat ii absolute veram exhibens,' Maros-Vasarhely, 1832. Translated into English by Halsted (Austin, Texas, 1891). * B. Riemann, ' Uber die Hypothesen, welche der Geometric zu Grunde liegen," Gottingen, Abh. Qes. Wiss., 13 (1866). The work was written in 1864, but was not published till after the death of the author. English translation by Clifford, Nature, 8 (1873). NON-EUCLIDEAN GEOMETRY 5 straight line is finite, and is, without any exception, uniquely determined by two distinct points, is due to Klein.1 The method inaugurated by Saccheri has now been applied to most of the axioms or fundamental assumptions which lie at the basis of the Euclidean system, and a number of non- Euclidean geometries, many of them of considerable interest, have emerged. We shall be exclusively concerned, however, with the ' classical ' non-Euclidean geometries, Hyperbolic (Lobachevsky-Bolyai) and Elliptic (Riemann-Klein). While the development of Hyperbolic geometry in the hands of Lobachevsky and Bolyai led to no apparent internal contradiction, a doubt remained that contradictions might yet be discovered if the investigation were pushed far enough. This doubt was removed by the procedure of Beltrami,2 who gave a concrete interpretation of non-Euclidean geometry by Euclidean geometry, whereby the straight lines of the former are represented by geodesies upon a surface of constant negative curvature (surface saddle-shaped at every point. The ' pseudosphere ' or surface of revolution of the tractrix about its asymptote is a real surface of this description). Any contradiction in non-Euclidean geometry was thus shown to involve a contradiction in Euclidean geometry, and so both geometries must stand or fall together as d priori systems. Several other concrete representations have been obtained, and it is proposed to discuss the most important of these. § 1. We shall confine ourselves almost entirely to the representations of plane non-Euclidean geometry, but the extensions to three dimensions will be indicated. We shall also consider for the most part only those representations in 1 F. Klein, ' Uber die sogenannte Nicht-Euklidische Geometric,' Math. Ann., 4 (1871), 6 (1873). French translation in Ann. Fac. sc., Toulouse, HO (1897). 2 E. Beltrami, ' Saggio di interpretazione della geometria non-euclidea,' Oiorn. Mat., Napoli, 6 (1868). Extended to n dimensions in 'Teoria fondamentale degli spazii di curvatura costante,' Ann. Mat., Milano (2), 2 (1868). Both translated into French by Houel, Ann. 6c. Norm., Paris, 6 (1869). 6 CONCRETE REPRESENTATIONS OF which points are represented by points. To a point may correspond a single point or a system of points. In the latter case the system of points must be regarded as a single entity, and a curve which corresponds to a curve passing through a point P must pass through all the points which correspond to P. The representation is in fact effected by a point- transformation. The straight lines of the geometry will be represented by a system of curves depending upon two para- meters ; and in general any curve of the system must be uniquely determined when it has to pass through two distinct points. In addition to such considerations, which belong to analysis situs, it will be necessary also to establish the relations between the metrical properties of the geometry and those of its representation ; we must determine the function of the positions of two points which corresponds to their distance, and the function of the positions (or parameters) of two curves representing straight lines which corresponds to the angle between them. The distance and angle functions are not independent, for a circle may be denned either as the locus of a point which is equidistant from a fixed point, or as the envelope of a line which makes a constant angle with a fixed line, or again as the orthogonal trajectory of a pencil of lines. THE CAYLEY-KLEIN PROJECTIVE METRIC § 2. The simplest representation which suggests itself is obtained by representing straight lines by straight lines. The projective properties of non-Euclidean geometry are identical with those of Euclidean geometry if we take into account imaginary and infinitely distant elements. Pro- jective geometry is independent of the parallel-postulate. It is only in regard to metrical properties that there is any dis- tinction between non-Euclidean geometry and its representa- tion by the straight lines of ordinary geometry. Now Cayley 1 1 A. Cayley, 'A Sixth Memoir upon Quantics,' Phil. Trans., 149 (1859), Math. Papers, vol. ii. Cayley wrote a number of papers dealing specially with non-Euclidean NON-EUCLIDEAN GEOMETRY 7 showed — and his methods were elaborated by Klein l — that the metrical properties of figures are projective properties in relation to a certain fundamental figure, the Absolute, which in ordinary plane geometry is a degenerate conic consisting of the line infinity and the pair of imaginary points (circular points at infinity) through which all circles in the plane pass, but in non-Euclidean geometry is a proper conic, real in Hyperbolic, imaginary in Elliptic geometry. In the language of group-theory this is explained by saying that the group of motions, Euclidean or non-Euclidean, is a sub-group of the general projective group, and is characterised by leaving invariant a certain conic.2 § 3. In ordinary plane geometry the metrical properties of figures are referred to a special line, the line infinity, u, and two special (imaginary) points on this line, the circular points at infinity, w, «'. The line infinity appears in point-coordinates as an equation of the first degree, u=Q, while every finite point satisfies the geometry, but although he must be regarded as one of the epoch- makers, he never quite arrived at a just appreciation of the science. In his mind non-Euclidean geometry scarcely attained to an independent existence, but was always either the geometry upon a certain class of curved surfaces, like spherical geometry, or a mode of representation of certain projective relations in Euclidean geometry. 1 Loc. cit., p. 5, foot-note 1. Klein has written a great deal relating to non-Euclidean geometry, and was one of the first to apply it, especially in the conform representation, to the theory of functions. His Erlanger Programni, Vergkichende Betrachtungen uber neuere geometrische Forschungen, 1872 (English translation in Bull. Amer. Math. Soe., 2 (1893) ), gives, in very condensed form, a number of representations of non-Euclidean geometry, especially in relation to Lie's theory of groups. 2 The following elementary account of the Cayley-Klein representation was published in the Proc. Edinburgh Math. Soc., 28 (1910). A simple exposition from the point of, view of elementary geometry was given by Professor Charlotte A. Scott in the Bull. Amer. Math. Soc. (2), 3 (1897). An analytical treatment is also given in her treatise on Modern Analytical Geometry (London, Macmillan, 1894). The literature of this repre- sentation is very extensive, as the Projective Metric, or, what comes to nearly the same thing, the use of Weierstrass' coordinates (see p. 28, foot-note 1), whereby the equation of a straight line is of the first degree, forms one of the most useful means of studying non-Euclidean geometry. 8 CONCRETE REPRESENTATIONS OF identity w=const. In trilinear coordinates, for example, if a, 6, c are the sides of the triangle of reference, A, u=ax+by+cz=2&. The circular points appear in line-coordinates as an equation of the second degree, (ow'=0, while every ordinary line satisfies the identity «<•>'= const. In trilinear coordinates '=f +-r}2+?-2r)£> cos A-2#cosB-2& cos C In rectangular cartesian coordinates, made homogeneous by the introduction of a third variable z, the equation of the line infinity is z=0, while for finite points 2=1. The line- coordinates of the line lx+my+nz=Q are I, m, n, and in general 12+ w2= constant. When the equation is in the ' perpendicular ' form, for example, the constant is unity. But for the line infinity 1=0 and ra=0 so that Z2+m2=0, and this is true also for any line y=±ix+b, i.e. for any line passing through one or other of the points of intersection of the line z=0 with the locus x2+y2=0. Now an equation of the second degree in point-coordinates or in line-coordinates represents a conic. But the equation j2_j.m2=Q represents a degenerate conic consisting of two (imaginary) pencils of lines, since l2+m2 decomposes into linear factors. Similarly z=0 as a point-equation, when written z2=0, represents a degenerate conic consisting of two coincident straight lines. These conies are just one conic considered from the two different points of view of a locus and of an envelope, for the reciprocal of the equation l2+m2=cn2 is c(x2+y2)=z2. When c=0 the point-equation represents a circle of infinite radius z2=0, and the line-equation Z2+m2=0 represents the two pencils of lines passing through the two points through which all circles pass. This degenerate conic is called the Absolute. If we now replace the degenerate conic by a proper conic, we get a more general form of geometry, which includes ordinary Euclidean geometry as a special case. It also in- cludes as special cases the geometries of Lobachevsky and NON-EUCLIDEAN GEOMETRY 9 Riemann, the former when the conic is real, the latter when it is imaginary. There are obviously other cases — for example, when the conic degenerates to two distinct lines — and there will be corresponding systems of geometry. Most of these geometries are very bizarre. In one, for example, the peri- meter of a triangle is constant. The only ones which at all resemble the geometry of experience are the three just mentioned. § 4. We have now to obtain the expressions for the distance between two points and the angle between two straight lines. As the absolute in ordinary geometry is less degenerate as an envelope than as a locus (the equation in line-coordinates being of the second degree) it will be simpler to take first the angle between two lines. The expression must be such as to admit of extension to the case of a proper conic. Now Laguerre x has shown that the angle between two straight lines can be expressed in terms of a cross-ratio. Consider two lines y—x tan 6, y=x tan 6', passing through 0. We have also through 0 the two (isotropic) lines, y=ix, y=—ix, which pass through the circular points. The cross-ratio of the pencil formed by these four lines is , , ,. t&nO—i . tan0+i (uu , tow )=— ... .-=- tan0'-» Hence 6'— 6=$ilog(uu', too/). We can now extend this to the general case. Through the point of intersection L of two straight lines p, q there are two lines belonging to the absolute considered as an envelope, viz., the two tangents from L. Call these x, y. The angle (pq) is then denned to be klog(pq, xy) where k is a constant depending upon the angular unit employed. 1 E. Laguerre, ' Note sur la theorie des foyers,' Nouv. Ann. Math., Paris. 12 (1863). B io CONCRETE REPRESENTATIONS OF It is usual to take k=$i so that the angle between two rays which form one straight line is Jilog l=£t . 2imr=mir. This corresponds to the circular system of angular measurement, and we see that the angle between two rays is periodic, with period 2w. The angle between two lines with undefined sense has, however, the period IT, If the two lines are conjugate with respect to the absolute, (pq, xy)=—l, and the angle is \TT. The two lines are therefore at right angles. An analogous definition is given for the distance between two points. On the line I joining two points P, Q there are two points belonging to the absolute considered as a locus, viz., the two points of intersection with I. Call these X, Y. The distance (PQ) is then defined to be Klog(PQ,XY) where K is a constant depending upon the linear unit em- ployed. § 5. To test the consistency of these two formulae for distance and angle it is sufficient to show that a circle, defined as the locus of a point equidistant from a fixed point, cuts its radii at right angles. Let the equation of the absolute, referred to two tangents OA, OB and the chord of contact AB, be xy=z*. In a line y=mx through O take the point P (x, y, 2). Let OP cut the conic in X, Y, and the chord of contact in M . Let X (or Y) and P divide OM in the ratios 1 : k and 1 : p. The coordin- ates of the points are : 0(0, 0, 1), M (1, m, 0), X(l, m, k), P(l, m, p). If (OP) is constant, P describes a circle, and we have the cross-ratio (OP, XT)= const. =p=!j£l2, where k^-k^^m. — Hence p= J -^- . *-/* Also px=z and py=mz. NON-EUCLIDEAN GEOMETRY n Eliminating ra we find the equation of the locus of P, which is a conic having double contact with the absolute at A and B. The equation of the tangent at P^y^Zj) is and that of OP is The pole of the line OP with respect to the absolute is (—xv ylt 0), and this lies on the tangent. Hence OP and the tangent are conjugate with respect to the absolute and are therefore at right angles. § 6. When the absolute is imaginary X, Y are conjugate imaginary points, and log (PQ, X Y) is a pure imaginary. In order that the distance may be real, K must then be a pure imaginary, and, as in the case of angles, we see that distance is a periodic function with period 2-n-Ki. By taking K=%i the period becomes IT, and we make linear measurement correspond with angular. This case will be seen to correspond to spherical geometry, but the period (the radius of the sphere being unity) is not TT but 2ir. This is exactly analogous to the case of two rays, or lines with defined sense. On the sphere two antipodal points define the same pencil of great circles, but with opposite sense of rotation. If we leave the sense of rotation undefined, then they determine exactly the same pencil, and must be considered identical, or together as forming a single point ; just as two rays, which make an angle TT, together form a single line. On the sphere two lines (great circles) determine two antipodal points or pencils of opposite rotations ; two points determine two rays of opposite directions. It is convenient thus to consider antipodal points as identical, or we may conceive a geometry in which this is actually the case. This is the geometry to which the name elliptic is generally confined, the term spherical being retained for the case in 12 CONCRETE REPRESENTATIONS OF which antipodal points are distinct.1 In the Cay ley-Klein representation spherical geometry is conveniently excluded, since two lines only intersect once. § 7. Consider next the case where the absolute is a real proper conic. This divides the plane into two distinct regions which we may call the interior and the exterior, and it is of no moment whether the conic be an ellipse, a parabola, or a hyperbola. It is convenient to picture it as an ellipse. If the points P, Q are in different regions, then (PQ, XY) is negative and log (PQ, X Y) is a complex number of the form a+(2n+l)iir, or simply a+iir, to take its principal value. a is zero only when (PQ, X Y) = - 1. K log (PQ, X Y) also will in general be complex whatever be the value of K. Of course it is possible to choose K=a—itr, which would make the distance real, but for points in the vicinity of Q the distance (PQ) would still be complex. On the other hand, if P, Q are in the same region, (PQ, X Y) is either real, when X, Y are real, or purely imaginary, when X, Y are conjugate imaginary points. Then by taking K either real or a pure imaginary we can make the distance between two points in the same region real when measured along a certain class of lines, purely imaginary when measured along another class : these are the lines which do or do not cut the absolute. Hence we are led to consider certain points and lines as ideal, Suppose we consider points within the absolute as actual points. The line joining two actual points always cuts the absolute, and we must take K real. Then all points outside the absolute are ideal points, for the distance between an exterior point and an interior point is complex (or purely imaginary in the case of harmonic conjugates). If Q lies on the absolute, while P does not, (PQ, XY) is either zero or infinite and log (PQ, X Y) is infinite. Hence the absolute is the assemblage of points at infinity. Two lines cutting in an 1 Some writers have distinguished these two geometries as single or polar elliptic and double or antipodal elliptic. NON-EUCLIDEAN GEOMETRY 13 actual point 0 make a real angle if A; is a pure imaginary, since the tangents from 0 are conjugate imaginaries. This then completes the representation of Hyperbolic Geometry. Actual points are represented by the points within a real proper conic. The conic itself consists of all the points at infinity, while points outside it are ideal. § 8. If now we consider points outside the absolute as actual points, there are two cases according as K is taken to be real or imaginary. In the first case the distance between two points will be imaginary if the line joining them does not cut the absolute. Such a line must therefore be considered ideal, and we get in any pencil of lines with an actual point as vertex a class of ideal lines and a class of actual lines, and these are separated by the two tangents to the absolute. As these tangents are real, k must now be taken to be real, and we get a system of angular measurement of an entirely different nature from that with which we are familiar. The period of the angle is now 2iirk which is imaginary, and complete rotation about a point becomes impossible. If the line q is a tangent to the absolute log (pq, xy) is infinite. The angle between two lines thus tends to infinity as one line is rotated. Further, if the line PQ touches the absolute log (PQ, XY)=0, i.e. (PQ)=0, or the distance between any two points on an absolute line is zero. This curious result can be found to hold even in ordinary geometry if we consider imaginary points. If the line PQ passes through one of the circular points, so that yl— y^=i(xl~ x2), then § 9. We have now to examine if the logarithmic expression for the distance between two points holds in ordinary geometry. In this case the two absolute points X , Y on any line PQ coin- cide, and (PQ, XY)=l. The distance between any two points would thus be zero if K is finite. As the distance between any two points must, however, in general be finite, it follows that we must make K infinite. I4 CONCRETE REPRESENTATIONS OF Let PY=PX+€ where e is small. Then and (PQ)=K\og (PQ, Let K approach infinity and e approach zero in such a way that Kf. approaches a finite limit X. Then Now to fix X we must choose some point E so that (PE)=l, PW the unit of length. Then 1=X . =^ — =•= JrJi. . Jf/Ji. ^j iT>n\ PX . EX PQ XE XQ i (PQ)= PE •PX:QX=PE^PQ=( If we take P as origin =0, Ul oo 1 which agrees with the ordinary expression since '——=1. 0*1 It will be noticed that this case differs in one marked respect from the case of elliptic geometry. In that system there is a natural unit of length, which may be taken as the length of the complete straight line — the period, in fact, of linear measurement ; just as in ordinary angular measurement there is a natural unit of angle, the complete revolution. In Euclidean geometry, however, the unit of length has to be chosen conventionally, the natural unit having become infinite. The same thing appears at first sight to occur in the hyperbolic case, since the period is there imaginary, but, K being imaginary, iK is real, and this forms a natural linear standard. (Of. § 27 (3).) § 10. It still remains for us to consider the cases in which the absolute degenerates as an envelope to two coincident points and as a locus to two straight lines which may be real, coincident or imaginary. In these cases k is seen to be infinite, NON-EUCLIDEAN GEOMETRY 15 and it appears as in the analogous case just considered that there is now no natural unit of angle available, as the period is infinite. A unit must be chosen conventionally. The geometries in the case in which k is infinite or real present a somewhat bizarre appearance, and are generally on that account excluded from discussion, the objection being that complete rotation about a point is impossible, and the right angle has no real existence. Yet, if we go outside the bounds of plane geometry, such geometries will present themselves when we consider the metrical relations subsisting on certain planes, ideal or at infinity. Let us consider the case of hyperbolic geometry of three dimensions. Here the absolute is a real, not ruled, quadric surface, say an ellipsoid, and actual points are within. Actual lines and planes are those which cut the absolute, and the geometry upon an actual plane is hyperbolic. But an ideal plane cuts the absolute in an imaginary conic, and the geometry upon such a plane is elliptic. A tangent plane to the absolute cuts the surface in two coincident points and a pair of imagin- ary lines. The geometry on such a plane is the reciprocal of Euclidean geometry, i.e. the measurement of distances is elliptic while angular measurement is parabolic. In this geometry the perimeter of a triangle is constant and equal to IT, just as in Euclidean geometry the sum of the angles is constant and equal to TT. Now if we make use of the theorem that the angle between two planes is equal to the distance between their poles with respect to the absolute, we see that the geometry of a bundle of planes passing through a point on the absolute is Euclidean. The sum of the three dihedral angles of three planes whose lines of intersection are parallel is therefore always equal to TT, a result which was obtained by Lobachevsky and Bolyai.1 1 A complete classification of all the geometries arising from the Cayley-Klein representation in space of n dimensions will be found in the author's paper, ' Classifica- tion of Geometries with Projective Metric,' Proc. Edinburgh Math. Soc., 28 (1910). 16 CONCRETE REPRESENTATIONS OF § 11. An apparent extension of the Cay ley-Klein theory, elaborated by Fontene l for space of n dimensions, deserves mention. The absolute conic in the Cayley-Klein theory is the double conic of a transformation by reciprocal polars. If we replace this transformation by the general dualistic linear transforma- tion there arise two distinct conies having double contact, the pole conic or locus of points which lie upon their corresponding lines, and the polar conic or envelope of lines which pass through their corresponding points. Consider any line Z and a point A upon it. To A there corresponds a line a which cuts Z in a point A'. Thus a homography is established between pairs of conjugate points A, A' on the line Z. The double points Qj, H2 of this homography are the points in which I cuts the pole conic. The distance (PQ) between two points P, Q on I can then be defined as The distance between two conjugate points P, P' is constant for the line Z, but it varies for different lines. It may be called the parameter of the line. By allowing K to vary the parameter could of course be made the same for all lines ; but it is impossible to adjust the system so that it may represent a geometry with the necessary degrees of freedom. In fact, since a motion consists of a collineation which leaves the absolute invariant, and since the general collineation leaves just three points invariant, these points must be the points of contact of the two conies and the pole of their chord of contact. The general motion is therefore impossible, the only possible motion being a rotation about a definite point, the pole of the chord of contact. E. Meyer 2 has considered a further generalisation of these ideas by taking two independent conies as the absolute 1 G. Fonten6, 'L'hyperespace It (n - 1) dimensions. Pr&pri&is mttriques de la corrttaivon gentrale. Paris, Gauthier-Villars, 1892. 2 ' tiber die Kongruenzaxiome der Geometric,' Math. Ann., Leipzig, 64 (1907). NON-EUCLIDEAN GEOMETRY 17 figures for angular and linear metric. He remarks that in ordinary Euclidean geometry the conies which play the rdle of absolute are also distinct, namely the one is a double line and the other is a point-pair. We have seen above, how- ever, that these are just different aspects of the same degenerate conic. The double line is the locus, or assemblage of point-elements, the point-pair or pair of imaginary pencils is the envelope, or assemblage of line-elements. CONFORM REPRESENTATION BY CIRCLES § 12. We shall next consider a very useful representation which has important applications in the theory of functions, that in which straight lines are represented by circles.1 Since a circle requires three conditions to determine it, one condition must be given. Hence if the circle x*+ y*+ 2gx+ 2fy+c=Q represents a straight line, the constants, g, f, c, must be connected by a linear relation, which may be written 2gg'+2ff'=c+c'. But this relation expresses that the circle cuts orthogonally the fixed circle x*+y*+ 2g'x+ 2f'y+c'=0. Hence the circles which represent the straight lines of a geometry form a linear system cutting a fixed circle orthogonally. Similarly in three dimensions if planes are represented by spheres they will cut a fixed sphere orthogonally. § 13. Thus we find at once that there are three forms of geometry, according as the fundamental circle is real, vanish- ing, or imaginary. A difficulty, however, presents itself. Two orthogonal 1 An interesting account of this representation, from the point of view of elementary geometry, is given by H. S. Carslaw, Proc. Edinburgh Math. Soc., 28 (1910). The following account, which was suggested by Professor Carslaw's paper, appeared in the same volume. C i8 CONCRETE REPRESENTATIONS OF circles in general intersect in two points, which may be real, coincident, or imaginary ; and the point-pair thus determined will not determine uniquely one orthogonal circle, but a pencil of circles. Two such points are inverse points with respect to the fixed circle. We shall see in § 18 that the ' distance ' between a pair of inverse points is real or imaginary according as the fundamental circle is imaginary or real. In the former case we may either consider the two points as distinct (so that two straight lines will intersect in two points), or identify them ; and we get the two forms of geometry, Spherical and Elliptic. In the latter case it is necessary to identify the two points, otherwise we should have two real points with an imaginary distance ; thus we get Hyperbolic geometry. Alternatively we may agree to consider only the points in the interior (or exterior) of the fundamental circle. When the fundamental circle reduces to a point 0, one of the points of any point-pair is at O and we need only consider the other point, so that two lines always intersect in just one point. This geometry is Parabolic, and we shall see that it is identical with Euclidean geometry. When the fundamental circle is real, two orthogonal circles intersect in two points, real, coincident, or imaginary. This corresponds to the three sorts of line-pairs in Hyperbolic geometry, intersectors, parallels, and non-intersectors. When the fundamental circle is imaginary, two orthogonal circles always intersect in two real points, so that in Elliptic or Spherical geometry parallels and non-intersectors do not exist. When the fundamental circle reduces to a point 0, every orthogonal circle passes through 0, and they cut in pairs in one other real point which may coincide with 0. The latter case corresponds to parallels in Euclidean geometry. § 14. Next, to fix the representation, we have to consider the measurement of distances and angles. Let us make the condition that angles are to be the same in the geometry and in its representation, i.e. that the repre- NON-EUCLIDEAN GEOMETRY 19 sentation is to be conform.1 We shall find that this fixes also the distance function. First let us find how a circle is represented. A circle is the locus of points equidistant from a fixed point, or it is the orthogonal trajectory of a system of concurrent straight lines. Now a system of concurrent straight lines will be represented by a linear one-parameter system of circles, i.e. a system of coaxal circles. The orthogonal system is also a system of coaxal circles, and the fixed circle belongs to this system. Hence a circle is represented always by a circle, and its centre is the pair of limiting (or common) points of the coaxal system determined by the circle and the fixed circle. The distance function has thus to satisfy the condition that the points upon the circle which represents a circle are to be at a constant distance from the point which represents its centre. To determine this function let us consider motions. A motion is a point-transformation in which circles remain circles ; and further, the fundamental circle must be transformed into itself, and angles must be unchanged. § 15. The equation of any circle may be written 2 zz+pz+pz+c=0 where z=x+iy, p=g+if and z, p are the conjugate complex numbers. Now the most general transformation which pre- 1 C. E. Stromquist, in a paper ' On the Geometries in which Circles are the Shortest Lines,' New York, Trans. Amer. Math. Soc., 7 (1906), 175-183, has shown that 'the necessary and sufficient condition that a geometry be such that extremals are perpen- dicular to their transversals is that the geometry be obtained by a conformal transforma- tion of some surface upon the plane.' The language and his methods are those of the calculus of variations. The extremals are the curves along which the integral which represents the distance function is a minimum, i.e. the curves which represent shortest lines ; and the transversals are the curves which intercept between them arcs along which the integral under consideration has a constant value. Thus in ordinary geometry, where the extremals are straight lines, the transversals to a one-parameter system of extremals are the involutes of the curve which is the envelope of the system. In particular, when the straight lines pass through a fixed point the transversals are concentric circles. " Cf. Liebmann, Nichteuklidische Oeomelrie (Leipzig, 1905), §§ 8, 11. 20 CONCRETE REPRESENTATIONS OF serves angles and leaves the form of this equation unaltered az'+P - is1 yZ'+8 ' This is a conformal transformation since any transformation between two complex variables has this property. To find the relations between the coefficients in order that the fundamental circle may be unchanged, let its equation be x*+y2+k=Q or zz+fc=0. This becomes (az+j3)(^+)8)+&(yz+S)(yz+S)=0. Hence aj8+fcy8=0 and fc( aa therefore aa=88, so that ^=i=-&=-i=l. 8 * ft ky We have a=xS and a=xS, and also a=-r-8, A therefore 1*1 = !• The general transformation is therefore 2 , where |X| = 1. By any such homographic transformation the cross-ratio of four numbers remains unchanged, i.e. (zfr, z3z4)=(z'1z'2, zV4). 1 The only other type of transformation possible is _ ff __ — —_ - ^ ^ _ __ --- ^ j •yz' + d y/ + fl but this only differs from the former by a reflexion in the axis of x, 2=2*, z=i*. 2 When, as is often taken to be the case, the fundamental circle is the z-axia, the conditions are simply that the coefficienta a, (3, y, 8 be all real numbers. NON-EUCLIDEAN GEOMETRY 21 To find the condition that this cross-ratio may be real, let 0V be the amplitude, and ry the modulus of zf— zjf then (ZZ ZZ ->-r« ^2* *'»-'M + '"-'») IZjZjj, z3z4j— — — - e ru rza Hence we must have and the four points zx, z2, z3, z4 are concyclic. § 16. Now to find the function of two points which is in- variant during a motion ; the two points determine uniquely an orthogonal circle, and if the transformation leaves this circle unaltered it leaves unaltered the two points where it cuts the fixed circle. Hence if these points are x, y, the cross- ratio (zjZg, xy) for all points on this circle depends only on Zj and za. If the distance function is (PQ)=/)(z122> xy)\ or> as we may write it, /(zl5 za), then for three points P, Q, R, (PQ)+(QR)=(PR), 01 f(*v z2)+f(z2, z3)=f(zlt z3). This is a functional equation by which the form of the function is determined. Differentiating with respect to Zj, which may for the moment be regarded simply as a parameter, we have f,(9 . QY d (PX\_ft(9 . RY d (PX\ * (2i' Za) • QX ' dz-1(pYr/ (z» ZS)-RX' ^(PY) ' Hence f'fr, z2) QX RY/PX RY\ (PX (?r\_(z1z3> xy) f'(*i> **)~QY RX~\PY RXt • \PY QX)-(zjZ2, xy) ' and (z^, xy)f'\(zlzz, xy)\=const.=fj.. Integrating, we have /(z1,za)=/tlog(z1z2, xy)+C. The constant of integration, G, is determined =0 by substitut- ing in the original equation. Hence , XY), (PQ, X Y) being the cross-ratio of the four points P, Q, X, Y 22 CONCRETE REPRESENTATIONS OF on the circle, i.e. the cross-ratio of the pencil 0(PQ, XY) where 0 is any point on the circle. § 17. The expression for the line-element can now be found by making PQ infinitesimal. We have, by Ptolemy's Theorem, PX . Q Y=PQ . X Y+PY . QX. Hence cfc^log (l+ Let OP (Fig. 1) cut the circle PXY again in R and the fixed circle in A, B. Then R is a fixed point so that PR is constant. Also =-=& fixed ratio=e, PX PY and PR.XY=PX.RY+PY.RX=2e.PX.PY. Therefore -&-*•— T>V=J^ an(* *s therefore a function of the tr JL . JT I rti position of P alone. FIG. 1 To find its value we may take any orthogonal circle through P, say the straight line PR. XY AB Then Hence PX.PY~PA. O.. . / J, d*- NON-EUCLIDEAN GEOMETRY 23 § 18. The distance function is thus periodic with period 2ip.ir. If P, P' are inverse with respect to the fixed circle -/* log =/* log (-D- and When Q is on the fixed circle (PQ)=cc . The fundamental circle is thus the assemblage of points at infinity. If the fundamental circle is imaginary, k is positive and p. is purely imaginary and may be put =i. Then if inverse points are considered distinct their distance is TT and the period is 2ir, but if inverse points are identified the period must be taken as TT. If the fundamental circle is real, k is negative and p. is real and may be put =1. Then the period must be taken as iir and inverse points must be identified, otherwise we should have two real points with an imaginary distance. In this geometry there are three sorts of point-pairs, real, coincident, and imaginary, or actual, infinite, and ultra-infinite or ideal. § 19. Now if we change x, y into x', y' with the help of an additional variable z' by the equations x y then x' so that (x, y) is the stereographic projection of the point (x', y', z') on a sphere of radius R. Obtaining the differentials dx', dy', dz', we find Hence R2=-p.2. Hence when k is positive and /* purely imaginary and =iR, the geometry is the same as that upon a sphere of radius 24 CONCRETE REPRESENTATIONS OF B, and the representation is by taking the stereographic projection. When k is negative the sphere has an imaginary radius, but such an imaginary sphere can be conformly represented (by an imaginary transformation) upon a real surface of constant negative curvature, such as the surface of revolution of the tractrix about its asymptote (the pseudosphere).1 When k is zero \L must be infinite and the sphere becomes a plane. Let 2.^/—k—. Then xz+yz By the transformation r'=s 6' =6 this becomes dsz=dr'z+r'zde'z=dx'z+dy'z. Hence when k is zero the geometry is the same as that upon a plane, i.e. Euclidean geometry, and the representation is by inversion, or reciprocal radii. § 20. Let us now return to the consideration of motions and investigate the nature of the general displacement of a rigid plane figure.2 In ordinary space the general displacement of a rigid plane figure is equivalent to a rotation about a definite point, and this again is equivalent to two successive reflexions in two straight lines through the point. Now the operation which corresponds to reflexion in a straight line is inversion in an orthogonal circle. The formulae for inversion in the circle zz+pz+pz— fc=0, which is any circle cutting zz+k=0 orthogonally, are x'+g)z+(y'+f)z y+f (x+g)z+(y+f)z 1 Cf. Darboux, Theorie des surfaces, viL, chap. xi. Also Klein, Nichteuklidische Geometric, Vorlesungen. 2 Cf. Weber u. Wellstein, EncyUopiidie der Elementar-Mathematik (2. Aufl. Leipzig, 1907), Bd. 2, Abschn. 2. Also, Klein u. Fricke, Vorksungen iiber die Theorie der auto- morphen Functionen (Leipzig, 1897), Bd. 1. NON-EUCLIDEAN GEOMETRY 25 or, using complex numbers, _(pp+k)(z'+p) (z'+p)(z'+p) ' Whence z=*b^. z'+p A second inversion in the circle zz+qz+qz— k=0 gives 2 This will not hold when the circle of inversion is a straight line, 6==3/e«*e This combined with an inversion gives ~z"+p Now these transformations are always of the general form z^fJlM, where X =1. /3z'+ a. In fact, this transformation is always of one or other of the two forms z=z'e"* (when /S=0) or *= (by dividing above and below by /3). Hence the general displacement of a plane figure is equivalent to a pair of inversions in two orthogonal circles. § 21. In the general transformation there are always two points which are unaltered, for if 2' =2 we have the quadratic equation /322+ (a- Xa)a+ &X/3=0. These form the centre of rotation, and the circles with these points as limiting points are the paths of the moving points. There are three kinds of motions according as the roots of this quadratic are real, equal, or imaginary, or according as 26 CONCRETE REPRESENTATIONS OF the centre of rotation is real, upon the fundamental circle, or imaginary. The first case is similar to ordinary rotation. In the second the paths are all circles touching the fundamental circle. In the third the paths all cut the fundamental circle ; one of these paths is an orthogonal circle, the other paths are the equidistant curves ; the motion is a translation along a fixed line. § 22. It would appear that the representation by circles is a sort of generalisation of the Cayley-Klein representation, since a straight line is a circle whose centre is at infinity. When the circles degenerate in this way, however, the fixed circle becomes the line infinity, and the geometry degenerates to Euclidean. It is of interest to deduce the general Cayley-Klein repre- sentation from the circular one, but this cannot be done by a conformal transformation. Abandoning the conformal representation, the transforma- tion which changes circles orthogonal to xz+yz+k=Q into straight lines is k The points (r, 6), ( — , 6) are both represented by the same point, so that this transformation gives a (1, 1) correspondence between the pairs of real points which are inverse with respect to the circle xz+yz+k=0 and the points which lie within the 2 2 circle x2+y2+ =0, since for real values of r, r'z< — -. Every A* /.' point upon the circle r2+fc=0 is thus to be considered double. To a pair of imaginary points corresponds a point outside the new fixed circle. Any circle, not orthogonal, is transformed into a conic having contact with the circle krz+pz=0 at the two points which correspond to the intersections of the circle with the fixed circle r2+k=0. In fact, any curve in the r'-plane which cuts the fixed circle NON-EUCLIDEAN GEOMETRY 27 at a finite angle is represented in the r-plane by a curve cutting the fixed circle orthogonally, and any curve in the r-plane which cuts the fixed circle at a finite angle other than a right angle corresponds in the r'-plane to a curve touching the fixed circle. Let the equation of a curve in the r'-plane be f(r', 0')= 0. Then But Therefore dr' dr'ldd'' d£_df dr = (f*-t)« df , , dl_dj dr'~dr ' dr' 2p(r*+ k) ' dr' ' dd'~d6 ' dd dr'~ 2p(rz+k)'dr' der dd Hence when rz+k=0, r-,= oo unless -j-=Q, which proves the dr dr results. § 23. This transformation receives its simplest expression through the medium of the sphere. Let a point Q be projected stereographically into P and centrally upon the same plane or a parallel plane into P' (Fig. 2). FIG. 2 Then and r=OP= 28 CONCRETE REPRESENTATIONS OF therefore r'=^*> which agrees with the former equation if cz—k and cc'——p, so that c'*=P*=k', say. /c Hence as the representation by circles corresponds to stereographic projection, the representation by straight lines corresponds to central projection. The transformation from the sphere to the plane is in this case given by the equations x where Then* § 24. To determine the distance and angle functions in this representation we have first the relation between the angles from § 22, tan 0'= - tan A . J^~k^ . - = - tan A . r*~ = - tan <6 . J -£- 2p(r*+k) r rz+k v r'*+k' 1 It may be noticed that the line-element can be expressed in terms of x', y" alone Thus expressing z', dz' in terms of x', y' by means of the equation x'* + y'2 + z'3 = S?, we have , „ _ Bf(dx^ + dy'1) - (y'dx' - x'd^ B'-x"-y'« Here x', y', - are the so-called Weierstrass' coordinates. Let the position of a point P on the sphere be fixed by its distances £, ij from two fixed great circles intersecting at right angles at Q, and let QP=p, all the distances being measured on the sphere along arcs of great circles. Then , , On the pseudosphere the circular functions become hyperbolic functions. (See Killing, Die nichteulclidiachen Raumformen, Leipzig, 1885, p. 17.) NON-EUCLIDEAN GEOMETRY 29 where is the angle which the tangent at P to the curve f(r, 0)=0 makes with the initial line. Draw the tangents P'Tlt P'TZ from P' to the circle (Fig. 3) Fio. 3 and let LOP'T^OP'T^a. Also draw P'X' parallel to the or-axis. Then , I~-V~ tana=v-7o — r/— — * v r* Therefore _ _ sin (a+^')~sin X'P'TZ ' sin =P'(X'0, TiTj. Thus the true angle is given by <^=| log (OX', TM. Hence the angle between two lines P'X', P'Y' through P' is given by •log (OY', TW-'log (OX', T.T^log (XT, TJTj. ' Next to determine the distance function ; let P, Q be- come P', Q' (Fig. 4). The orthogonal circle PQXY becomes a straight line P'Q'X'Y', and OPP', OQQ', etc., ar» collinear since angles at 0 are unaltered. 30 CONCRETE REPRESENTATIONS OF We have then (PQ)=H\og (PQ, XY)=n\og (pf But PX sinXOP PY_smYOP OP ~ sin OXP' OP sin 0 YP , PZ QY_BmXOP sin YOQ sinQFP emOXQ PY ' QX~ sin YOP ' sin XOQ ' sin OXP ' sin 0 YQ i.e. (PQ, XY)=(P'Qr, X'Y')(QP, XY) therefore (PQ, XY)*=(P'Qf, X'Y'). Hence we have the true distance (PQ) given by >'Q', X'Y')=(P'Q'). FIG. 4 Then the line-element can be obtained in a manner similar to that of § 17. We find as before that (PQ, XY)=1+ „ r JL . but in this case PX . PY=x*+y*+k' and X 72= -4^'(dxz+ dyz)+ (ydx-xdy)*\l(dxz so that ^._^.y(^+%V(yfe-^)s. Comparing this with the expression in § 23 we find NON-EUCLIDEAN GEOMETRY 31 § 25. Finally, this representation may be transformed projectively (distances and angles being unaltered as they are functions of cross-ratios), and we get the usual generalised representation in which the fixed circle or absolute becomes any conic ; straight lines are represented by straight lines, and distances, and angles in circular measure, are expressed by the formulae XY) where X, Y are the points in which the straight line PQ cuts the conic, and x, y are the tangents from the point of inter- section of the lines p, q to the conic. GEODESIC REPRESENTATION ON SURFACES OF CONSTANT CURVATURE § 26. It has been seen that both the Cayley-Klein represen- tation and the conf ormal representation by circles are derivable by projection from a sphere, real or imaginary, on which the non-Euclidean straight lines are represented by great circles. By Gauss' Theorem the sphere may be transformed, or limited portions of the surface may be deformed, into a surface of constant measure of curvature, in such a way that geodesies remain geodesies and are unaltered in length. The effect is that of bending without stretching ; the geometry therefore remains the same. To Beltrami 1 is due this representation of non-Euclidean geometry upon a surface of constant cur- vature, and it is the only representation in which distances and angles are represented unchanged. § 27. While this representation is of the first importance in non-Euclidean geometry, it has to be distinctly understood 1 Loc. tit., p. 5, foot-note 2. 32 CONCRETE REPRESENTATIONS OF that it is only a representation. A vast deal of misconception has grown around it. The following points have been most generally misunderstood :— (1) There is an essential difference between Riemann's geometry and the geometry on the surface of a sphere.1 The former is a true metrical geometry of two dimensions, and is no more dependent upon three dimensions z than ordinary geometry is on the ' fourth dimension.' The geometry on the surface of a sphere, on the other hand, is a body of doctrine forming a part of ordinary geometry of three dimensions. (2) The fact that there is in ordinary space only one uniform real surface other than the plane has led certain critics 3 to reject Hyperbolic and Elliptic geometries as false and absurd, while they admit Spherical geometry only as a branch of ordinary geometry of three dimensions. This view is not so common now since the investigations of Pasch, Hilbert, and others on geometries defined by systems of axioms have become better known. (3) The term ' curvature,' especially when extended to space of three dimensions, has given rise to much confusion, and has led to the notion that non-Euclidean geometry of 1 Cf. P. Mansion, ' Sur la non-identite du plan riemannien et de la sphere euclidienne,' Bruxelles, Ann, Soc. scient., 20 B (1896), a reply to Lechalas in the same volume. See also B. Russell, ' Geometry, non-Euclidean," Encyd. Brit. (10th ed.), p. 669d. 2 This statement must not be confused with the result that plane projective geometry, which is free from metrical considerations, and in which the Euclidean and non-Euclidean hypotheses are not distinguished, cannot be established completely without using space of three dimensions. The theorem of Desargues relating to perspective triangles, which is proved easily by projection in space of three dimensions, is incapable of deduction from the axioms of plane projective geometry alone. Thus there are two-dimensional but not three-dimensional non-Desarguesian geometries. In the same way the theorem of Pascal for a conic, or, in the special form, the theorem of Pappus, when the conic reduces to two straight lines, from which Desargues' theorem can be deduced, is in- capable of deduction from the axioms of plane projective geometry alone. In this sense plane geometry is dependent upon three dimensions ; but it is only necessary to make some additional assumption, Pascal's theorem or an equivalent, in order to construct plane geometry without reference to three dimensions. 3 Cf., e.g., E. T. Dixon, The Foundations of Geometry (Cambridge, Bell, 1891), p. 140. NON-EUCLIDEAN GEOMETRY 33 three dimensions necessarily implies space of four dimensions.1 The truth is that Beltrami's representation, as he himself expressly states, breaks down when we pass to three dimen- sions, and it is necessary, in order to obtain an analogous representation, to introduce space of four dimensions. The geometry, however, is a true geometry of three dimensions, having its own axioms or assumptions, one of which is that there exists no point outside its space. The term ' curvature ' is therefore without meaning. The constant K2 which occurs in the Cayley-Klein formula, and which corresponds to the measure of curvature of the surface upon which the geometry may be represented, has been called on this account the measure of curvature of the space, but as this is so mislead- ing the term is now generally replaced by ' space-constant.' When it is finite it gives a natural unit of length like the natural angular unit. In Elliptic geometry it may be replaced by the length of the complete straight line ; in Hyperbolic geometry where K2 is negative iK can be constructed as follows : 2— Take two lines OA, OB at right angles, and draw A'B' so that A'E' || OB and E'A' || OA ; then draw an arc OL of a limit- curve through 0 perpendicular to OA and E'A' ; the arc OL=iK. Another natural unit based upon K is the area of the maximum triangle, which has all its angles zero, the limit being —irK2. (4) Confusion has also existed with regard to the compari- son of spaces with different space-constants. As there can be no comparison between one line and another unless they are in the same space, it appears clear that it is meaningless to 1 For example, S. Newcomb, ' Elementary Theorems relating to the Geometry of a Space of three Dimensions and of uniform positive Curvature in the Fourth Dimension^' J. Math., Berlin, 83 (1877). Clifford attempted, playfully no doubt, but with a certain seriousness, to explain physical phenomena by periodic variations in the curvature of space (Common-sense of the Exact Sciences, chap, iv., § 19). Helmholtz also, by his popularisation of the results of Beltrami and Riemann, did a good deal to promulgate this view especially among philosophers. Cf. Russell, toe. cit, * See Engel, Leipzig, Ber. Oes. Wiss., 50 (1898), p. 190. E 34 CONCRETE REPRESENTATIONS OF speak of different spaces of the same type but with different upace-constants. It is exactly analogous to the obvious absurdity of speaking of spaces in which the total angle at a point is of various magnitudes. The angle may be repre- sented by different numbers, 4, 360, 6*283 . . ., and so on, according to the arbitrary unit which is adopted ; so long as we are dealing with one space this angle has a constant magnitude, but there is no possibility of comparing magni- tudes when the objects are in different spaces. There are the three types of space according as K* is positive, zero, or negative. For elliptic geometry, for example, different positive values of Kz mean simply a different choice of the arbitrary unit of length. (Cf. Russell and Whitehead, Encycl. Brit, (llth ed.), article 'Geometry,' section vi., 'Non-Euclidean Geometry,' p. 725d.) REPRESENTATION OF PROJECTIVE METRIC BY APPARENT MAGNITUDE AS SEEN FROM A VIEW-POINT § 27. It has been indicated in § 6 that spherical and elliptic geometries of two dimensions are capable of representation as the geometries of a bundle of straight lines or a bundle of rays through a fixed point. The former is the geometry of ' visual space,' the latter is the geometry of the infinitely distant elements. For a plane or a line through the fixed point gives a line or a point at infinity, and the angle between two lines is represented by the distance between the infinitely distant points. The absolute for the geometry at infinity is the imaginary circle at infinity. ' Visual geometry ' is a two-dimensional geometry exactly analogous to this except for the existence of antipodal points ; i.e. as every astronomer knows, Visual geometry is the same as Spherical geometry. It is the geometry which would be constructed by a being endowed only with monocular vision and without powers of locomotion. NON-EUCLIDEAN GEOMETRY 35 § 28. A somewhat analogous representation for geometry of three dimensions has been devised by E. M'Clintock and modified by W. W. Johnson.1 We have seen that the geometry on the surface of a sphere gives, by central projection on any plane, a representation by straight lines with the Cayley-Klein projective metric. On every plane, with the exception of those through the centre of the sphere, a definite metric is thus established. To eliminate these exceptional planes M'Clintock proceeds in this way. A fixed point 0 is taken in space, and the metric on any plane through this point is defined to be that upon a tangent plane to the sphere in which 0 corresponds to the point of contact. The metric upon any other plane at a distance r from 0 is then defined to be that upon a plane parallel to the tangent plane, and at a distance r from it on the opposite side from the centre, the foot of the perpendicular from 0 corresponding to the foot of the perpendicular from the centre of the sphere. This procedure is modified in an elegant manner by John- son. Assume a ' central point ' 0 and a linear magnitude c corresponding to the radius of the sphere ; then the projective measure of a segment is its apparent magnitude viewed from a point P at a distance c from 0 measured in a direction perpendicular to the plane through the given line and 0. All lines in this plane have the same view-point, or pair of view- points. Consider any line I, and let the plane through 0 perpendi- cular to I cut I in A. Draw a circle with centre A passing through P, P'. Any point on this circle will also be a view- point for the line I. Hence a line has a view-circle. Consider any plane a, and take a line I in it. Construct the view-circle of I, whose centre is A and whose plane passes 1 E. M'Clintock, ' On the non-Euclidean Geometry,' New York, Butt. Amer. Math. Soc., 2 (1892), 21-33. W. W. Johnson, ' A Case of non-Euclidean Geometry,' find., 158-161. 36 CONCRETE REPRESENTATIONS OF through 0. Let A' be the foot of the perpendicular from 0 upon a, and let OA' cut the view-circle of I in Q, Q', Then, in Fig. 5, A'Qz^AQ2-p'2=AP2-p2+d2=c2+d2. Hence the points Q, Q' depend only upon the position of the plane and are independent of the line 1. Q, Q' therefore form a pair of view-points for all lines in the plane. Again, for all planes through I the view-points lie on the view-circle of I, and the FIG. 5 metric upon any line is the same, independently of the plane in which it may be conceived to lie. The measure of an angle is then defined to be its apparent magnitude viewed from the view-point of its plane. This representation is only suitable for Elliptic geometry. In Hyperbolic geometry c2 is negative, and the radius of a view-circle is real only if p2 >— c2 ; the view-points of a plane are real only if d2 >— c2. Hence for all lines and planes which do not cut the real sphere with centre 0 and radius +/—c the geometry is elliptic, and these lines and planes correspond to ideal or ultra-spatial elements. For the lines and planes which cut the sphere and which correspond to actual elements the view-points are imaginary. NON-EUCLIDEAN GEOMETRY 37 REPRESENTATION BY A NET OF CONICS § 30. We have next to consider a generalisation of the representation by circles, in which the circles are replaced by conies. The conies must form a linear system depending upon two parameters, i.e. a net. Further, to make the system correspond as closely as possible to the system of circles, which are conies passing through the two circular points, we shall suppose the net to be a special net passing through two fixed points, X, Y. The general equation of a system of conies passing through two fixed points may be written S+ (px+qy+rz)a=0 where S is an expression of the second degree, a of the first degree, and p, q, r are parameters. The parameters must be connected by a linear homogeneous relation, hence the variable line px+qy+rz=Q must pass through a fixed point Z. Taking XYZ as the triangle of reference, the equation reduces to the form axy+bz2+z(px+qy)=0 where p, q are now the two parameters of the net. The conic degenerates to two lines, one through X, the other through Y, if pq—ab. It degenerates to the line z=0 and a line y=mx, passing through Z, for infinite values of the parameters. § 31. Consider a line y=mx through Z. This cuts a conic of the system where amx2+ bz2+ zx(p+ qm)=0. By choosing p and q suitably it may be made to touch the conic. The condition for this is (p+qm)2=4:abm. Eliminating p+qm we obtain (amx2+ bz2)2=±abmz2x2, or (axy—bz2)2=Q. The locus of points of contact of tangents from Z to the 38 CONCRETE REPRESENTATIONS OF system is therefore a double conic, which touches ZX and Z Y at X and Y. Every line through Z is therefore cut in involution by the system of conies, and the double points of the involutions lie on the conic axy=bzz. Further, on each conic of the system there is an involution formed by the pencil with vertex Z, and the double points of these involutions are the points of intersection of the conies with the conic axy=bzz. We have, then, what we require, two absolute points on every conic which represents a straight line, and these absolute points lie on a fixed conic. We may therefore call the conic axy=bzz the Absolute. § 32. The conic axy+ bz2+ z(px+ qy)=0 cuts the absolute where (2axy)*b=axy(px+qy)z, which gives x—0 or y—0 or 4:obxy= (px+ qy)z. According as the points of intersection are real, coincident, or imaginary, the conic represents a line with hyperbolic, para- bolic, or elliptic metric. The condition that the points of intersection be coincident is If a or 6 vanishes all lines are parabolic. When a=0 the absolute becomes a double line z2=0, and every conic of the system breaks up into this line and a variable line px+qy+bz=Q. The representation is then by straight lines, and if X, Y are an imaginary point-pair the geometry is Parabolic. If X, Y are the circular points the geometry is Euclidean, and the representation is identical. When 6=0 the absolute breaks up into two lines x=0, y=0, and every conic of the system passes through the three points X, Y, Z. If X, Y are an imaginary point-pair the geometry is again Parabolic, and if X, Y are the two circular points the representation is by circles passing through a fixed point. NON-EUCLIDEAN GEOMETRY 39 If the absolute is not degenerate we may get lines of all three forms. If X, Y are real the absolute is real. We may suppose a, 6, which are real, to have the same sign, then the conic represents an elliptic or a hyperbolic line according as pq^-ab. If X, Y are imaginary the triangle of reference has two imaginary vertices, but we may take as real triangle of reference a triangle self -conjugate with regard to the absolute. The equation of a conic of the system may then be written X#2+ p.y2+ za+ z(px+ qy) = 0 where X, /A have the same sign, and the equation of the absolute, found by the same method as before, is The absolute is therefore real or imaginary according as X and (A are both positive or both negative. The discriminant in this case is If X, [i are both negative this is negative, and all lines are therefore elliptic when the absolute is imaginary. The equation of a conic of the system may be written Hence when X and ji are both positive the conic is real only when A >0, so that, when the absolute is real and X, Y are an imaginary pair, all real conies represent hyperbolic lines. The following is a summary of the results : — X, Y are imaginary, and the absolute is (1) A real proper conic, with the point Z in its interior. Hyperbolic geometry. (2) An imaginary conic. Elliptic geometry. (3a) A double line XY. Parabolic geometry, with repre- sentation by straight lines. (36) A pair of imaginary lines ZX, Z Y. Parabolic geometry, with representation by conies passing through Z. 40 CONCRETE REPRESENTATIONS OF When X, Y are real there are conies which represent hyperbolic, parabolic, and elliptic lines, and the measure of angle is hyperbolic. When X, Y are coincident the measure of angle is parabolic. § 33. In the representation by circles the points X, Y are the circular points, while Z is the centre of the fixed circle. The general representation by conies in the case where X, Y are imaginary is, of course, at once obtainable by projection from the representation by circles. A real conic and a point inside it can always be projected into a circle and its centre. All that is necessary is to make the centre correspond to the given point and the line infinity to the polar of this point. From this we deduce at once the distance and angle functions in this representation. The angle between two lines is \i times the logarithm of the cross-ratio of the pencil formed by the tangents to the two conies at their point of intersection and the lines joining this point to X, Y. Two points P, Q determine a conic cutting the absolute in U, V ; the distance (PQ) is then jx times the logarithm of the cross-ratio (PQ, UV) of the four points on this conic. A circle is represented by any conic passing through X, Y. § 34. In the circular representation we saw that motions are represented by pairs of inversions in orthogonal circles. In the representation by conies there is an analogous trans- formation. Any line through Z is cut in involution by the system of conies, the double points being on the absolute. The transformation by which any point is transformed into its conjugate is a quadric inversion.1 The conies of the system are transformed into themselves by such a trans- formation, while the points of the absolute are invariant. To find the equations of transformation, take XYZ as triangle of reference, and write the equation of the absolute 1 On quadric inversion see C. A. Scott, Modern Analytical Oeomelry, pp. 230-236. NON-EUCLIDEAN GEOMETRY 41 xy=z*. P is found from P' as the intersection of ZP' with the polar of P'. The polar of P'(x'y'z') is xy'+x'y=2zz', and the equation of ZP' is xy'=x'y. Hence r . „ . ~_ 1 . 1 . 1 *V • M • 2 J . — J * — j . § 35. We may similarly establish a quadric inversion with regard to any conic of the system. Let the conic cut the FIG. 6 absolute in 7, J. Draw the tangents at I, J to the absolute, cutting in 0. Then 0 is to be taken as the centre of inversion. The same point 0 is obtained by drawing the tangents to the conic at X, Y. (See Fig. 6, where the absolute is the ellipse and for clearness X, Y are taken to be real.) Hence XY is the polar of 0. The conic and the absolute with the points Z and 0 simply exchange roles, and the conic is left invariant by the transformation, while the absolute is transformed into itself. 42 CONCRETE REPRESENTATIONS OF The inverse of a conic is in general a curve of the fourth degree, but if the conic passes through X, Y the inverse is also a conic passing through X, Y. In fact, taking OXY as the triangle of reference and representing the fixed conic by the equation xy=z2, the equation of any conic passing through X, 7 is czz+fyz+gzx+hxy=Q, and this is transformed into hzz+fyz+gzx+cxy=0. Also any conic whose equation is of the form z2+fyz+gzx+xy=0 is transformed into itself. One of these is the absolute. Let its equation be z2+ ayz+ bzx+ xy — 0. (The coefficients of yz and zx cannot be zero since the fixed conic does not in general touch the absolute.) The point Z is the pole of XY, i.e. 2=0, with respect to the absolute, hence its coordinates are (a, 6, —1). The absolute and a conic of the system have a pair of common chords, one of which is z=0. To find the other we have to make the equation X (z2+ ayz+ bzx+ xy)+ czz+fyz+ gzx+ hxy=Q break up into z=0 and another line. Hence X= — h, and the equation of the other chord is (g-bh)x+ (f-ah)y+ (c-li) =0. But this chord is the polar of Z with respect to the conic. The equation of the polar of Z is (g-bh)x+ (f-dh)y+ (2c-ga-fb)=0. Hence ga+fb=h+c, Avhich is the condition which must be satisfied by the co- efficients in order that the conic czz+fyz+ gzx+ hxy=Q may be a conic of the system. Since the relation is symmetrical in c and h the inverse is also a conic of the system. Hence by quadric inversion with regard to any conic of NON-EUCLIDEAN GEOMETRY 43 the system the absolute is transformed into itself, and any conic of the system is transformed into a conic of the system. A single quadric inversion is thus analogous to a reflexion, while the general motion is produced by a pair of quadric inversions. These results could also be obtained by projection, for quadric inversion, in the case where the points X, Y are imaginary, can be compounded of ordinary inversion in a circle and a collineation. By a quadric inversion the pencil of lines passing through Z, which, together with the line XY, form a pencil of conies of the system, is transformed into a pencil of conies passing through 0. Hence we may extend the result of § 31 and say that every conic of the system is cut in involution by any pencil of conies of the system, the double points being the points of intersection with the absolute. Like the representation by circles, this representation admits of immediate extension to three dimensions. Planes are represented by quadric surfaces passing through a fixed conic, C. Two such quadrics intersect again in another conic. The linear metric is referred to an absolute quadric also passing through C, such that, if Z is the pole of the plane of C with respect to the absolute, any quadric which represents a plane cuts the absolute in a plane section, which is the polar of C with respect to the quadric. REPRESENTATION BY DIAMETRAL SECTIONS OF A QUADRIC SURFACE § 36. We shall briefly describe one other representation, due to Poincare.1 In this representation straight lines are represented by diametral sections of a quadric surface. 1 H. Poincare, ' Sur les hypotheses fondamcntales de la geometric,' Paris, Butt. Soc. math., 15 (1887), 203-216. Cf. also H. Jansen, ' Abbildung der hypcrbolischen Geo- metrie auf ein zweischaliges Hyperboloid,' Hamburg, Mitt. math. Ges,, 4 (1909), 409-440. 44 CONCRETE REPRESENTATIONS OF Project the quadric stereographically, i.e. with the centre of projection 0 on the surface. The two generators through 0 give two fixed points X, Y, and any plane section is projected into a conic passing through X, Y. The points at infinity on the quadric project into a fixed conic, also passing through X, Y, and the pole of X Y with respect to the fixed conic is a point Z, which is the projection of the centre C of the quadric. The tangents at infinity, i.e. the asymptotes, of a diametral section, pass through C, and their projections therefore pass through Z. Hence the projection consists of the net of conies which we considered in the last section. The angle and distance functions can therefore be deduced. At a point P a pencil is determined by the tangents to the diametral sections and the two generators, which correspond in the projection to the two lines passing through X, Y. The angle between the lines represented by the diametral sections is then proportional to the logarithm of the cross-ratio of this pencil. In a diametral section a range is determined by two points and the two points at infinity, which correspond in the projection to the intersections with the fixed conic. The distance between the two points is then proportional to the logarithm of the cross-ratio of this configuration on the diametral section. A circle corresponds in the projection to any conic passing through X, Y, i.e. it is represented by any plane section. If the quadric is ruled the points X, Y are real and the measure of angle is hyperbolic ; or parabolic if the quadric degenerates to a cone. The geometry is Hyperbolic, Parabolic, or Elliptic accord- ing as the quadric is a hyperboloid of two sheets, an elliptic paraboloid, or an eUipsoid. § 37. If the quadric is projected from the centre, diametral sections become straight lines ; the points at infinity give again a fixed conic, the section of the asymptotic cone ; and any plane section projects into a conic having double contact NON-EUCLIDEAN GEOMETRY 45 with the fixed conic, so that the representation is by the Cayley-Klein projective metric. The close connection between the representation by dia- metral sections of a quadric surface and that by diametral sections of a sphere is now apparent. There is an apparent gain in the generality of the repre- sentation if the centre of projection 0 be chosen arbitrarily. The tangent planes through 0 to the asymptotic cone project into two straight lines cutting in Z, the projection of the centre. These lines are tangents to the conic which corre- sponds to the points at infinity, and the points of contact are X, Y. A plane section projects into a conic passing through X, Y, and its asymptotes project into the tangents at the points of intersection with the fixed conic. For a diametral section these tangents pass through Z. Thus we obtain once more the same representation by a net of conies through two fixed points, and there is no gain in generality by this general projection. The extension of this representation to non-Euclidean geometry of three dimensions requires Euclidean space of four dimensions. The representation is by diametral sections of a fixed quadratic variety, which must not be ruled, i.e. a tangent 3-flat must cut the variety in an imaginary cone. The geometry is Hyperbolic or Elliptic according as the variety cuts the 3-flat at infinity in a real or an imaginary quadric. DUNCAN M'LAEEN YOUNG SOMMEEVILLE ON THE ALGEBRAICAL SOLUTION OF IN- DETERMINATE CUBIC EQUATIONS PART I § 1. Theorem. If a particular non-zero solution of a homogeneous indeterminate cubic equation be known, then an algebraical solution can in general be found. Let (Xv Xz, . . ., Xn)=Q (1) be a homogeneous indeterminate cubic in n variables Xlt Xz, . . ., Xn, and let it have a particular non-zero solution, say Jf1=a1, Xz=az, . . ., An=aB, (2) so that <£(«!, az, . . ., an)=0 (3) where by hypothesis alt az, . . . an do not all vanish. Now make the substitutions X^Xjr+a^ Xz=xzr+az, . . ., Xn=xnr+an (4) and equation (1) becomes on expansion in powers of r A^+A^+Ajr+^a^ a2, . . . aj=0, (5) where A3, Az, A1 are homogeneous integral functions of xlt xz, . . ., xn of the third, second, and first degree, respectively. The term in equation (5) independent of r vanishes by (3). The coefficient of r can be made to vanish by solving the equation A^Q (6) which being linear and homogeneous in xlt xz, . . . xn can always be solved. Let the value so found for xn say in terms of xlt xz, . . . xn,1 be substituted in Az and A3, which will in general be finite functions of xlt x2, . . ., »n-i« The equation (5) is then identically satisfied by taking r=-A'tfA's (7) where A'z, A'3 are what A2, A3 become when xn is expressed in terms of xlf xz, . . ., xn_^ by (6). The values of xn and r, given 48 ON THE ALGEBRAICAL SOLUTION OF by (6) and (7), when substituted in (4), furnish the solution. Moreover, since the original equation (f>(Xi, Xz, . . ., Xn)=Q is homogeneous, we can make the solution integral by multiplying the value of each of the roots given by (4) by the algebraical quantity A'3 and by the numerical quantity which is introduced from the fact that the value of xn is in general fractional ; and since A'2, A'3 are integral homogeneous functions of xlt x2, . . ., xn_v it follows that the solution presents the roots Xlt X2, . . ., Xn as rational integral homogeneous functions of the third degree in n— 1 variables xlt xz, . . ., xn_r If any of the quantities ait a2, . . ., an instead of being numerical are arbitrary literal quantities, they will appear in the final values for Xlt X& . . ., Xn as variables, and will therefore alter the number of variables in, and the degree of, the final solution. § 2. If the equation <£=0 is not homogeneous, and integral solutions be required, some care in the choice of particular solutions and in a suitable preparation of <£ must be exercised to secure this end. An example of this is given in Question 6 below. § 3. This process is naturally open to failure when the equation under consideration admits only of solutions of a certain type. An example of this is the equation x3+y3=2z3, which admits only of solutions of the type (k, k, k), or of the type (k,—k, o),1 and the application of the method furnishes only the same type of solution. § 4. It is to be remarked that if <£=0 is a homogeneous cubic in three variables, the solution does not present the roots as functions of two variables in accordance with § 1 ; for <£=0 may be regarded as a non-homogeneous cubic in two independent variables, and the solution will not present the roots as functions of even one unknown, but is again par- ticular, and being in general fractional and distinct from the 1 Euler, Elements of Algebra, part ii. chap. xv. § 247. Fourth edition. 1828. INDETERMINATE CUBIC EQUATIONS 49 assumed solution, in this way an infinite number of fractional solutions is found. The reason why the method fails to give an algebraical solution in this latter case is because equation (6) gives a linear and homogeneous relation between xl and a;2, and when it has been satisfied equation (5) is no longer an indeterminate one between r and xl but gives a unique determinate value for the single variable formed by their product XjT. Thus both values of the variables Xlt X2 in (4) are particular. (See Question 5 below.) Again, if an indeterminate equation of any degree can, by regarding certain of the variables as constant, be considered as an indeterminate cubic in two or more variables of which a particular solution is known, then another solution can be found, but since it will present the roots as functions of the variables which were for the time regarded as mere coefficients, it is clear that the solution will always be algebraical.1 Finally, if <£(.X\, X2, . . ., Xn)=0 be an indeterminate cubic of which a particular solution is known, namely, and if ^(Ylt Y2, . . ., Yr) be a function of the third degree, not necessarily homogeneous but containing no constant term, of r variables Ylf Y2, . . ., Yr, none of which are identi- cal with any of X1} X2, . . ., Xn except those whose value for the particular solution of =0 is zero, then the equation (Xlf x2, . . ., xn)=o can be solved. For it is evidently an indeterminate cubic in n+r variables having the particular solution YI= Y2= . . . = Yr=0, Xl=a1, X2=a2, . . ., Xn=an. An example of this is given in Question 2 below. § 5. It is to be further remarked that if <£=0 is a quadratic indeterminate, then equation (5) does not contain A3r3, and solutions are obtained by simply taking r=—A1/A2, and 1 See Part II, Section I : On the Algebraical Solution of the Equation Question 1. 50 ON THE ALGEBRAICAL SOLUTION OF these as before may be integralised if the equation be homo- geneous. On the other hand, if <£=0 be an indeterminate biquadratic, in at least four unknowns, then equation (5) will contain an additional term Atrl where A± is homogeneous and of the fourth degree in the unknowns. When equation (6) is solved and the value of one of the variables so determined is substituted in Az, A'2=Q becomes an indeterminate quad- ratic in at least three unknowns. If, therefore, rational solutions of -42=0 can be found, equation (5) is solved by taking r=— A3/A4, and it is clear that in general the solutions of the final equation <£=0 will be numerical or algebraical according as those of A2 are algebraical or numerical. It is, however, exceptional for the subsidiary equation A2=Q to yield rational solutions.1 § 6. The above convenient method for the solution of indeterminate cubics will be illustrated by some typical examples. From these it will appear that there are few problems, if any, in indeterminate cubics to which it is in- applicable, or in which the results furnished are less general than those of another process. QUESTION 1. Solve algebraically the equation (i) Let w=3, so that we have to solve P03=P^+P^+P^ (1) Here we may take as our particular solution Po^P^X, P2=-P3=(i (2) Making then the substitutions PQ=x0r+\, Pj^r+X, P2=z2r+[ji, P3=z3r-[A, (3) equation (1) takes the form (av-+ X)3= (X or, on expansion (x(?-x13-x23-x3*)r3+ 3(Xz02- Xa^2- j + 3(X2:r0-X2a;1-!Ji2z2-(x2a;3)r=0. (4) 1 See the writer's paper: 'On the Algebraical Solution of the Indeterminate Equation XA'4 + ft Y* = vZ* + p V* ' : in course of preparation. INDETERMINATE CUBIC EQUATIONS 51 To make the coefficient of r vanish we may take *o= (^2*1+ V***+ f^s) A2 (5) and equation (4) is then satisfied by taking r=3(xs12+ [*z22-f*:r32-XV)/(V-*i3-*23-*33) (6) Substituting the value of x0 from (5) in (6) we derive If we nowput A-(X2a;1+[ji2a;2+(x2a;3)3-X6(a;13+a;23+a:33) (8) equations (3) take the form A • P0= +3x(X2z1+ ^xz+ [*2*3)Sx V+ I* A • P= A • P2=A(a;2r+!i)=(x(X2a;1+!Ji2a;2+(A2X3)3-XV(a;13+^23+a;33) + 3x A • P= (9) As equation (1) is homogeneous, (9) is the integralised form of its algebraical solution and presents the roots as rational functions of five variables xlf xz, x3, X, ;/. ; of the third degree in xlt xz, xa, the ninth in xlt xz, xa, jx, and the tenth in xlt xz, As a numerical example, a?1=a;2=X = l,— a;3=p(.=2 gives, on removal of the common factor, 33+43+53=63, the lowest solution which exists. (ii) Let -»=4, so that we have to solve • P03=P13+P23+P33+P43 (!') Here we may take as our particular solution P0=P1=X, P2=-P3=[z, P4=0 (2y) Making then the substitutions P =x r+ X P =x r-\- X P —x r-\- u. P =x T— y. PA=X*T, (3') equation (!') takes the form (x0r+ X)3= (XjT+ X)3+ (xzr+ (i)3+ (x3r— [x)3+ (x^r)3 52 ON THE ALGEBRAICAL SOLUTION OF or on expansion (#03— xf— x23— xa3— #43)r3+3(o;02X— o^X — x22[i.+x3z^)r2 + 3(a;0X2-:r1X2-cc2[A2-a:3!A2)r=0 (4') To make the coefficient of r vanish we may take z0= (X%+ 1^2+ f^ag/X2 (5') and equation (4') is then satisfied by taking r=3(X*12+ [^22- [LX3*-ix(?)l(x* 3 [ /y* 3_j_ /v» o 1 />» Q\ •*• 1 *— * V**''! * ~T~ / ** \ " t*/i |~ wt. «X/o~j~ JX *^"l/ " V 1 l^ *^2 1^ *vo ] *X/£ 7 AT2= A'( A'P3= A'(a;8r- pi)= - fi(X2x1+ iA2+ !.2a:3)3+ X (9') As before (9') is the integralised form of the algebraical solution of (!'), and presents the roots as rational functions of six variables xlt xz, xs, xt, X, JA ; the roots being of the third degree in xlf x2, xa, x^ the ninth in xv xz, xs, xt, (x, and the tenth in xv xz, x3, x±, X, [>.. As a numerical example Xj=a;2=a;3=a;4=X=(jL=l gives 493=473+243+l3+l3. (iii) In general, the assumptions 1 1 Of. Mathetnatics from the Ediu-ational Times, Now Series, vol. iv. No. 15225. INDETERMINATE CUBIC EQUATIONS 53 v)3+(xi x2mr-p)3 are sufficient to give algebraical solutions of equation (A) for the two cases n=2m and n=2m+l. In the former case the roots will be functions of the 2m quantities xlt x2, . . ., x2m, and of the ra quantities X, p, . . ., p ; in the latter the roots will be functions of the 2m+l quantities xlt x2, . . ., x2m+l, and of the ra+1 quantities X, jz, . . ., p. QUESTION 2. Solve the equation xy(x-y)=-kz3 (1) knowing a particular solution, say x—a, y=b, z=c so that a&(a-6)=Xc3 (2) Put x=Xjr+a, y—xzr+b, z=x3r+c, (3) and (1) on expansion and rearrangement becomes (x1—x2 + (ab x1—xz+a—b axz+bx1— 3Xc2ic3)r=0 (4) Hence, making the coefficient of r vanish by taking x3—(ab x^—Xz+a—b ax2+bxj)/3^c2, (5) equation (4) is satisfied by taking AX3 — XiX2(Xi X2) - 9a6(o. - ft)(a8 _ - 6) V - 3ab\5a3 - 6a26 + ^)x^ + 3a26(2a3 - Gab* + Sft^z,' - a'(26 - a) V on substituting the value of x3 given by (5) and replacing Xc3 by ab (a—b) from (2). Hence, if we call the denominator of r, A, we find, finally, ^y=(x2r+b)=b(2a-b)3(bx1-ax2)3 ^z=(x3r+c)=-(a+b)(2a-b)(2b-a)(bx1-ax2)3 which, it is to be remarked, is not an algebraical solution, but 54 ON THE ALGEBRAICAL SOLUTION OF gives a second solution when one is known. Thus if X=6, then x=3, y=2, 2=1 is a known solution, and (B) gives another solution #=128, y=5, 2=20, from which we may derive a third, and so on ad infinitum. If, on the other hand, the original equation had been in which y.lt y.z, . . ., \t.n are given coefficients and Vv F2, . . ., Fn variables, since Fj= F2= . . . = Fn=0, x=a, y=b, z—c is, by hypothesis, a particular solution, the substitutions (3) and V,=t,r would have led to an equation in no wise differing from (4) except that the term SnA3 would have appeared in the coefficient of r3. Hence, subtracting 27x2c6(2[A/s3) from the denominator of r in (6), it is clear that the final solution of (!'), omitting the common denominator, would be Vs=-9ab(a-b)(a?-ab+b2)X2ts, >x where X is written for bxv—axz, and this solution is obviously algebraical. Thus if we put n—1, ^=5, the equation which has the obvious particular solution F = 0, x = 3, y = 2, 2=1, has the algebraical solution F= -31 z= -20X3-4860f. Thus X = 4, t = 1 gives the solution F = 1512, x = 3597, y = 382, 2 = 1535. QUESTION 3. Solve the equation xy^-^^tf + W, (1) knowing a particular solution say x = a, y = b, u = c, v = d. This equation is of the fourth degree in x, y, u, v, but by putting y = b it becomes bx(xi - b2) = uz + 2tf (2) INDETERMINATE CUBIC EQUATIONS 55 which is an indeterminate cubic in x, u, v having the particular solution x = a, u = c,v = d. Hence, to find an algebraical solu- (3) (4) (5) tion, we put x = xp + a, u = x3r + c, v = x3r + d and (2) on expansion and rearrangement becomes + (3a'26 - b3x, - 2d«2 - 4dx3)r = 0. Hence, making the coefficient of r vanish by taking + 4(c2 equation (4) is satisfied by taking r = (xi2 + 2xi-3abx1z)/bx3 = [(3a26 - 63)2 - 24a6(f]a;12 - 4c(3a26 - SbdFx* on substituting the value of xs given by (5). Hence, we find x = [{ (3a26 - 63)3 - 16abd*}x* - 4c(3a2& - bs)x'zx u = [Sbcd2x* + { (3a2b - 63)2 - - 4c(3a26 - - b3) + 32bdt}xl3 + 4(3a26 - 63)(3c2 + 2=0 (3) Hence, making the coefficient of r vanish by taking x^-d^/Xa2)^ (4) equation (3) is satisfied by taking r=-3(Xa»1a+ !%12)/(Xz13+ ivfl--9HtolQ*f-vVfar (5) Now since x-L= — (\ibzl'kaz)yl, and equation (5) gives merely a particular value of rylt it is clear that the values of x and y obtained from (2) will also be particular, viz. : z=a(l+n&3)/(xa3-n&3), 2/=-6(l+Xa3)/(Xa3-(A63). (6) In fact it is clear from the above and the solution of the first part of Question 2, that no greater generality would have been obtained by starting with the equation xZ3+!x73=I/Z3 (7) knowing the particular solution X=a, Y=b, Z=c. INDETERMINATE CUBIC EQUATIONS 57 For if in (6) we replace X, jx, a, b by X/i/, \LJV, ale, b/c respec- tively we shall obtain for (7) that if Xa3+[x&3=vc3, (8) then xXHuPWZ3, where Z=a(ft63+vc3), 7=-&(Xa3+^c3), Z=c(Xa3-(x63). (9) For example, if the equation were then a= — 1, 6=1, c=l ; X=5, !A=6, v=l, and consequently X=-7, 7=4, Z=-ll. From this solution we may proceed to derive a third, and so on ad infinitum. Cor. If in (8) we put X= \j.=v, we derive that if as+63=c3 then i a3(63+c3)3-63(c3+a3)3=c3(a3-63)3. QUESTION 6. Express 5, 17, and 41 algebraically, each as the sum of five integral cubes.2 Taking the case of 5 first, since 5=53+23— 43— 43— O3, if we assume 5= (£-4)3+ (af-4)3+ (_af+2)8+ (&£)3+ (-&£+ 5)3 (1) it is clear that, on expansion, the coefficient of f3 will be unity, while the constant term disappears. If, therefore, we can make the coefficient of £ vanish by expressing both a and 6 integrally in terms of some unknown, then we shall obtain an integral algebraical value for £ which will render (1) an identity. Now (1) on expansion is £3+3(-4-2a2+5&2)^2+3(16+12a-256)f=0. (2) Hence to make the coefficient of £ vanish we must have 256=12a+16, and this is done integrally by taking a =25t+ 7, 6= 12^+4. Substituting these values in (2), we find for the required value of £ 1590<2+660H-66. 1 This result is due to Tait. See Chrystal's Algebra, part i. chap, xiv., Ex. xx. No. 2. 2 Cf. Mathematics from the Educational Times, New Series, voL IT. No. 15225. H 58 ON THE ALGEBRAICAL SOLUTION OF Hence, finally, substituting these values of a, 6, f in (1), we find the identity 5= (1590*2+ 660*+ 62)3+ (39750*3+ 27630<2+ 6270Z+458)3 - (39750<3+ 2763C«2+ 6270*+ 460)3+ (19080*3+ 14280*2 + 3432<+ 264)3- (19080*3+ 14280<2+ 3432* + 259)3. Thus t=Q gives 5=623+4583-4603+2643-2593, while i— — 1 gives 5=9923-179323+179303-79683+79733. By a similar process we obtain 17EE-(108Z2-48£+4)3+(864*3-816;2+240J-22)3 - (540£3-456*2+ 126t- 13)3+ (540«3-456«2+ 126<-9)3 -(864*3-816<2+ 240^-21 )3, and41 = -(246*2-264f+76)3+(1968i3-3096*2+1680<-310)3 -(1968«3-3096f2+1680f-311)3+(738«3-1284f2 + 762Z- 157)3- (738f3- 1284«2+ 762f- 159)3. t=0, and t=l give respectively 17==_43_223+133-93+213, = -643+ 2663- 1973+ 20P-2673, and 41=-763-3103+3113-1573+1593, = -583+ 2423-2413+ 593-573. INDETERMINATE QUARTIC EQUATIONS 59 PART II FOB the purposes of Diophantine Analysis, biquadratic equations may be divided into two classes according as they do or do not admit of representation, by a perfectly general transformation of their variables, as indeterminate cubics. Thus, for example, the equation by the perfectly general transformation P-^x+y, Pz=u—v, P\=x—y, P'z=u+v, becomes i.e. x3y+xy3=v?v+uv3, which is an indeterminate cubic in x and u ; while, on the other hand, the equation does not seem capable, by a perfectly general transformation of its variables, of being represented as an indeterminate cubic in any number of variables. From what has been already shown,1 it is clear that the former class of equations admits of an algebraic solution (at least when the number of variables exceeds 2) by a process universally applicable, and it is to this class the present paper is confined, though the methods of solution will not be restricted to that already given. The biquadratic equations of the second class require special artifices for their solution and a separate paper will be devoted to them.2 1 In the writer's paper, Part I. 2 Viz., Part III. 6o ON THE ALGEBRAICAL SOLUTION OF SECTION I — On the algebraical solution of the equation § 1. Before attempting to solve this equation for all values of n and r, it will be convenient first to give solutions for a few particular values of n and r. QUESTION 1. Solve in integers the equation P1*+P24=PV+P'24. (1) Let P1=z1+23, P2=z2— z4, P'i=z1— z3, P'2=z2+z4 and the equa- tion becomes Zjfa+XjZf^zfzi+zff (2) This is an indeterminate cubic in zx and z2 and if any par- ticular solution is known, another can be found, but as it presents the new roots as functions of the coefficients z3 and z4, it will be an algebraical solution. Thus, putting z1=a;1r+?/1, Z2=x2r+y2, (2) becomes or, on expansion and rearrangement, -a;2z43)r+(t/13z3+2/1z33)-(i/23z4+7/2z43)=0. (3) To make the term independent of r vanish, we must have which is equivalent to knowing a particular solution of (1) since it may be written and to satisfy this we may evidently take 2/i+z3=-2/2-z4» 2/i-z3=:>/2-z4> «'•«• yi=-«4» y2=-«3 (5)- In order to make the coefficient of r in (3) vanish, we must take i.e. ^2=[(3y12z3+z33)/(3i/22z4+z43)]a:1 =[z3(3z42+za2)/z4(3z32+z42)]a:1 (6) on substituting for ylt y2 their values given by (5). INDETERMINATE QUARTIC EQUATIONS 61 Equation (3) is now identically satisfied by taking 8(<4 say dr=3z4(z42-z32)2(3z32+z42), where rf=a:1(z42+ z32)( 18z42z32-z44-234) Hence rfz1=a;1^r+^1=3a;1z4(z42-Z32)2(3z32+z42) +*1z4(z42+z32)(18z42z32-z44-z34) and by symmetry ^2=2^3 Hence dP1=d(z1+z3)=z1[2z4(z4«+ Thus, finally, we have the algebraic identity + [a;7 on writing x for z4 and y for z3. For example x=l, y=2 gives 764+12034=11764+6534, and x=l, y=3 gives 1334+1344=1584+594. 1 This identity ia due to Euler (Commentationes Arithmetical, vol. ii. p. 289), who obtained it by a different method. 62 ON THE ALGEBRAICAL SOLUTION OF § 2. The equation P14+P24=P'14+P'24+* is always soluble whatever be the value of k, provided a particular solution be known. For by putting P^x+y, P2=u—v, P\—x—y, and P'2=u+v it becomes 8x3y+ 8xy3=8u3v+ Sui^+ k which is an indeterminate cubic in x and u. We proceed to solve the case where fc=P'34— P34. QUESTION 2. Solve the equation P14+P24+P34=P/14+PV+^/34 (1) First method. We may assume (X1r+a)i+(xzr+b)*+(x3r+c)*=(x1r+d^+(xzr+e^+(xar+f)* (2) where by hypothesis a4+tf+c*=d*+e*+f*. (3) On expansion and rearrangement (2) becomes ayt- i?^f*xa)r=Q. (4) To make the coefficient of r vanish we must take *3H(a3-rf3)*i+ (&3-e3)o;2!/(/3-c3). (5) Equation (4) is then satisfied by taking * - 2[{(a - d>,» + (6 - ifrfK/* - c3)'^ +/c + e») - {(a3 - on substituting for xa its value given by (5). These values of x3 and r when substituted in (2) render it an identity and constitute a solution which is clearly algebraical. We may satisfy equation (3) in several ways. Thus we may put d=—a, f=b, e=c. Hence a;3=[2a3a;1+ (63-c3)x2]/(63-c3), and INDETERMINATE QUARTIC EQUATIONS 63 As a particular case put x1——x2=a—c=l, 6=2, so that ar3=-5/7, r= -21/13. Hence substituting in (2) the foregoing values we obtain 84+284+474=344+344+414. In the same way we may derive other algebraical solutions by putting d=±b, e=±c,f=±a, etc. Second method. In the identity replace a by x^ and 6 by Then t.e. 4\4 /T_4_2?/.4\4 fx 2\4 'l2 4\4 4\4 4_2« 4\4 f~ 2\4 (1) Taking the first only of these equations and integralising we have (2) - which is an identity of the kind required. Thus we have *: * » y-a f^j^f^ivt^jy 1 2 1 1 34+74+84=l4+24+94 1 1 2 1 84+164+334=44+244+314 of which the former is the second smallest solution which exists, the smallest being 24+44+74=34+64+64. Cor. In equation (2) replace xlf ylt xz, yz by their recipro- cals and multiply each root by x-^-y^-x^-y^. We thereby obtain 64 ON THE ALGEBRAICAL SOLUTION OF . (3) Adding now corresponding sides of (2) and (3) and omitting the terms which are common to both sides we obtain For example, xv=x^=y^=\, ^=2 gives 24+ 84+ 184+ 334= 64+ 144+ 244+ 314. Again, we have identically Multiplying corresponding sides of these equations together we obtain which is an algebraical solution of the equation P14+P24+P34+P44+P54=P'14+P' Again, we have identically 4 24 ~ * ~ Hence by subtraction we get r,4 — «.4\4 -w 7; Also — s— — = — — — s- =4. * INDETERMINATE QUARTIC EQUATIONS 65 If then we multiply corresponding sides of these last two equations together it is clear we shall obtain an identity of the form PS+PJ+ . . . +P74=PY+P'24+ . . . +P'74. The foregoing processes obviously admit of infinite com- bination and repetition. Third method. We have identically and (u+v)*+(u-v)*+(2v)*=2(u2+3v2)2. Hence we shall have (x+y)*+ (x-y)*+ (2y)*=(u+v)*+ (u-v)*+ (2w)« (1) provided x2+3y2=u2+3v2 (2) Now the most general solution of (2) is given by h arbitrarv Hence the most general solution of (1) is given by [(3xa+l)v+(3x2+2x-l)i/]4+[(3x2+l)t;+(3xa-2x-l)i/]* Thus X=— y=2, v=l gives 84+ll4+194=l4+164+174. Cor. 1. From the foregoing we may derive the solution of For it is clear that the integer N, which is equal to the product (a12+3612)(a22+3622) . . . (ar2+36r2), is expressible in the form p2+3q2 in 2r~1 ways, and therefore as above 2N2 is expressible in the form (x+y)*+(x— y)*+(2y)* in the same number of ways. In practice, where an arithmetical result merely is desired, it is easier to proceed as follows. Selecting the smallest r 66 ON THE ALGEBRAICAL SOLUTION OF mimber,7, which isof the forma2+362, since 74=492=(l8+3-42)2, we have \ (3) Hence it is clear that we may replace x and y by -^-- and x~t y respectively (or by x~ y and - x~ y respectively, etc.) without altering the value of the left-hand side. Thus the next repetition gives /65a;+39y\* /39z-l%\« .„ ~72~~7 +( V~) ' and the process may be continued ad infinitum. Also we observe that we may interchange x and y without altering the left-hand side so that we have also 72— Thus for example if we put in (3), (4), (5), (6), x=l, y=3 we obtain 494+ 1474+ 196*=844+ 1194+ 2034=94+ 172*+ 1814 =284+ 1614+ 1894=1014+ 1034+ 2044. Cor. 2. Changing the sign of y in equation (3) we get INDETERMINATE QUARTIC EQUATIONS 67 Hence, subtracting (7) from (3) we derive x-5y)*+ (Sx+ 3-3z+3-:e4)4+(6z2-6-a;4)4 = (3z3+3z2-3z-3-x4)4+(3z3-3z2-3z+3+x4)4+(6z2-6+a;4)4 +J6:r(z2-l)j4. (1) Thus x—\, z=2 gives 84+ 1 74+ 284= 104+ 184+ 194+ 264. Since the quantity x only occurs in the form a;4 in (7) it is clear that we may replace it as before by Sz,,4— Syr4 and obtain an identity of the form. which holds for all values of n and r, except w=r=0. SECTION II — On the algebraical solution of the equation § 1. As in the previous case we shall first give solutions for a^few particular values of r. QUESTION 1. Solve in integers the equation xPjM-nP^xPY+txPY. (i) Assume as before X(*1r+o)*+ i>.(xzr+b)*=\(xlr+c)*+ v.(xzr+d)* (2) where Xa4+[A&4=Xc4+!Ad4 (3) 70 ON THE ALGEBRAICAL SOLUTION OF On expansion and rearrangement, (2) becomes 4(Xaz13+ [xfo;23-Xca;13-(idz23)r3+6(Xa2z12+ n&^-Xc2^2 -\td?xf)ra+tyla*x1+v.lfar-l(Pxl--\iLcPxjr=0 (4) To make the coefficient of r vanish we take Xl=u.(d3-b3)xzma3-c3). (5) Equation (4) is then identically satisfied by taking 3[X(c'-oa)g1«+ j*(d2-&2):r22J on substituting for xt its value given by (5). Hence x r, - 2[tA2(a-c)p3-63)3+X2(6-rf)(a3-c3)3] i.e. if we write A for 2|>2(a-c)(d3-&3)3+ X2(6-rf)(a3-c3)3] Also r r+??_3x(a3-c3)[|x(c2-a2)(rf3-63)2+X(^-62)(a3-c3)2] , =(d-&)(3d+6)(a3-c3)3X2+3(c2-a2)(a3-c3) (rf3-63)2X(ji+26(a-c)(d3-63)3(A2. Also * r,c_3ix(rf3-63)[^(c2-a2)(rf3-63)2+X(^-&2)(a3-c3)2 Also ,_-ix---- rf INDETERMINATE QUARTIC EQUATIONS 71 =(^-6)(c?+3&)(a3-c3)3X2+3(c2-a2)(a3-c3) Hence omitting A and the factor (a— c)(6— d) common to XjT+a, xzr+b, x^r+c, xzr+d we have that if Xa4+n&4=Xc4+[^4, (3) then also x[2a(a2+oc+c2)(a3-c3)2X2-3(6+d!)(a2+ac+c2)(a3-c8) = X[2c(a2+ac+c2)(a3-c3)2X2-3(6+^)(a2+ac+c2)(a3-c3)(d3-63) that is to say, if one solution of equation (1) be known, another can be found. This second solution will not be algebraical, nor if we attempt to satisfy (3) by putting c2=a2, d!2=62 will it be anything but nugatory. If, however, any solution of (3) other than c2=a2, d2=b2 be used, a second solution, in general distinct from the first, will be found and so on ad infinitum. Thus, for example, to solve the equation P14+3P24=P'14+3P'24, starting from the particular case 24+3-34=44+3-l4, we derive the following : — ct b c d JV+SfVofa'+VV 2 -3 4 1 234+3-274 = 37H-3-1* 2 3 4 -1 6614+3-1474 = 127H-3-5034 2 3 4 1 11414+3-25894=32174+3-17694 -2 3 4 1 37034+3-56914=65174+3'47534 72 ON THE ALGEBRAICAL SOLUTION OF Again, putting d—0, we have that if Xa4+ [i&4=Xc4, then also X[2a(a2+ac+c2)(a3-c3)2X2+364(a2+ac+c2)(a3-c3)X(i + (3c+a)&V]4 + (x[6(a2+ac+c2)(a3-c3)2X2+3(c+a)65(a3-c2)X|i+26V2]4 =X[2c(a2+ac+c2)(a3-c3)2X2+364(a2+ac+c2)(a3-c3)64X(i + (c+3a)&V]4 + ^[36(a2+ac+c2)(a3-c3)2X2+3(c+a)(a3-c3)&5X!A]4. Thus to solve P14+5P24=P'14+5P'24, since !4+5-24=34+5-04, we derive a 6 c Pj* + 5P24=PV + 5-P'24 1 2 3 194+5-2814=4174+5'1174 -1 2 3 714+5-1014=1474+5-634 § 2. The equation XPj4+ [iP24= is always soluble whatever be the value of I, provided a particular solution be known. For by putting P-^ Pz=u—v, P'l=x—y, and P'z=u+v, it becomes which is an indeterminate cubic in x and u. We proceed to solve the case where Z=v(P'34— P34). QUESTION 2. Solve the equation xP14+!*P24+I,p34=xPY+I,p'24+J'P/34> (i) knowing a particular solution, say Xa4+[/.&4+j/c4=Xd4+(z.e4+i'/4 (2) Here we may assume for equation (1) X(Xir+ a)4+ \».(x2r+ 6)4+ v(xar+ c)4= X(x1r+ d)*+ i>.(x2r+ e)* +"(*V+/)4 (3) which on expansion and rearrangement becomes To make the coefficient of r vanish we must take =0 (4) (5) INDETERMINATE QUARTIC EQUATIONS 73 Equation (4) is then satisfied by taking _ 2[{X(a - d) on substituting for a;3 its value given by (5). These values of x3 and r when substituted in (3) render it an identity and constitute a solution which is clearly algebraical. To satisfy equations (3) the solutions d?=a2, e2=62, /2=c2 obviously make r zero and therefore lead to no new result, but we shall presently show how solutions of a different character may be obtained whatever be the values of X, (i, and v. § 3. We shall now show how the equation may be solved by a single formula which holds for all values of r except zero. Let us first solve the equation ^l4+^24=^/l4+^/24+"(2P)4. (1) Putting Q^a+b, Q2=c-d, Q\=a-b, Q'2=c+d, P=ax, (1) becomes Ia3b+lab3=u.c3d+v.cd3+2va*x* (2) If now we take d=\ab3/\nc3, (2) is satisfied by taking i.e. a=x6([A2c8-X268)/2[xVc8rc4 (3) so that rf=X264(!x2c8-X268)/2[A3j/c11a;4. Hence, omitting the common denominator, and replacing x throughout by xjc2, we have as a solution of (1) Thus we have the identity +!Ji[2iJL3rc4a;4-X2&4((A2c8-X268) 4+!i[2(Ji3j/c4a;4+X264([A2c8-X268 4 (4) 74 ON THE ALGEBRAICAL SOLUTION OF If in this we replace #4 by x*— y* we obtain - X264( [i2c8- X268)]4+ if 2x |i6c( ; tx2c8- X268)?/]4 »4-?/4)+X264 4 (5) Equation (5) is an algebraical solution of the equation for all finite values of X, JA, and v. Thus, putting c=x=2, b=y=\, equation (5) becomes - X2(256(A2-X2)]4 (6) For example X=2, |i=l, ^=3 gives 34+ 2- 334+ 3- 144= 1 74+ 2- 234+ 3- 284. r=r r~r Again, if in (4) we replace vx* by Svra;r4— 2vryr*, we obtain an equation 4-Si'r2/r4)+ X264(jx2c8-X268)]4 *) (7) which is an algebraical solution of the equation X1P14+X2P24+ . . . +Xr+JP?+2=X1P/14+X2P'24+ . . . + Xr+2P r+2 for all values of r except r=0. Fhially, if in (7) we put all the ?/'s]equal to zero, we obtain X[6c3|X[i((i2c8-X268)+ 2[*32j/ra;r4S]4 -X264([jL2c8-X268)]4 = X[6c3|X (i( [X2c8- X268) - 2[x32JVrr4j]4 X268)]4 (8) INDETERMINATE QUARTIC EQUATIONS 75 which is an algebraical solution of the equation where X, (i, vj, ^2, v3, . . ,,vr are arbitrary coefficients, positive or negative, which are unrestricted except that >. and \>. are both finite, and the v's do not all vanish — a result of consider- able generality. 76 ON THE ALGEBRAICAL SOLUTION OF PART III § 1. The equation has been shown by Euler x to be insoluble when w=2, and there is reason to believe that it is also insoluble when w=3, although no demonstration has ever been given.2 The case w=4 does not appear to have been solved either algebraic- ally or otherwise, but the present writer has discovered one numerical solution, which shows that the equation is soluble. When n exceeds 4 there is no great difficulty in obtaining algebraical solutions, but these are of a very specialised character, on account of the particular assumptions made as to the forms of the roots, and are not at all to be regarded as typical of the general rule. It will be convenient to commence with the cases n=5, 6, and 7 (§§ 2-4) from the formulae for which it will be shown algebraical solutions for all values of n greater than 7 may be deduced, then to proceed in the light of these results to discuss the case w=4 (§ 6), and finally to give algebraical solutions for all values of n greater than 2 of the equation transformed by replacing P04 by P02 (§ 7). § 2. QUESTION 1. Solve the equation P0^=P^+ P24+ P34+ P44+ P54. We have identically ( and 1 Elements of Algebra, Fourth Edition, 1828, Part n., Chap. xiii. §§ 206-208. * Cf. Euler, Commentationes Arithmetics, vol. I., xxxiii. § 1 ; vol. n., Ixviii. § 3. INDETERMINATE QUARTIC EQUATIONS 77 Hence we shall have (w2+v2)4=(%2-v2)4+ (2uv)*+(x+y)*+ (x-y)*+ (2y)* (1) provided 2uv(u2-v2)=x2+3y2 (2) Now if for the moment we regard v as a constant, equal to v' say, this equation may be written which, being a non-homogeneous indeterminate cubic in u, x, y, can be solved algebraically, if a particular solution is known.1 But a particular solution of it is obviously u—2, v'=l, cc=3, y=l, and others are easily found, for example, (u, v, x, y)=(7, 6, 3, 19), or (7, 6, 15, 17), or (7,6, 27, 11), or (7, 6, 33, 1). Hence an algebraical solution may be found. To solve (2) we may therefore put u=Xjr+u', v=v', x=xzr+x', y=x3r+y' (3) where we suppose (u', v', x', y') to be a particular solution of (2), i.e. 2ii'v'(u'2-v'*)=x'2+3y'2 (4) Making the substitutions (3), equation (2) then becomes 2(x1r+u')v'[(xlr+u')2-v'2]=(x2r+x')2+3(x3r+y')2 or, on expansion and rearrangement according to powers of r, in virtue of (4), (2v'x1*)r3+ (6u'v'x12-x22-3x32)r*+ 2[x1v'(u'2-v'2) + 2u'2v'x1-x&'-3x3y']r=Q (5) To make the coefficient of r vanish we must have say, x^Zu'W-v'^-x'^IZy' (6) Equation (5) is now satisfied by taking r= (a;,1 + 3ay - SuVa;, s)/2n V _ -W + (3u'°-v' - vy-xS - 2x'(3u"v'-v'*)xlxt+tfixt'> (7) on substituting the value of x3 given by (6). 1 See the writer's paper, Part I. 78 ON THE ALGEBRAICAL SOLUTION OF Hence we have ' - 2x'(3u'« - , _ , _ 6« W + MSn'* - 1/2)2 .,r ! , = ---»- «-^tt-x1-aia 18y'3o^s /2- v'22- ISu ' In these equations u', v', x', y' must have such values as satisfy equation (4). Thus if we take u'=2, v'=l, x'=3, y'=\, we shall have u= (97 x^- 66:^2+ 1 2x2z)/Qx1z, v=l, ?/=(953a;13-981a;12a;2+330a;1a;22-36.T23)/18a;13. These give, finally, the identity [(97o;12-66a;1a;2+ 12a;22)2+ (6a:12)2]4=[(97a;12-66^+ 12cc22)2 -(6V)2]4 +[2a;1(1007a;13-726a;12a;2+132a;1a;22)]4+[2^1(899a;13-1236a:12a;2 +528a;1a;22-72a;23)]* (8) The solution w=2, v=\, x=3, y=l gives from (1) 54=3*+44+44+24+24 and w=7, v=Q, a;=3, y=19 gives 854=844+ 384+ 224+ 164+ 134. If in (8) we put ^=1, a;2=2, we derive 2054=1664+ 1564+ 1334+924+744. It is to be remarked that if we have any solution of the form P04=/Y+P24+ (x INDETERMINATE QUARTIC EQUATIONS 79 we may derive others from it of the same form by multiplying by (p2+32+322)4 is expressible in the form 2(AZ+W2)2, where A = (p2- 3qz)x+ Qpqy, B= 2pqx- (p*-3q2)y, (A and B having in general a variety of values depending on the composite character of x*+3yz and of p2+3qz). Hence (9) becomes \P0(P*+ 3gT=iPi(F!+ 3?2)|4+ \Pz(p*+ 322)S4+ 2(^2+ 352)2 + (2J5)4 and since p and q are arbitrary this will give an infinity of solutions. In practice, where an arithmetical result merely is desired, we proceed as follows. Starting from any solution of the required form, say 54=44+34+44+24+24 =44+34+2-122. we multiply this by 74 say, 7 being the smallest integer of the form p2+ 3<72, and obtain 354=284+214+2-122(l2+3-42)2 =284+214+2(242+3-22)2 =284+ 214+ 264+ 224+44. A second application of this process gives 2454= 1964+ 1884+ 1474+ 1424+464. Similar results may be obtained by multiplying by 134, 194, etc. The defect of all the foregoing solutions lies manifestly in the assumption that P3+P4=P5, a restriction which such , identities as 314=304+ 174+ 104+ 104+ 104, 3134=3124+ 904+ 754+ 704+ 304, etc., show to be unnecessary. 8o ON THE ALGEBRAICAL SOLUTION OF § 3. QUESTION 2. Solve the equation Po^P^+P^+Pg^P^+Ps^Pe4. (1) In a former paper 1 there occurs the identity [6c3(c8-68+ 22z,,4)]4+ [2c*2x, 4-&4(c8-68)]4 =[6c3(c8-68-2Sa:n4)]4+[2c4Sa;B4+64(c8-68)]4+[26c(c8-68)]* From this we see that if 68-c8=2SxB4 (2) then [2c4Sxll4-64(c8-68)]4=[6c3(c8-&8-2Sa:n4)]4+ [2c4Sa;n4+64(c8-68)]4+[26c(68-c8)]4(a;14+a;24+ .... +«,*) i.e. on making use of (2) and dividing each root by c8— &8, a result which is immediately obvious from the fact that we have identically (&4+ C4)4= (&4_ C4)4+ 8&4C4(68+ C8) = (&4-c4)4+ (26c3)4+ 864c4(68-c8). Now the equation (2) is soluble algebraically when n is of the form 2r+2. For, if in the identity we replace x by ic4 and y by 4«/4 we derive (z4+4t/4)4-(z4-4y4)4=2(2zi/)4(:e so that multiplying each side by (a;4+42/4)4+(a;4— 4?/4)4 we have + (*4-42/4)4] (4) Hence as a solution of equation (1) we have, putting 6=x4+4?/4, c=z4-4t/4 in (3) P6= 8xy3(x*+ 4?/4) (a;4- 1 Viz., Part II. INDETERMINATE QUARTIC EQUATIONS 81 Thus we have the identity (z4-4t/4)3]4 + [8a*/3(z4+41=2v23 will convert u^+lGv^ into M224+ 212v224, which by (7) is expressible as the sum of four rational biquadrates (so that «272+212(2v23)24 is expressible as the sum of four or of six biquadrates), and since as before the sum of two of these four biquadrates will be w28+16v28, multiplied by a factor, it is clear that the successive substitutions u2=u3s, vz=2v33, and in general ur=ur+13, vr—2vr+l3 will enable the right-hand side of (8) to be expressed simultaneously as sums of biquadrates successively increasing by 2, i.e. we shall have a biquadrate equal to the sum of any even number of biquadrates greater than four. There is no great difficulty in writing down, in accordance with the above formulae, a biquadrate equal to the sum of 6+ 2n biquadrates, but such formulae will only give arithmetical results of a high order of magnitude, the reason being that they do not give a biquadrate merely equal to the sum of 6+ 2n biquadrates, but furnish a special kind of biquadrate which possesses the peculiar additional property of being expressible simultaneously as the sum of every even number of biquad- rates greater than four up to 6+ 2n. 84 ON THE ALGEBRAICAL SOLUTION OF § 4. QUESTION 3. Solve the equation Pf-Pf+Pf+Pf+Pf+Pt+ff+Pi* (1) In the identity (H- 1 )«=(«-!)«+ 8*+ 8*» put f=(o;4+2/4+z4)/8 and we obtain 4+y4+24+8\4/a4+y4+24_8y 4 4 4 (x4+y4+s4)3 (2) 8 / \ 8 / 64 Let us now choose x, y, and z so that To do this, since we have identically it will be sufficient to take x=2ab+b2, y=a?—bz, z=az+2ab, for since this makes xz+xy+yz=(az+ab+bz)z, we shall have 1 (2a6+62)4+(a2-62)4+(a2+2a6)4=2(a2+a6+62)4. Making these substitutions (2) becomes J L 4 + (a2+2a6)4+2(a2+a6+62)12/16. (3) Now since 2 - (3) may be written [(o2+a6+ 62)4+4]4=[(a2+a6+62)4-4]4+ (8a6+462)4+ (4a2-462)4 This is an algebraical solution of equation (1) and it may be integrated by multiplying each root by (cz+cd+d2), or 1 This solution of the equation 2P04=P!4 + P24 + P34 involves the assumption P1= P2 + P3, a restriction which the identity 2-4848134 = 575528' + 155873* + 1 167454 shows to be unnecessary. INDETERMINATE QUARTIC EQUATIONS 85 simply by taking as a particular case c—a, d=b, in which case it becomes [(a2+a6+62)4+4]4=[(a2+a6+62)4-4]4+(8a6+462)4-f(4a2-462)4 + (4a2+8a6)4+24(a2+a6+62)8[(a2-62)4+(2a6+62)4+(2a6+a2)4]. For example a=2, 6=1 gives 24054= 23974+ 7844+ 4904+ 2944+ 324+ 204+ 1 24. Another solution, closely allied but giving smaller results, may be obtained thus. We have identically (a2+462)4=(a2-462)4+2-2%262(a4+ 1664). (4) Now put 6=z2, a=x2+3yz, and this becomes |>2+ 3t/2)2+4z4]4=[(z2+ 3*/2)2-4z4]4+ (2z)4[(a;2+ 3*/2)4+ (2z2)4] x [(x+y)*+(x-y)*+(2y)*\ (5) For example, we have when x=y=z, 54=44+44+34+24+24, x=0, y=z, 134=124+84+64+64+54+44+44, £=3, y=z=l, 374=354+244+124+124+44+24+24, z=2, y=z=I, 534=454+424+284+144+124+84+44. Also, since the equation can be solved algebraically for all values of r greater than 2 (see § 7 infra), it follows that by putting 6=z2, a=Q0 in (4) that we can get algebraical solutions of the equation where s=7+2w. Again, write equation (5) in the form [(a;2+ 3?/2)2+4z4]4= [(z2+ 3i/2)2-4z4]4+ (4yz)*[( + (2z)*(x*+3y*)*[(x+y)*+(x-y)*]+(4:z*)*[(x+y)*+(x-y)*] (6) Now, an algebraical solution of the equation £14+g24=P14+P24+ . . . +Pr4 (7) has been found 1 for all values of r greater than 2 by means of a single formula. If then, we choose x and y so that we can, on replacing (x+y)*+(x—y)* by P^+P2*+ . . . +Pr4, 1 See § 3 of Part II., Section I. 86 ON THE ALGEBRAICAL SOLUTION OF in one only of the two places in which it occurs on the right- hand side of (6), say the second, write equation (6) as [(x2+ 3i/2)2+4z4]4= [(x2+ 3t/2)2-4z4]4+ (4i/2)4[(a;2+ 3y2)4+ (2z2)4] This is therefore an algebraical solution, by means of a single formula, of the equation P0*=PS+PZ*+ . . . +P,4, for all values of s greater than 6. As before, the defect of all these solutions is the unnecessary peculiarity possessed by one (or more) of the roots of the biquadrates on the right-hand side of the equation, viz., that it is always equal to the sum of the roots of two of the remaining biquadrates. § 5. From the foregoing results other solutions may be obtained for various values of n. For if we have solved, arithmetically or algebraically, the equations then each of the biquadrates on the right-hand side of the equation may be expressed as the sum of s biquadrates so that this equation gives a biquadrate equal to the sum of r+k(s— 1) biquadrates, where fc=0, 1, 2, 3, . . ., r. This is obvious and calls for no special exemplification. Also, for particular values of n, other independent formulae may be found. For example, let w=16. Then since (2u+ v)*+ (u+ 2v)*+ (u- v)*= 18(u?+ uv+ v2)2, we have, on putting u=2xy+yz, v=x2—yz, so that (x2+4:xy+yz)*+(2x2+2xy-y2)*+(x2-2xy-2yz) INDETERMINATE QUARTIC EQUATIONS 87 Now we have identically (a4+ 64+ c4+ d4)4— (a4+ 64+ c4- d*)*= 8d*(a*+ 64+ c4) [(a4+64+c4)2+d8] 4-d4). (1) Hence if we put a=xi+4:xy+y2, b=2x2+2xy—y*, c=x2—2xy—2y2, d=2(x2+xy+y2) equation (1) will become (a4+ &4+ c4+ d4)4= (a4+ &4+ c4-<24)4+ (2d3)4(a4+ 64+ c4) which gives a biquadrate equal to the sum of 16 biquadrates. Again, since we have identically and we have on multiplying corresponding sides of these equations together y\ (2) Hence if we can express x*—y* as the sum of r biquadrates, then equation (2) will give a biquadrate equal to the sum of l+4(r+l) biquadrates. Thus for example if z=(a4+4&4)4+ (a4-4&4)4, y=(a4+4&4)4-(a4-4&4)4, then by § 3, Question 2, x*—y* will be equal to the sum of 5 biquadrates, so that equation (2) will give a biquadrate equal to the sum of 25 biquadrates. These examples of the extension of the results of §§ 2-4 must suffice, for, with the increase in the magnitude of n, diminishes, naturally, the difficulty of solving the equation. § 6. We come now to the case w=4, i.e. to the equation P04=P14+P24+P34+P44. (1) As the assumption that P0 and Pt are respectively the 88 ON THE ALGEBRAICAL SOLUTION OF sum and the difference of two squares has led to solutions in all cases where n is greater than 4, it is natural to try this assumption here. Now if then P04-P1*=(w2+t£)4-(«*-08)4=8M8va(***+v*)' (2) Now every bi quadrate is of the form 5n or 5n+l ; hence three of the roots on the right-hand side of (1) are multiples of 5 always. If we assume that Pt is the root prime to 5, then from (2) 8wV(M4+*;4) must be divisible by 54. But this is impossible so long as u and v are both prime to 5. Hence, since P0 and Pl are both prime to 5, one of u, v is a multiple of 5 and the other prime to 5, and uzvz must therefore be divisible by 54. Since uzvz is always to be divisible by 54, this suggests that possibly u and v are both squares, the one always divisible by 5, and the other always prime to 5. This again suggests as a suitable transformation u=(x2+y2)2, v=(xz—y2)2, or on analogy with the results of Question 2, M=(z4+4«/4)2, v=(x4— 4«/4)2, each of which manifestly satisfies the required condition so long as x and y are both prime to 5. Hence we are led to the assumptions P0=(a;2+i/2)4+(a;2-i/2)4, P1=(x2+2/2)4-(x2-2/2)4 (3) or P0=(*4+4t/4)4+(a;4-42/4)4, P1=(a;4+4?/4)4-(a:4-42/4)4 (4) of which the latter is merely the specialised form obtained from the former by writing in it a;2 instead of x and 2y2 instead of y, It now remains to choose P2, P3, P4 in such a way as to make P24+P34+P44 a homogeneous symmetrical function of x and y of the 32nd or 64th degree according as we take assumption (3) or (4). This we may do in a variety of ways, but probably the simplest forms which present themselves would be Pz=2(x*-y*)[2xy(x2+y*)],] P= 2 or Ps=2(x*-y*)(xy*) and the forms obtained by replacing x by xz and y by 2y2. INDETERMINATE QUARTIC EQUATIONS 89 Hence we have as our final trial equations +\2(x*-y*)\*[\2xy(x2+y2)>f+\2xy(x*-y*)\*+ (z4-*,4)4] (5) =[(z4+4y4)Ma (*4+ 4«/4)j4+ J4 (7) /4)4- (a;4-4t/4)4]4 + j(2(a;8- 16«/8)»4[(z8+4a;y+ 16*/8)4+ (8z V)4+ (8*y )4] (8) Of these the first two are immediately to be rejected since they imply (see § 7, equation (5), infra) and [( +J2(a;8- 16i/8)j4(a;16+ 224»V+ 256«/16)2 respectively, equations which are known to be impossible.1 The remaining two agree in giving, the former when x=2 and y=I, the latter when x=y, (54+34)4=(54-34)4+(30)4(214+24+84) or on removal of the common factor 2 from the roots 3534=2724+3154+304+1204, a result which direct calculation will verify. Neither of the equations (7) or (8) however seems to yield any more solutions for other values of x and y ; and they must therefore be regarded as, at the best, only more or less likely approxima- tions to an algebraical solution. N.B. — Hence collecting the results of §§ 2, 3, and 6 we have 3534= 3154+ 2724+ 1204+ 304 =3004+ 2724+ 1804+ 1504+ 1354+904 =2724+2524+2344+ 1984+ 1894+ 1304+364+304 =3004+ 2724+ 1804+ 1504+ 1354+ 724+ 724+ 544+ 364+ 364 ; etc. 1 Euler, Elements of Algebra, I.e. 51 go ON THE ALGEBRAICAL SOLUTION OF § 7. Finally we may solve the equation for all values of n greater than 2, when a particular solution is known. To do this we may employ the method of Diophantos, and take We have then to solve the equation (y0r*+x0r+a0)*=(x1r+al)*+(xzr+az)*+ . . . +(xnr+an)*. (2) Now if a02=a14+«24+ . . . +an4, it is clear that on expansion of the square and biquadrates in (2) we can make the coefficients of r and r2 both vanish by solving linear equations for x0 and y0, and therefore on division by r3, (2) becomes a linear equation in r. The value of r found from this equation with the values of x0 and y0 already found substituted in it makes (2) an identity and it is an algebraical solution which clearly presents on integralisation the roots of the biquadrates as functions of the fourth degree in the n variables xv x2, . . ., xn. It is further clear that the equation *oJY=^l4+^24+ • • • +^Pn* (3) is in general soluble by the same method if a particular solution is known ; thus if X0=n, Xt=X2= . . . =Xn=l, we may take as a particular solution P0=a2, P1=P2= . . . =Pn=a. For particular values of n, other special solutions of (1) and (3) may be found which will give in general different numerical results from the foregoing. Thus take the case of (1) when n=3. We have identically (a2w2+6V)2+ (b*u*-aW)*=(au)*+ (bu)*+ (av)*+ (bv)*. (4) Let us now choose a, 6, u, v so that To do this we have necessarily a=x2—y2, b so that 2xyu=(x2+y2)v. INDETERMINATE QUARTIC EQUATIONS 91 Hence we may take ii:=x ~T~y > v=^2ixy and with these values equation (4) becomes (x8+ 14#42/4+y8)2=(a:4— yt)*+[2xy(x2+yz)~\*+[2xy(xz— «/2)]4. (5) Thus x=2, y=l gives 4812=124+154+204. Cor. The foregoing result evidently amounts to this, that if «262+62c2=c%2, then (a2-62+c2)2=a4+64+c4. Again, to solve (3) when X0=2, n=3, X1=X2=X3=1, since «*+ &4+ (a+ 6)4= 2(a2+ ab+ 62)2, replacing a and b, each by its square, we have a*+ 68+ («2+ b2)*=$[a*+ 64+a2+62]2. If then we put a=xz—y2, b=2xy, this becomes ;4] (6) by (5) above. Thus a:= 2, y=l gives 38+48+58=2(124+154+204)=2-4812. The last part of (6), which is simply a solution of a8+ 68+ c8= 2(a464+ 64c4+ c%4), is immediately obtained otherwise. For if a, b, c be integers connected by the equation a2+62=c2, we have, on squaring each side of this equation and on again squaring we obtain a8+ &8+ c8= 2(a464+ 64c4+ c4a4) which is the required result if we put a—xz—yz, b=2xy, c=x2+y2 to make «2+62=c2. Corresponding results may naturally be obtained by squaring any identity of the form + \iy2=vz2. But for all values of n greater than 4, a single algebraical 92 ALGEBRAICAL SOLUTION OF EQUATIONS formula can be found to satisfy equation (1). For we have identically If then we put t=x2+xy+y2 this becomes [(x*+xy+y*)z+zi]*=(xz+xy+y*)*+ (z*)*+z*[x*+y*+Jc+y*]. (7) Since x and y are arbitrary we may take (as in § 4 supra) x=Qi, y=Qz, where Ql and Qz are algebraical quantities satisfying the equation where r is greater than 2. Hence (7) becomes + . . . +P*) which gives a square equal to the sum of 5 or any greater number of biquadrates. ROBERT NOEEIE THE PROBLEM OF PARTITION OF ENERGY, ESPECIALLY IN RADIATION 1. AGREEMENT seems to be nearly as remote as ever regarding the manner in which deviation from the condition of equi- partition of energy amongst the various freedoms of an apparently conservative system, in apparent equilibrium, is brought about. The well-known discrepancies which occur between the actual ratios of the principal specific heats of gases and their theoretical ratios calculated, on the assumption of equipartition, from the multiplicity of freedoms which radiational phenomena make evident in the case of even monatomic gases, make the fact of extreme deviation from equipartition very evident. It is fully recognised that, in many very special cases, dynamical freedoms may be entirely inoperative. So one way of avoiding the difficulty consists in asserting that the special freedoms made evident in radiational phenomena are in- operative in ordinary thermal phenomena. Such a mode is unsatisfactory apart from the specification, by analogy at least, of an appropriate mechanism ; for the doctrine of equipartition does not permit mere partial inoperativeness — the inoperativeness must be total. Another method, adopted by Jeans, consists in regarding a final condition of statistical equilibrium between matter and ether, with consequent equipartition of energy amongst the freedoms, as unattainable in finite time ; so that the practical ' steady ' conditions, which subsist in experimental tests, and are the result of a steady supply of energy in one form compensating an equal steady loss in an other form, give rise to that non-equable 94 THE PROBLEM OF PARTITION OF partitioning of energy amongst wave-lengths which is expressed by Planck's well-corroborated law. A third method, that of Planck, locates the source of non-equipartition in the intrinsic nature of energy itself, which is postulated to be atomic, the ultimate unit being so large that it may only be manifested in relation to many degrees of freedom, some freedoms absorbing no units, others one unit, and so on. 2. Planck's postulate has the merit of leading to a well- supported expression for the distribution of energy amongst the various wave-lengths in ' natural ' radiation ; it has the possible demerit of necessitating discontinuities of motion on molecular, atomic, or, at any rate, on ' freedomal ' scale. Yet it may be that the seeming demerit is not real, the discontinu- ities vanishing as a matter of statistics. Sir J. Larmor, in his recent Bakerian Lecture (Proc. R.S., 1909, vol. Ixxxiii.), modifies and amplifies Planck's treatment in such a way as to get rid of the assumption of the finitely atomic nature of energy. Indivisibility of an element of energy is replaced by an unalterable ratio of the element of energy of any one type to the extent of a ' cell,' of correspond- ing type, in which that element is contained. The actual element itself may be infinitesimal, so motional discontinu- ities become infinitesimal. A ' cell ' replaces the ' degree of freedom ' of the previous treatment, and each cell is of equal opportunity or extent as regards an element of disturb- ance, which may pass from one cell to another of a different type, the amount of energy associated with it being possibly altered in the process. Thus — in analogy with the passage of heat energy in diminished (or increased) amount from a region of high to one of low temperature (or conversely), in association with the necessary performance (or absorption) of external work — we have the transformation of radiation from one wave-length to another in association with the performance or absorption of work. These postulates lead to Planck's law, the constants only having important modifications of meaning. ENERGY, ESPECIALLY IN RADIATION 95 3. Jeans has recently discussed Larmor's view, and arrives at the conclusion (Phil. Mag., Dec. 1910) that it is neither possible to avoid finiteness of the element of energy nor ultimate discontinuity of ether structure in relation to radia- tion if Planck's law corresponds to the true final condition of equilibrium. That is to say, radiation can only be regarded as capable of existing in the ether in amounts which are multiples of a finite unit. While Jeans' own view (§ 1) must be recognised as indicat- ing a possible solution of the fundamental difficulty regarding the partition of energy, it is not possible, because of our ignorance of the intrinsic nature of matter, of ether, and of the connection between these, to be quite certain that Larmor's view, or even Planck's, is inadmissible. It is not inconceivable that the nature of these entities may impose identity between the distribution which obtains in the steady state under experimental conditions, and that which would obtain in the final state of a strictly conservative system. I venture there- fore to indicate the following mode of considering the problem. It leads to an expression which differs slightly in form from that of Planck, but which can practically be identified with it throughout the range of observed wave-lengths, and which with it reduces to Rayleigh's form when the wave-length is of suitable magnitude. 4. Interchange of energy amongst freedoms of the same type constitutes ordinary transmission of energy of the type involved ; interchange of energy amongst freedoms of distinct types constitutes that transmission of energy which is ordinarily called transformation. When different subsystems, in the equilibrium condition, are freely open to interchanges of energy, a universal generalised temperature or potential, possessing a definite statistical value throughout the total system when that system possesses a definite total amount of energy, must exist. Let there be altogether v subsystems, let JVj . . . N, be 96 THE PROBLEM OF PARTITION OF the respective numbers of freedoms in these subsystems, and let Cj . . . cv be the respective capacities of these subsystems for energy. If P is the equilibrium value of the universal potential, the amounts of energy, E± . . . Et, in each subsystem are c^P . . . c^P respectively. As in Boltzmann's treatment, the equilibrium state is the most probable state ; and so, following Planck's modification of that treatment, the probability being estimated by the number of ways in which cP units of energy can be contained in N freedoms, we obtain as the condition of equilibrium the equation ?TlogX^Pd.CmP=() ... (1; l^t* Now, in the condition of statistical equilibrium, as in the approach to it, there is constant transmission of energy from one subsystem to another ; and the energy tends to accumu- late in those subsystems from which the rate of transmission is slowest. Hence the total rate of transmission tends to a minimum. So, rmcJP being the rate of transmission from the subsystem ra, we have, if the r's are constant, ^rmd.cmP=0 .... (2) These two conditions imply nothing more regarding the potential P than that it is statistically uniform throughout the total system. It might be slowly varying with time. If we further add the condition of conservation of energy, we get ^d.cmP=0 .... (3) The three equations give where a and 6 are functions of P alone. The simplest admis- ENERGY, ESPECIALLY IN RADIATION 97 sible conditions are a=aP"1, b=/3P~l, where a and y8 are absolute constants, in which case (4) becomes 5. To apply this expression to the case of radiation we have to evaluate Ejrm—EM the energy transmitted per second per unit range of wave-length in the neighbourhood of wave- length X. We must therefore either appeal to experiment for the determination of the appropriate forms to be given to Nm and rm in terms of X, or we must determine these by means of suitable assumptions regarding ether and matter and their connection. Thus if we assume that the fractional rate of transmission of the energy content of each freedom is identical per vibration, the fractional rate of transmission per unit of time is proportional to the frequency, so that we can write /3rm=y^~l, y being an absolute constant. The value of Nm, when the frequency is not too small, is given by Ray- leigh's reasoning (Sc. Papers, vol. iv. p. 484, or Phil. Mag., xlix. p. 539, 1900) as A^'\ where A is a universal constant. Hence ~' . (5), 6 r -1 an expression which, with Wien's displacement law holding, gives the well-known experimental result that the maximum energy is proportional to the fifth power of the absolute tem- perature provided that the latter be identified with P. The expression becomes identical with Planck's so long as aX is negligible relatively to y. We must therefore recognise that this restriction holds throughout the range of wave-length to which Planck's formula is applicable. Outside that range the quantity E^ becomes very small. If, within that range, Px becomes large relatively to y, N 98 PROBLEM OF PARTITION OF ENERGY the expression (5) reduces to Ay'1. PX"4, which is the form given by Rayleigh as applicable when Px is sufficiently large while X is not too large. When X is very large with P not too small, (5) takes the form Aa~l . PX"5. 6. The distinction between energy transmitted by, and energy stored in, definite freedoms is of fundamental import- ance. The equation (4a) shows that there is not universal equi- partition of the energy allotted to all freedoms except under the condition that $rm is negligible relatively to a ; while, on the other hand, there is equipartition universally amongst the energies transmitted per unit of time if ftrm is large relatively to a and small relatively to P. The ratio of the energy transmitted per unit of time per degree of freedom to the energy stored in that freedom takes here the place of the element of energy, and there is no limita- tion upon its finitude. There is necessarily equipartition of energy amongst all freedoms for which it has one and the same value. WILLIAM PEDDIB ON THE ACCURACY ATTAINABLE WITH A MODIFIED FORM OF ATWOOD'S MACHINE INTRODUCTION A CAREFUL determination of g by means of the ordinary type of Atwood's machine does not, as a rule, lead the average student in a physical laboratory to a better result than 930 or 940 cm/sec2. From the point of view of successful teaching, it is somewhat unfortunate that, after bestowing reasonable care and attention to his work, a student should be unable to obtain a result approximating satisfactorily to what he knows to be the correct figure. Not unnaturally, he takes it for granted that the actual numerical result obtained from his experimental labours is quite immaterial as long as the processes involved are clearly comprehended, and to him experimental physics is anything but an exact science. On the other hand, to set before the ordinary student a compli- cated apparatus specially designed for reaching an accuracy of O'l per cent, would be proceeding to the other extreme, and one could hardly expect much benefit to be derived from its use. But even the student who has already had some experi- mental training, and who has realised that quantitative relationship is just as important as qualitative, could not do any better in this case, for the defects are inherent to the method usually followed of timing the fall through a distance of 150 or 200 cm. with a metronome or stop watch : in his interests, at any rate, a more accurate procedure should be adopted. The purpose of the present paper is to show how the usual type of Atwood machine may very readily be modified so as 99 ioo ACCURACY OF ATWOOD'S MACHINE to give synchronous chronographical records of both time and distance at various points of the fall. The apparatus was, in fact, devised in the course of a research upon the wind-pressure law and the efficiency of air-drags, for the purpose of obtaining accurate time-distance curves for the fall of a parachute. Some results of this research will soon appear ; but as the calibration of the apparatus showed that it could be used with tolerable accuracy for the determination of g, it seemed desirable to publish a description of it, along with a few examples of the degree of accuracy that may be reached with it. The results here appended are not to be taken as giving the limits of accuracy of the method, for, unfortunately, the construction of the friction rollers in the apparatus used is not quite satisfactory, with the result that friction is somewhat variable and in need of constant evaluation : with a more carefully constructed apparatus friction would be much smaller, and its constancy more assured. DESCRIPTION OF APPARATUS A six-spoked aluminium wheel, cut with a V-groove in the rim, and mounted on a steel spindle, runs on four aluminium friction rollers. These rollers have conical sockets and are mounted on points screwed into a brass plate, so that if the steel spindle were completely homogeneous and were laid on the rollers, electrical contact would be complete from one side of the apparatus to the other during one whole revolution of the wheel. At one end of the spindle, however, a semi- cylindrical portion of the steel is removed and replaced by an identical piece of hard ebonite, the whole being then turned true in the lathe ; it follows that at a certain point during each revolution the ebonite is in contact with both friction rollers on which it rests, and at that moment electrical contact is no longer possible between the two sides of the apparatus. Hence, by connecting with a chronograph, a single record is To Relay and Chronograph To Chronograph 102 ON THE ACCURACY ATTAINABLE WITH obtained of the distance fallen through during each revolution of the wheel (being the distance equal to the effective circum- ference of the wheel and string), and of the time taken to describe that distance. A sensitive relay is interposed between the revolving wheel and the chronograph, so that a very small current may be used, and no sparking occurs during motion. In the experiments given in this paper a three-pen motor-driven chronograph was used in conjunction with a clock beating half-seconds, and times were easily obtained to xjy^th of a second. The third pen served to record the actual moment at which motion began. Its magnet is connected in parallel with a small electro-magnet held in an adjustable stand. When an experiment is about to be made, the pan with the smaller load, to the bottom of which is soldered a small iron disc, is brought down to this release magnet, and, if necessary, its height is adjusted until the point of no contact in the wheel revolution is just reached. Paper is inserted between the magnet and the pan to ensure immediate release ; the current is then broken, setting the system in motion and recording the initial point of the experiment on pen No. 3 ; every complete revolution of the wheel is recorded on pen No. 2, while the half seconds are marked by pen No. 1. The string used was a strong silk fishing line, fitting well into the groove and continued beneath the pans to form an endless loop. The effective fall for one pulley revolution was determined by attaching a 10-metre tape to one of the pans and reading against a fixed point the distances covered between successive markings of the chronograph while the pulley was slowly rotated. The mean of fourteen separate measurements gave an effective fall of 38-92 cm., and conse- quently a mean pulley radius £>=6-194 cm. The constancy of these individual measurements showed that the string did not slip appreciably in the V-groove. A FORM OF ATWOOD'S MACHINE 103 DYNAMICAL EQUATIONS AND DATA Putting L=\oad on each side, including pans and string, w= driving weight, P=weight of revolving pulley, p= effective radius of pulley, fc=radius of gyration of pulley, a= observed acceleration, a=radius of spindle, and a sin X= effective friction radius, the friction moment becomes 2L+P+w(l— -) asinX, and we V 9' readily obtain the well-known result sin •(I) w Frictional retardation, a', is determined by observing the time taken to come to rest after communicating a certain speed to the system symmetrically loaded. This is also done on the chronograph, it being now necessary to observe and record on pen No. 3 the moment at which motion ceases. To get as near as possible to the same conditions of load as those obtaining in the actual a experiment, it is well to observe a' with a load L'=L+™ on each side, and in that case a! is given by 2 a P 2L+w+P-z P* Hence equation (1) reduces to the very simple form . . . (2) w In this form, viz. Driving Force minus Frictional Force equal to 104 ON THE ACCURACY ATTAINABLE WITH Effective Force, the meaning of the equation can be grasped even by a student whose dynamical knowledge is small. With the present apparatus it was found necessary to determine a several times before and after each determination of a, ; with an apparatus of more satisfactory construction this would probably be unnecessary. To get k, the radius of gyration, the pulley wheel was removed from its position on the friction rollers and was attached to bi-filar suspensions. Three separate determina- tions gave £=4-221, 4-205, 4.230, giving a mean &=4-218. This, with the weight P=44-0 g, and the radius p=6-194 cm., £• gives for the equivalent mass of the pulley P— 2=20-5 g. The inertia of the four friction rollers was found from their dimensions and their weights to be one-tenth of that of the pulley wheel itself. As their angular speed is less than one- tenth that of the pulley, their total kinetic energy is less than one-thousandth of the kinetic energy of the pulley, and has therefore been left out of account in the subsequent calculation of g. rz A graphical evaluation of P-^ made in the usual way ftfl from the results appended, by plotting - — , against 2L+w a+a and reading off the intercept on the load axis, led to a value 21-2 g. A graphical method may also be adopted for ascertaining the fraction of a revolution at the beginning in finding a, and at the conclusion in finding a'. If R is the number of revolu- R tions from and to rest respectively, we have™ in each case a constant, and so x, the unknown fraction of a revolution, can be at once obtained by plotting (0, 1, 2, 3, etc.) R against Tl, T\, T\, T\, etc., where T0, Tv etc., are the times of x,l+x, 2+x, etc., revolutions. In the a measurements this fraction is A FORM OF ATWOOD'S MACHINE 105 reduced to the smallest possible value by initial adjustment of the level of the release magnet, but one has no control over its value in the a! determinations. As a rule, however, it is not necessary to evaluate x, the incomplete part of a revolution, in the determination of a and a' ; it is sufficient to plot squares of times from beginning and end respectively against number of complete revolutions, and the products of the slopes of the resulting straight lines into twice the effective distance of one revolution at once give the acceleration and retardation required. This was the procedure adopted in obtaining the results communicated in this paper : the experimental points, as long as the speed did not become excessive, lay exactly on a straight line, whose slope could easily be found to 1 in 1000. The weight of the two pans and string used in the following experiments was 87-5 g, while additional loads of 50, 100, 150, and 200 g were added to each side. In each case two driving weights were tried, viz. 10 g and 20 g. The chronograph clock was carefully calibrated, giving a mean nominal second equivalent to 0-995 true seconds. The maximum fall available was about 700 cm., but although records were obtained for the complete fall in every case, the s— tz curves showed some curvature for the last feAv metres of fall, and in the case of large acceleration this curva- ture was quite pronounced, the acceleration in every case diminishing as the speed increased. This is obviously due to the resistance of the pans to motion through the air, and, in fact, the apparatus is used chiefly for the determination of these resistances with larger surfaces. The resistance was allowed for in the present experiments by using for the deter- mination of a only the first part of the fall, where the speed was low and the graph was accurately straight ; should it be desired to use it over greater distances, it would be well to dispense entirely with pans, and to use weights made in the form of rods, so that the area presented normally to the direction of motion would be a minimum. o io6 ON THE ACCURACY ATTAINABLE WITH RESULTS The following are eight different values of g obtained by the method indicated above. The experiments are grouped in pairs, at the beginning and end of which a' was determined a few times, the mean of each set being taken as holding for the actual experiments. VALUES OF g DETERMINED WITH MODIFIED FORM OF ATWOOD'S MACHINE ZL gram. a' cm/sec2. w gram. 2L + w gram. 2£ + «. + j£ gram. a cm/sec-. g cm/sec2. 187-5 3-49 10 197-5 218 41-37 978 20 207-5 228 82-00 975 ..976 287-5 3-64 10 297-5 318 27-16 979 20 307-5 328 56-19 981 ..980 387-5 3-56 10 397-5 418 19-96 983 20 407-5 428 42-35 983 ..983 487-5 3-31 10 497-5 518 15-65 982 20 507-5 528 33-75 978 ..980 Mea n value of g 980 DISCUSSION OF RESULTS Considering the magnitude of the frictional correction for the particular apparatus used in these experiments and the slight uncertainty in its numerical value, the remarkably good mean value obtained for g must be regarded as somewhat A FORM OF ATWOOD'S MACHINE 107 fortuitous. That there are irregularities is evident from the 1+P- above values of a', which, as a varies as should decrease asymptotically as the load increases ; but it would also seem that the effect of these disturbances can be determined by a proper evaluation of friction for each indi- vidual experiment. No doubt, more carefully constructed friction rollers would prove more regular in action ; but as the accuracy here attained is more than sufficient for the author's immediate purpose, he did not think it necessary to have another wheel constructed in order to be able to test this point further. It is, perhaps, of interest to examine under what conditions greatest accuracy may be attained. For a given absolute possibility of time measurement a should be as small as possible in order that it should be known with the highest relative accuracy, but then the difficulty arises that a' is a large fraction of the total, and any slight uncertainty in its value affects the result accordingly. On the other hand, to increase a so as to make a! relatively small would entail less accuracy in the time measurement, the square of which is involved, and, moreover, air resistance at such comparatively high speeds would be- come appreciable even during the earlier stages of the fall. It is, perhaps, significant that the worst value amongst the foregoing results is that obtained from the greatest accelera- tion. What should be attempted, therefore, is the reduc- tion of the absolute value of of to a minimum. Since a'=ga sin P 1 — JL u -> , for a given load L', with P fixed by considerations of stability, the desired result will be attained io8 ACCURACY OF ATWOOD'S MACHINE by making a and X a minimum, and k as nearly equal to p as possible. This means that the supporting pivots must be thin and well lubricated, and that the mass of the pulley must be concentrated in the rim, the spokes being as light as possible, consistent with the load they have to bear. In conclusion, it might be well to draw attention to the fact that the only modification in the usual type of ATWOOD pulley necessary to adapt it to the foregoing method is the very simple one of forming the wheel spindle partly of metal and partly of non-conducting material. Before the plan described above was adopted, the effect was tried of coating half of one end of the spindle with a very thin layer of hard varnish. This answered the purpose sufficiently well for one or two experiments, but the varnish soon cracked in places, giving rise to confused records on the chronograph. Some- thing more permanent is required. When the apparatus is to be used as an ATWOOD machine for determining the acceleration due to gravity, an inking chronograph is not at all essential ; in fact, the accuracy of the time measurement obtained with the simpler forms, in which a smoked plate travels or a smoked drum revolves in front of a vibrating tuning fork, would reach the order of 1 /500th second. Most laboratories now possess such a chronograph in their equipment, and most students are called upon to use it at some stage of their laboratory experience ; and as equation (2), containing the dynamics of the method, is extremely simple, there is no reason why any junior student should be unable to apply it ; the extra knowledge of experi- mental work required is but small, while the resulting gain in accuracy is great. JOHN PATRICK DALTON THE DEVIATION OF THE OSCILLATIONS OF A VISCOUS SOLID FROM THE ISOCHRONOUS LAW THE fact of the distinct departure from isochronism of the torsional vibrations of a metallic wire has been known since the classical researches of the Russian physicist Kuppfer were published in the middle of last century. And that fact, amongst others, exhibits the essential difference existing between the origin of the internal dissipation of energy in this case and that which is effective in cases of true viscosity, in spite of the other fact, established by Kelvin and very rigidly corroborated by Tomlinson, that the logarithmic law applies to the decay of the small oscillations of a metallic wire. When the oscillations are large the logarithmic law of decay is widely departed from, and the range of oscillation y is, to a high degree of approximation, related to the time x by the condition yn(x+a)=b, where n, a, and b are constants throughout a large series of oscillations (see Mr. Ritchie's paper in this volume, p. 113). Further, the departure from the sine law during any single oscillation is very marked. The time of inward motion from the maximum elongation to the zero point is greater than the time of outward motion from the zero point to the maximum (Trans. Roy. Soc. of Edin., 1896). In this note I propose merely to indicate, by the aid of a diagram, the nature and extent of these deviations ; a full descriptive and theoretical 109 no OSCILLATIONS OF A VISCOUS SOLID account will be communicated to the Royal Society of Edinburgh. The wire to which the diagram refers Avas one of soft copper, No. 19 B.W.G., and 22'5 cm. in length. The oscillator was in the form of a brass ring, from which brass teeth, arranged at equal angular intervals, projected downwards. These teeth, as the ring revolved, made contact with radial mercury pools in an ebonite plate below. These pools were also arranged at equal angular intervals, but the interval between them was different from that between successive teeth, so that the principle of the vernier came into play. The contacts so made completed electric circuits, by means of which chrono- graphic records were obtained of the instants at which the contacts occurred. It was possible in this way to make records of the position of the oscillator at successive intervals of 2° throughout its range. In the special experiment here described much fewer observations were sufficient, and so records were taken at intervals of 12°. The full curve in the diagram is drawn through the recorded points, and represents the course of an oscillation from the first positive to the first negative elongation, negative values being plotted as if they were positive. In that single half- oscillation the amplitude dropped by nearly one-third of its initial value. Times are represented as angles, the complete time of the half-oscillation being 180. The time of the inward oscillation exceeds that of the outward oscillation by nearly one-fifth of the mean value of the two. The dotted curve represents the course which would have been followed had the drop of amplitude been due to a true viscous resistance. It follows that any representation of the outward or inward motions separately, either as unresisted simple harmonic, or as viscously resisted harmonic, motion, is of no real value. Yet the comparative accuracy of these empirical representa- tions is of some interest. If the inward motion is represented § § ,\ § § .X X § ^ ii2 OSCILLATIONS OF A VISCOUS SOLID as the first quarter of a simple harmonic motion, the following results are obtained : — 1 0-950 0-804 0-578 0-293 O'O 1 0-948 0-795 0-568 0-280 O'O the first row giving the calculated values. If it were repre- sented as a decaying simple harmonic motion, with logarithmic decrement 0-00441, calculated and observed values are re- spectively : — 1 0-969 0-816 0-675 0-531 0-393 0-248 0-0995 0 1 0-974 0-825 0-681 0-534 0-390 0-241 0*0990 0 For the accuracy which has been obtained in the observa- tions I am greatly indebted to the skill of Mr. J. Linton, mechanic to the Department of Physics, University College, Dundee, who constructed the apparatus used, with the exception of the chronograph ; and to that of Mr. Ednie, mechanic to the Physiological Department, University oi Edinburgh, who constructed the chronograph. I have also to record my indebtedness to the Carnegie Trustees for a grant which was in part expended on the construction of the chronograph. WILLIAM PEDDIE THE DISSIPATION OF ENERGY AND OTHER EFFECTS OBSERVED IN TORSIONAL OSCILLATION IN the determination of the law of decrease of torsional oscil- lations of an iron wire, when the range of oscillation is large in comparison with the palpable limits of elasticity, an equa- tion of the form yn(x+a)=b has been shown by Dr. Peddie (Phil. Mag., July 1894) to give close representation of results where — y=the range of oscillation, a;=the number of oscillations since the commencement of observations, n, a, 6= quantities, constant for any one experiment, depend- ing on the initial conditions of the experiment and the previous treatment of the wire. The present work has been undertaken to find if this equation can with equal accuracy be applied in the case of wires of brass and other materials, and to find the effect pro- duced on the constants of this equation by altering the initial conditions of the wire by change of temperature and by fatigue induced in the wire by repeated extensional or torsional strains. METHOD OF CALCULATING THE CONSTANTS The method described by Dr. Paddie in a second paper on the same subject (Trans. Roy. Soc. Edin., 1896) was employed for the determination of the quantities n, a, and &. Since n log y+log (.T+a)=log b, 114 THE DISSIPATION OF ENERGY then if log (x+a) be plotted against log y, the corresponding points will lie on a straight line which intersects the axis FIGURE 1— Brass at 275° C. a=100 n=-86 =? \ \ -i \ \ \ \ N \ \ \ \ \ , \ •80 •90 Logy 1-0 1-1 along which log y is measured at an angle whose tangent is n, provided that the proper value of a be inserted. The actual value of this constant to be added to x depends upon the interval which elapses after starting the experiment until the first reading is taken. A rough idea of the value of a to be taken is got from the curve with scale readings as ordinates and number of oscillations as abscissae, when the distance IN TORSIONAL OSCILLATION 115 from the y axis of the line which the curve approaches asymp- totically gives the value of a. If a wrong value of a be taken, the points in the curve of log y against log (x+a) will not lie in a straight line, a curve convex to the origin being obtained if the value of a be too large, and a curve concave to the origin if the value of a be too small. This is seen 1 to be the case in figure 1, when with a=90 the curve is concave, and with a= 1 10, convex. The value of a which gives the straightest line is taken, and from the tangent of the angle included by the line and the axis along which log y is measured n is found, and 6 can then be got by substitution. METHOD OF CONDUCTING THE EXPERIMENT The wire under consideration was suspended from a clamp attached to a torsion head, and at the other end was clamped, symmetrically and horizontally, a heavy lead ring of large moment of inertia. To the outer surface of this ring was fastened a scale divided into millimetres. The vibrations of the apparatus were damped out, and the torsion head then carefully turned so that no pendulum oscillation should be set up in the wire. Exterior disturbances were also, as far as possible, avoided. Readings of successive maxima ranges of oscillation were taken by means of a telescope with cross wires inserted, the crossing point being fixed in the same horizontal plane as the lead ring, at a distance of about 6 feet from the scale. It was found convenient to miss the first reading, and to take readings at the end of every oscilla- tion after the first until ten oscillations had been completed, and thereafter to take readings after every fifth oscillation. Except in the case of tin wire, in which case the oscillations died down with extreme rapidity, the readings were extended over a hundred oscillations. The zero of the scale was found by taking successive readings to right and left at intervals, 1 The scale readings y on the diagrams correspond to a rotation through 1° per '175 cm. of scale. n6 THE DISSIPATION OF ENERGY and the average of these values was then taken. A curve drawn with the scale readings as ordinates and the number of swings as abscissae showed by means of the waviness of the curve if ordinary pendulum oscillations had been appre- ciably started in the apparatus. The values of log y were then plotted against those of log (x+a), and when the proper value of a had been found, so that the points lay practically on a straight line, the constants were obtained. CONFIRMATION OF THE EMPIRICAL LAW Wires of nine different metals were tested, brass, copper, aluminium, tin, zinc, silver, german silver, platinum, and nickel. Of these, brass, tin, zinc, silver, german silver, and nickel were found at the ordinary temperature to give close agreement, over the very large range of oscillations taken, to the general law, in each case a suitable value of a being found which caused all the points to lie on a straight line. It was found, however, that in the case of the remaining metals, and especially in the cases of aluminium and copper, no one value of a could be found to bring all the points into one line, an s-shaped curve being obtained in general. When this was first observed, it was thought that the law did not hold in such cases, or at least that it did not hold over the range taken. In attempting to straighten, in this case, one part of the curve, however, it was found that, with a certain value of a the points could be brought to lie on two straight lines in- clined at an angle not differing much from 180°. It was further found that this could not be done in every case with the same value of a for the two portions, but, by choosing a slightly different value of a, in every case the points could be brought to lie on two straight lines. The doubling of the line, as will be seen when the metals are considered separately, was found to depend upon controllable conditions, e.g. in brass it occurred when the metal had been brought to a certain temperature in the neighbourhood of 375° C. In IN TORSIONAL OSCILLATION 117 most cases it was found that the value of n was greater in the line drawn through the points corresponding to the smaller oscillations. EXPERIMENTS ON BRASS WIRE In the present series of experiments brass was the material most studied in detail, and, for the purpose of experiment, lengths of brass wire, approximately one millimetre in diameter,1 were used. The length was in each case chosen so that, from clamp to clamp on the torsion apparatus, there should be exactly one foot of wire. It was found in a subse- quent experiment, however, that change of length had no effect on the constants a and n, although b might differ considerably. In an experiment on 6 inches of brass wire, the values of a and n were found to be equal to those got with 12 inches of the same wire. The reason can readily be seen, as follows. If we postulate that the loss of potential energy in a breaking down of molecular groups is proportional to a power of the angle of torsion, we can approximately write (Peddie, Phil. Mag., July 1894) the loss of energy per swing in the form —kydy=pymdx. Now, in a wire of half length, k is doubled for the same value of y ; and the loss of energy, with the same y at half length, is half of what it would have been in the wire of whole length at 2y. But in the wire of whole length at 2y the loss is p2mymdx. Thus -2kydy=p2m-lymdx -kydy=p2m-2ymdx, *"-! 1 -0975cm. n8 THE DISSIPATION OF ENERGY The empirical law was found to hold over a very long range at the ordinary temperature. In all cases the points lay, with the proper value of a, in straight lines. The following Table gives some of the results got with wires each one foot long :— Best Value of a Value of n Value of 6 140 •79 947 130 •76 765 120 •78 535 160 •74 783 150 •79 1035 180 •78 1242 TABLE OF HEATING EFFECT Temperature Value of a Value of n 275' C. 100 •86 339 120 1-02 358 60 1-26 368 40 1-60 372 20 1-80 375 14 1-80 and 1'90 377 6 and 5 2-15 and 1-90 400 5 and 0 2-10 and 2'30 413 2 2-40 and 2'60 466 4 2-30 and 2 '80 485 3 and 2 2-15 and 2-70 535 4 2-30 and 2 -55 625 1 and 0 2-60 and 3 -20 675 2 2-40 and 2-90 IN TORSIONAL OSCILLATION 119 FIGURE 2 — Showing relation of (n) and (a) with Temperature. 3'4 140 1-o \ 130 3-0 \ 110 2-8 \ 100 L e 3 y 2-6 •^ 90 2-4 V s s 80 \ s . 1 2-2 / _, — * 70 2-0 \ •, s /' ' 60 1-8 \ x* & s •iO 1-6 } f 40 1-4 7 \ 30 12 / f \ W / \ 1-0 n x 10 •^ 270° 290° 310" 330° 350° 370° 380" 410° TIME EFFECT 430° 450° 470° 490° In these temperature experiments, the plan adopted was to raise the brass wire up to the temperature indicated, and to remove it from the furnace at once, as it was found by previous tests that the change was a sudden one. Wires were introduced for \ hour, \ hour, 5 minutes, and 1 minute, respectively, at a temperature of 400° C., and in each case the same values of n got. LOWERING OF TEMPERATURE A length of original l wire was allowed to remain in liquid air for over half an hour and subsequently tested, but this seemed to have no effect upon the value of n ; and this result 1 This term will be used in subsequent pages to denote a length of wire cut from a coil as supplied by the dealer. 120 THE DISSIPATION OF ENERGY was also got on treating in the same manner a portion of wire already heated past 375° C. EFFECT OF EXTENSIONAL STRAIN A length of the original wire was hung with a 14-lb. weight attached in a long vertical shaft, and portions cut from it were tested at intervals of days or weeks, but no change was observed. The weight was then replaced by one of 28 Ibs., and latterly of 56 Ibs., but still the same values of n were observed. The wire after heating to 400° C., however, could be easily elongated by a pull ; and on subsequent testing it was found to give results tending to approach those obtained with an unheated length of wire. A wire 12 inches in length was raised to 400° C. and then stretched till rupture occurred. The value of a was found to have changed to 100, whilst n was found to be 1'34, the points again all falling into one straight line. The diameter was now 0*90 mm., and the change of length 3 inches. Thus stretching is followed by a reversion to the original conditions of the unheated wire ; and it is quite probable that, with more careful stretching, the value of n could be reduced to its original amount. The effects of repeated extensional and torsional strains on brass are treated subsequently. WIRES OF VARYING DIAMETER It was found that a wire, of diameter 1-2 mms., gave results similar to those got for that of diameter 1*0 mm. The exact values got were a=95, %=-70, &=439. A thicker wire, of diameter 1*65 mms., gave the results a=85, ?i= -66, 6=360. In this case the readings had to be taken very quickly, since the oscillations died down with extreme rapidity, and IN TORSIONAL OSCILLATION 121 thus the accuracy could not be so great as in the former experiments. This was found to be even more apparent in the next experiment, with a wire of diameter 2*0 mms. It was found impossible to take readings with any degree of accuracy with 12 inches of this wire, and so a double length was taken. Distinct pendulum oscillations were also of more frequent occurrence when the thick wire was used, and the curve obtained showed a wavy appearance. By drawing a straight line through the observed points, so as to eliminate the disturbing effect, the following results were obtained : — a=90, w=-84, 6=427. This value of 6, as shown on page 117, can be compared with the values got with the wires already discussed. For it was shown that in a wire of half length Thus the value of b will be got for that of double length by 6=6'-2" =238. The value of 6, then, although not exactly under control, is seen to decline steadily with increases of diameter, thus :— Diameter a n b 1 mm. About 100 •70— -80 700—1000 1'2 mms. 95 •70 439 1-65 „ 80 •66 360 2-0 „ 90 •84 238 The values of a and n, then, are largely independent of change of sectional area, just as they were of change of length. This is to be expected, in accordance with the original assump- tion that the loss of energy for a given material depends solely on a power of the angle of distortion. Q 122 THE DISSIPATION OF ENERGY ANNEALING FROM HIGH TEMPERATURES A length of brass wire heated to 535° C. was annealed and then tested. The values got, o=l, w=2-30, a=l, w=2-70, show no change from those for wire heated to 535° C., and quickly withdrawn from the furnace. On that occasion the values got were a=4, w=2'30 and 2*60. An experiment when the wire was annealed from 485° C. also showed no change. RESULTS WITH VARIOUS OTHER WIRES Wire Best value of a Value of n Effect of Heating Aluminium Copper German Silver Zinc 0 and 100 (two lines) 0 and 15 (two lines) 110 0 1-34 and -20 3 -80 and 1'60 1-05 •60 Tendency of wire to assume form of one straight line. No visible change. Still one line, values being a=20, « = 1-70. Silver 65 •45 Tin 3 •82 Platinum 28 and 30 •95 and 1 '05 (two lines) THE EFFECTS OF FATIGUE ON TORSIONAL OSCILLATIONS In the preceding observations it was noted that the appli- cation of a large extensional force had a great effect in some cases, notably in the case of several of the metals after having IN TORSIONAL OSCILLATION 123 been raised to a red heat in the electric furnace. The under- noted work was undertaken to find the effect produced by a repeated application of an extensional force and by a repeated application of a twist to one end of the wire, the other end being held firm, thus tending to induce fatigue in the wires, the supposition being that such treatment might have an effect on the mode of oscillation when tested immediately afterwards. The latter step is necessary, since it has been shown (Peddie, Trans. Roy. Soc. Edin., vol. xxxix., 1897- 1898) that rapid partial recovery from fatigue is possible. The apparatus used for imparting these continued strains is a modification of an earlier machine used by Dr. Peddie in these investigations, and was designed by him, the cost of construction being defrayed by means of a Royal Society grant given for the purpose. The accompanying photograph shows the complete apparatus, with the battery and rheostat introduced into the circuit. It consists essentially of an electrically oscillated pendulum, which by means of a series of toothed wheels at its mid-point twists one end of the wire introduced, the other end of it being firmly clamped. EXPEBIMENTAL RESULTS Brass wire, similar to that used in the temperature experi- ments, is the only wire yet tested, and experiments included the finding of the effects on the constants n, a, and 6 of the equation yn (x+a)=b after subjecting the wire (1) to rotational strain in the original unheated state ; (2) to rotational strain after. the wire had been raised to a red heat and cooled, i.e. after the points obtained by plotting log (x+a) against log y could not be fitted into one straight line ; (3) to extensional strain in the 124 THE DISSIPATION OF ENERGY FRONT VIEW OF APPARATUS : D — H, WIRE FOR EXTENSIONAL STRAIN ; A — B, WIRE FOR ROTATIONAL STRAIN IN TORSIONAL OSCILLATION 125 END VIEW OF APPARATUS: D— C, WIRE FOR EXTENSIOHAL STRAIN 126 THE DISSIPATION OF ENERGY unheated state ; and (4) to extensional strain after raising to a red heat and then allowing to cool. 1. Wires were fatigued in this manner for periods vary- ing from five minutes to three and a half hours, and subse- quently tested. The values got were as follows : — Time of Fatigue Best Value of a Value of n Calculated Value of b 5 mins. 100 •78 1000 1 hour 180 •78 834 2% hours 70 •78 560 sj „ 100 •76 484 The value of n is seen to remain constant over this range of fatigue, whilst that of a oscillates about 100, the values of a when large being more uncertain than when small, since in trying to find the best value of a, values of the latter differing by ten or twenty units may make very little difference in the slope of the line, or the possibility of the points lying along it ; b seems to be the only constant affected. It has already been seen that the value of 6 is very variable, but there appears to be a gradual fall in value here, however. Rota- tional fatigue, then, has little or no effect on the constants of the equation. Further work will be directed to ascer- taining if this fall in the value of b with increase of fatigue is invariable and therefore real. 2. It was shown above that after brass wire had been raised to a temperature above 375° C., two values of n were obtained, one value extending over one range of oscillation, and another over the remaining part. It was shown also that these values of n increased to a constant value. Fatigue, induced by rotational strain, is again found to have no effect on the constants a and n, as the following results will show : — IN TORSIONAL OSCILLATION 127 Value 9 Of n Time of Beat Value , * Fatigue of a 1st Part of Curve 2nd Part of Curve 0 mins. 0 2-52 3-00 5 2 2-50 3-20 10 1 2-51 3-10 15 0 2-53 3-02 30 0 2-81 3-00 60 0 3-36' 3-03 120 0 2-56 3-03 180 2 2-59 3-07 The behaviour of 6 after the wire has been heated and fatigued will be the object of further work. 3. A length of brass wire, clamped vertically between C and D, and fatigued for two and a half hours, gave the values a=90, ?i= -76, &=841. These are the values got for the original wire, and thus exten- sional fatigue has no effect on the constants when the wire is unheated. This was confirmed by further experiments. 4. After heating to a red heat and then cooling, exten- sional fatigue was found to have the same tendency as stretch- ing had — i.e. to tend to straighten the curve so as to allow of one straight line representing the results. A typical experi- ment gave the values a=30, /i=l-24, 6=524. This wire was fatigued for one hour after having been raised to 485° C. It was found that the pendulum made twenty-five complete oscillations per minute ; therefore, since the wire between C and D receives two pulls during one oscil- lation, the total number of pulls given was 3000. The weight 1 This value, 3 '36, got for the line lying along the points of the first oscillations, is quite abnormal, and the wire used in this experiment was examined to discover, if possible, the cause of the unusual value of n. A repeated experiment gave the same result. The wire was then halved, each half being tested separately, and finally the abnormality was traced to one of the quarters. 128 ENERGY IN TORSIONAL OSCILLATION attached to the end H of the lever was 3000 gins., and the ratio of the arms 3 to 1, the fulcrum I being half way between the cam P and the end C, while PH=PC. The pull exerted is therefore in this case 9000 gms. weight. The values got on testing this wire after heating to 485° C., and before fatigue was applied, were a=2, n=2-15 and 2-70 ; a then is seen to be raised to a value intermediate to those obtained from the fresh and heated wires. Similarly n has now an intermediate value. The effects of both forms of fatigue on the empirical equation as applied to other materials will be the object of further research. JAMES BONNYMAN RITCHIE WAVE IMPACT ON ENGINEERING STRUCTURES THE failure of a breakwater, formed of blocks of masonry or concrete, under wave impact, is usually initially due to under- mining or settling of the foundations, or to the displacement of some one or more of the face blocks, this permitting free access to the interior of the mass. Once any such serious displacement has taken place, total failure is usually a matter of comparatively short time. As is a well-known fact, the normal impact of a wave on the face or top of such a structure may produce the with- drawal of a block weighing several tons, the motion of the block being in the opposite direction to that of impact. Thus at Ymuiden, a ' header ' block in the seaward face of the pier, measuring 7 ft. in length and presenting a face to the waves of 4 ft. by 3 ft. 6 ins., was started forward to the extent of 3 ft. by the stroke of a wave compressing the air behind it.1 This block weighed about seven tons, and the top of it was at the level of low water. It had three courses of concrete blocks, each 3 ft. 6 ins. in thickness, resting upon the top of it. These were all set in Portland cement mortar, but the course to which the block in question belonged was built dry, as were those below it. A similar instance was noticed at East London (Cape Colony) where several adjoining blocks were forced outwards on the harbour side of the breakwater during a storm.2 Another example of this was noticed by 1 This description is from Shield's Principles and Practice of Harbour Construction* p. 83. The italics, however, are the author's. 2 Wm. Shield, Harbour Construction, p. 109. K 130 WAVE IMPACT ON Mr. Mallet on the sea-walls of the Dublin and Kingstown railway x during a severe gale, in which masses of water, deflected vertically upwards by the sea face of the wall, fell heavily upon the pavement inside. In this case many of the square granite pavement blocks were seen to jump vertically out of their beds at the instant of the fall of the sea upon them and were thrown landwards towards the line of way. The commonly accepted explanations of such a phenome- non will be gathered from the following quotations from authorities on harbour works. Mr. Vernon Harcourt notes :— ' The blocks at and below low water have open joints, into which air penetrates on the recoil of a wave and also fills any cavities behind ; and the succeeding wave, compressing the air inside, leads to the gradual forcing out of a block by the pressure from behind on the retreat of each wave during a storm.' 2 In another passage the same writer remarks, ' Waves striking against the outer blocks also compress the air in the open joints, which aids in the dis- placement of the inner blocks.' 3 A somewhat similar but wider explanation is to the effect that, ' The air or water confined within a joint, when struck by a wave, is converted into a very destructive agent. The air is compressed and forced along joints and seams, and, immediately upon the wave receding it expands and tends to loosen and push out pieces of the rock. When water is confined in joints the force of the wave stroke is transmitted by it, and is made to act over large areas on the same principle as a hydraulic press.' 4 Similar views were adopted by Captain D. D. Gaillard, U.S.A.,5 who, as a result of experiments, came to the con- clusion that the pressure due to the impact of an interrupted 1 Proc. Inst. C.E., vol. xviii. p. 113. 8 Vernon Harcourt, Civil Engineering applied to Construction, p. 486. 8 Vernon Harcourt, Harbours and Docks, p. 298. 4 Wm. Shield, Harbour Construction. e Wave Action in Relation to Engineering Structures, Ca.pt. Gaillard, p. 181. ENGINEERING STRUCTURES 131 jet of water on a normal plane is no greater than that of a continuous jet of the same velocity ; that waves therefore exert a continuous pressure over a definite period of time and not a sudden momentary blow, and that as a result of this continued pressure on a wall with open joints the wave forces water through the joints, and compresses the air in the interior. As the recession of a wave is sudden compared with its advance, the pressure is suddenly withdrawn from the outside, and the excess internal pressure results in outward displacement of the face blocks. Experiments at Roorkee 1 on the impact of jets on a normal plane, also led their authors to the conclusion that the maxi- mum pressure produced by sudden impact is the same as is exerted by a steady jet, and is given, within about 1 per cent., by ' H ' or vz 4- 2g feet of water where ' H ' is the effective head producing flow, measured above the point of impact. In these experiments the pressures were measured by a water- column, in communication with small orifices in the plane. The present investigation has been carried out in the engineering department of University College, Dundee, with a view of determining the magnitude of the effects which such actions as have been outlined, may produce, and of showing that still another phenomenon may be called into play during wave impact, and may produce much greater effects than are compatible with these theories of simple hydrostatic trans- mission of pressure, or of air compression. Pressure following impact — When a mass of water is hurled with velocity ' v ' against the face of a breakwater, the pressure on the face rises almost instantaneously to a value approximating to v2 -f 2g feet of water. Substantial confirma- tion of this is afforded by the results of experiments by the late Thos. Stevenson, who, measuring these impact pressures by means of a dynamometer with spring control,2 fixed at about 1 Proc. Inst. C.E., vol. be. p. 436. 1 Trans. Roy. Soc. Edinburgh, vol. xvi. p. 23. 132 WAVE IMPACT ON the level of f tide, obtained maximum values of 3£ and 3 tons per square foot at Dunbar and on the Banffshire coast. These values correspond to heads of 122 and 105 feet and to velocities of 89 and 82-3 feet per second respectively. This mass of water being diverted by the face of the breakwater wip then be projected upwards to a height approximately the same as that corresponding to this pressure. As the result of observa- tion it is known that on the breaking of a wave, during a storm, masses of water are, on occasion, hurled to heights in the neighbourhood of 150 feet, and even greater heights have been occasionally recorded.1 As, however, the breakwater causes an upward deflection of the air currents in such an onshore wind as commonly accompanies the production of large waves, this will probably account to some extent, for the extreme height to which the spray is often thrown, and there would appear to be some doubt as to whether, apart from this action, the height exceeds some 100 to 120 feet, except in very abnormal cases. Throughout the present paper calculations have been made on the assumption that this face pressure may attain a value of 6400 Ibs. (2-86 tons) per sq. foot, corresponding to a head of 100 feet and to a velocity of 80 feet per second. If water accumulates in the interior of a breakwater and is in communication with the sea-face through one or more crevices or open joints which are themselves full of water, such a face pressure may be transmitted hydrostatically to every portion of the interior. Moreover, as all portions of the face may not be exposed to direct wave impact at the same instant, this may give rise to an internal pressure tending to cause outward displacement of the blocks, which may thus attain an effective value of 6400 Ibs. per square foot of projected area of the block. Effect of air compression in the joints — Let the joint Fig. 1 a, have a length I feet, perpendicular to the sea-face, 1 Harbour Construction, Shield, pp. 80 and 20S. ENGINEERING STRUCTURES 133 and a cross-sectional area ' a ' square feet. Let this com- municate at its inner end directly with the surface of the enclosed water, and imagine a frictionless piston to form a definite boundary between the entrapped air and the impinging column of water. Let ' x ' be the distance to which this column penetrates on its first impact. If the column loses no energy during its entry to, and passage up, the joint, the energy given up by it when it has come to v2 rest equals 62'4axK- foot Ibs.1 20 Equating this to the work done on the air during com- pression, and assuming this, because of the rapidity with which it takes place, to be adiabatic we have .-- 2g -4 where pa is the initial atmospheric pressure (2120 Ibs. per square foot) and pt is the final air pressure. Substituting for Pi in terms of pa we finally get 62-43. £=^ and if^- = 100 this simplifies to The value of the ratio x + l, which satisfies this equation is independent of Z, and is equal to *81, in which case (1 V4 _ } =10'25 atmospheres, =21,700 Ib. per square foot, =9'7 tons per square foot. 1 This assumes fresh water of weight 62'4 Ibs. per cubic foot. As fresh water was used in the author's experiments this value has been adopted in these calculations. In the case of sea water the value would be 64ax - foot Ibs. 2(7 134 WAVE IMPACT ON Actually the maximum pressure attained will be less than this, because energy is lost by eddy formation at the entrance to such a passage (-5v2 -f 2g if the edges are sharp and normal to the face) and by friction during motion up the passage, while it is probable that except in a joint of very small cross-sectional area the motion of the water will be rather of the type indicated in Fig. 1, 6, than in Fig. 1, a. Such a type of motion, leading to backward displacement of the air over the top of the advancing column would produce further loss of energy in eddy forma- tion. It appears probable that in no case could the loss of energy, due to these various causes, amount to less than 25 per cent, of the initial energy of impact, and that it would normally amount to 50 per cent, or even more. The effect of such losses on the maximum pressure obtainable in this way is shown in the following table :— Initial head = 100 feet Velocity =80. ft. per sec. 4 Percentage loss of head 0 25 50 Max. pressure attained in J- tons per sq. ft. 9-7 4-9 2-13 Ratio of max. pressure to pressure on face 3-49 1-77 •77 In any case, however, should the face pressure be maintained for a sufficient length of time, water will be forced along the crevice until the internal pressure is equal to the face pressure. Experiments by the authors, the results of which are given at a later stage, showed that using a light piston to simulate the state of affairs, shown in Fig. 1, a, the maximum pressure attained by air compression was 2-48 times the face pressure, while without this piston it was 2*05 times the latter pressure. As the state of affairs in these experiments was particularly ENGINEERING STRUCTURES 135 suitable for the production of such pressures it is probably safe to say that in a sea-wall the maximum internal pressure due to air compression never exceeds twice the face pressure, Sfff<-*rr* ^. =?<; -I X~Y FIG. 1 i.e. does not exceed some 5-75 tons per square foot with a velocity corresponding to a head of 100 feet. Where the crevice opens into a cavity of any considerable size in the interior of the-sea wall, or where, as will be usually 136 WAVE IMPACT ON the case, the latter is sufficiently porous to allow of the escape of air under pressure, a further and indefinite reduction of pressure will follow. Possibilities of water-hammer action — If, during the advance of the impinging column, there should be any direct impact of water on water in a confined space, of the nature indicated in Fig. 1, c, the possibilities of water-hammer action become obvious, and the authors' experiments have been mainly devoted to ascertaining whether any such action takes place, and, if so, the magnitude of the resultant effects. As has been previously pointed out, experiments on the impact of jets on plane surfaces pierced with orifices com- municating with the pressure-measuring device, show a sudden rise in pressure of magnitude vz + 2g feet, with no hammer action. Further apparent confirmation of this is afforded by the results of the first experiments made by the authors. In these, a closed cylindrical cast-iron box, six inches in internal diameter and six inches deep, was provided with a cast-iron cover pierced with a series of orifices respectively £, J, f , £, f, £ , and 1 in. diameter. An indicator cock mounted on the side of the box, carried a M'Innes Dobbie steam-engine in- dicator with a ^ spring. All the orifices but one being plugged, the box was filled with water and the jet from a 1-in. diameter pipe, discharging with velocities up to approximately 20 feet, per second, was suddenly directed on to this orifice. In no case was any appreciable vibration of the indicator pencil, such as would indicate an action of the nature of water- hammer, noticed, the pressure rising suddenly to a value slightly less than that corresponding to the velocity of impact (vz -r 2<7 feet of water). A little consideration, however, shows that under circum- stances such as obtained in these experiments and in those previously noted, the absence of any indication of water- hammer action by the pressure recorder does not necessarily ENGINEERING STRUCTURES 137 prove that such action does not occur. The indicator or water column adopted as a measuring device is a system capable of vibrating naturally with a frequency n per second. If to this system is applied a steady force, or a periodic force whose frequency is much less than n, the actual and recorded pressures are sensibly identical. If, however, the applied force is periodic, and if its frequency 'a -=^, this velocity, for the experimental pipe line, is 3400 feet per second. Although valve closure was not instantaneous, it may readily be shown that if closure is complete before the dis- turbance produced reaches the open end of the pipe, i.e.. if the time is less than I -7- Vp or -^ second, the pressures produced are the same as would accompany instantaneous stoppage. The fact that up to a discharge of 25-7 Ibs. per minute (•y='60 f.s.) the pressure is equal to 48 v Ibs. per square inch for all velocities, shows that up to this point the time of closure was less than this, and indicates that for this discharge and corresponding valve opening, the time was approximately 35 second. Impact of a moving on a stationary column of water — When a column of water which is confined laterally, impinges on a stationary column of the same fluid, the magnitude of the hammer pressure may be shown to be one-half that attending sudden stoppage by means of a valve. On playing a jet into the open end of pipe ' C ' Fig. 2, except for the effect of the entrapped air a sudden rise in pressure, of magnitude 24 v Ibs. per square inch, would be attained throughout both the moving and the stationary columns. This pressure is indepen- dent of the length of either of these columns. Assume, as will be commonly the case in a sea-wall, that the length of the stationary column of water filling a cavity which may extend for some considerable distance in the interior of the mass, is greater than that of the impinging column, which is largely governed by the length of the crevice piercing the face. As in the experimental coil, a mass of water hurled at the open end forms a column which will, except as modified by the 1 Water-Hammer, ante tit., p. 16. 142 WAVE IMPACT ON presence of air, impinge on the end of the stationary column with velocity v. At the instant of impact let its length be I feet. Impact is followed by a rise in pressure at the junction of the columns, and a wave of compression (24 v Ibs. per square inch above normal) is propagated in opposite directions from this point. The wave traversing the impinging column reaches its free end after a time, I -f Vp seconds. At the same time the wave traversing the stationary column has also travelled a distance of I feet, and at this instant, that portion of the joint column nearest the open end of the pipe, and of a w2 Ibs. per «~^ ^ V (* ™~ „«,„ 2g sq. ft. 'Zero vel. 1 ^ 24 v Ibs. per sq. in. i ^Zero velocity and normal pressure «L;t. ^ if-v s* ft*L i i T~*.«I IA...A.JI. *c -*:.*, r ft « — — - . • -. -. M }< • — xui/ »«• 5- Open end Closed end FIG. 4 length 21 is under an identical state of pressure and velocity, this pressure being 24 v Ibs. per square inch above normal and the velocity v -f 2 towards the closed end. At the open end the stressed layers rebound outwards, the pressure falls to that obtaining at this point, i.e. toh-r 62'4 Ibs. per square foot, or to zero if the face pressure has been removed, and a wave of normal statical pressure and of zero velocity is propagated along the pipe in the rear of the pressure wave. The state of affairs after a further short interval of time 8t is then as repre- sented in Fig. 4. When the front of the pressure wave reaches the closed ENGINEERING STRUCTURES 143 end of the pipe the motion is suddenly stopped and the pressure rises to p=48 v Ibs. per square inch. After a further interval, 21 -=- Vp seconds, that portion of the column, of length 21, nearest the closed end is at rest under this pressure, the rest of the pipe being under normal pressure and zero velocity. Instantaneously, however, the end of this stressed column, which is more remote from the closed end, rebounds with velo- city v + 2 and pressure 24 v Ibs. per square inch, layer after layer following suit, until after a further interval, 21 -r Vp seconds, this is moving under this state of pressure and velocity towards the open end. The dilatation, under this tendency to retrograde motion, of those layers in contact with the closed end, causes the pressure to fall to normal, and a state of normal pressure and of zero velocity is now reflected to the open end, following the preceding pressure- velocity wave, whose state occupies a length, 21, of the pipe at each instant. When this former wave reaches the open end the pressure falls to that obtaining at this point, the velocity increases to 'v,' and a wave of normal pressure and of velocity 'v ' (towards this end) is reflected towards the closed end. On arriving at that portion of the pipe in which the pressure is normal and the velocity zero, the pressure becomes —24v Ibs. per square inch, and the velocity v -f 2 and a wave conveying this state over a length 21 -f Vp of the pipe, travels on towards the closed end. On its arrival the pressure becomes —48 v Ibs. per square inch, the velocity zero, and this state is reflected to the open end again. Arriving here, the pressure becomes suddenly normal, the velocity becomes v towards the closed end, and the whole cycle is repeated. The true pressure-time diagrams, as obtained respectively at this end, and at a point I' from the end would then appear as in Fig. 5, a and 6. Since in a number of the jet experiments I', the distance of the indicator from the closed end of the pipe, was greater than 21, the theoretical pressure diagram at this point in such cases would be as indicated in Fig. 5, c, the maximum pressure at 144 WAVE IMPACT ON this point in the pipe never exceeding 24 v. The diagram as recorded by the indicator will, however, in general show smaller maximum pressures than those actually obtained at this point, because of the extremely short interval of time during which these pressures are exerted. A close approximation « 1 2L 1 — ««xv?,., , ..„ - J i 48 v Ibs. per 1 1 sq. in. 1 4'-2Z/ sees a ' Vf ~r- 24 v. 24 v. <~_-J r 24 v Ibs. per sq. in. c Fio. 5 to the effect of the suddenly applied force on the indicator may be got by assuming it to be uniform over the time 41 4- Vp seconds. Let F be the magnitude of the force on the piston in Ibs. ; let — be the effective mass of the indicator u piston line and linkage, and let ' S ' be the stiffness of the ENGINEERING STRUCTURES F 145 indicator spring, so that -^ would be the displacement of the piston under a steady force F. The equation of motion now becomes 9 where ' x ' is the displacement of the piston at time ' t ' seconds after the first application of the force F, and the solution of this is This shows, that if the time ' t,' during which the force F is applied, is very short, the recorded pressure is less than the true pressure at the indicator in the ratio 1— cos\/?= . t, where t= I JST 4Z -r Vp seconds. The value of the term \J gr-= can be obtained, experimentally, since the frequency of the natural vibration 1 / IS of the indicator is equal to TJ-V 0™— w- Values of 'n' were ZTT W obtained by mounting the indicator on the cylinder of a gas engine, taking a diagram, and counting the number of oscilla- tions traced out during a definite portion of a revolution. The results of these observations were very consistent and gave the following results : — Spring 1 60 1 40 1 10 n 250 204 102 The validity of this reasoning was checked by a series of experiments on the pipe line modified as shown by the dotted 146 WAVE IMPACT ON lines in Fig. 2. Water was allowed to flow from an open tank through, the upper horizontal length I of the pipe, through the valve V. This valve being closed suddenly the corresponding hammer pressure was measured by the indicator at the lower end of the coil. The results of these experiments were very satisfactory, the pressures as calculated from the formula p=4c8 v Ibs. per square inch, and as obtained from the indicator diagrams with ordinates multiplied by the corresponding multiplying factor, agreeing, for values of I between 20 inches and 60 inches, within about 3 per cent. Evidently if Z=0, corresponding to normal impact on a joint which is full of water the value of the multiplying factor becomes infinitely great, and the indicator could only show the effect of a hammer pressure if this were also infinitely great. A further check was obtained by taking diagrams under similar conditions with T^ and -fa indicator springs. Since n for the ^j spring is twice as great as for the T10 spring, the pressures indicated with the former spring for values of I in the neighbourhood of 25 inches (cos xi • *— 0 for fa spring, =—1 for fa spring) should, if the reasoning is valid, be approximately twice those recorded with the weaker spring. The results showed that the pressures recorded by the stronger spring were higher in a ratio which varied from 1*75 to 2*2. Adopting the formula as giving substantially accurate results, it becomes possible to deduce, from an indicator diagram, the true pressures in the pipe when a jet is suddenly diverted into its open end, provided only that water-hammer actually does take place. A preliminary series of experiments, carried out to settle this point, showed that by slightly modifying the method of application of the jet, two distinct types of diagram could be obtained. The jet played instantaneously and normally on to the open end, usually gave a diagram identical with those obtained when the piston was in use (Fig. 6 a) indicating that ENGINEERING STRUCTURES 147 the phenomenon was due to air compression alone. If more gradually applied, so as to facilitate displacement of air by the incoming water, a very different state of affairs was indicated. Fig. 6, b, shows a diagram obtained under such Times pressure equivalent u8 to £- feet 20 of water Times FIG. 6 circumstances. The initial rise in pressure is no longer gradual but is as instantaneous as that accompanying a sudden valve closure, and this is followed by rapid negative and positive alternations of pressure, substantially of the form indicated by theory, on the assumption of hammer action. The experi- 148 WAVE IMPACT ON ments showed conclusively that under favourable circum- stances water-hammer is set up by wave impact on an open joint, and experiments were then carried out to determine the magnitude of the pressures obtained in the experimental pipe line. The velocity of the jet (12-6 feet per sec.) was practically constant throughout the whole series of experiments, and was as high as could be obtained from the available supply. This velocity is equivalent to a face pressure p0 of 154 Ibs. per square foot. The mean results of the experiments, corrected for the frequency of the applied force, are as follow : — (a) Experiments in which the phenomena are essentially due to air compression. Length of horizontal pipe in inches 70 65 16 4 Maximum pressure in With piston 367 310 288 •• Ibs. per sq. ft. Without piston 317 288 388 282 (6) Experiments in which the phenomena was due to water-hammer. (1) Jet played into horizontal pipe. Length of horizontal pipe in inches 70 65 56 16 4 Max. pressure, Ibs. per sq. ft. 504 632 763 1450 3040 ENGINEERING STRUCTURES (2) Jet played into vertical pipe. 149 Length of empty portion of vertical pipe in inches 23 14 8 2 1 ' Max. pressure in Ibs. per sq. ft. 1220 2160 2340 11000 14400 The diminution in the magnitude of the maximum pressures given in these tables, as the length of the passage is increased, is undoubtedly due partly to the reduction in the velocity of impact which is produced by the greater resistance to flow and the greater effect of the entrapped air in the longer passages, and partly to the fact that while the multiplying factor has been deduced on the assumption that the length I of the impinging column is identical with that of the passage, this can only be approximately true in the shortest of the passages. Unfortunately the true length I cannot be ascer- tained with any degree of accuracy, and while it is certain that the maximum pressures attained are, except in the shortest of the passages, greater than are given above, no very close approximation to their actual value can be obtained. Under favourable circumstances they may, however, be expected to approach the maximum values obtained for the shortest passages, and for safety such values should be considered possible. It is evident, moreover, that with values of I less than the least of those experimented upon, the effect of the entrapped air would become increasingly small, and the results appear to show that with an infinitely small value of I the pressure would ultimately approximate to the value corresponding to sudden stoppage of a column moving with the velocity of impact. In these experiments this would be 87,500 Ibs. per square foot, or approximately six times the pressure obtained when I is 1 inch. 150 WAVE IMPACT ON In certain of the experiments an air valve, J inch diameter, was mounted at the upper side of the horizontal pipe at its inner extremity, and was left open during impact. Where the diagrams showed the phenomenon to be due to air com- pression, the effect of this in reducing the maximum pressures was very marked. Where water-hammer occurred the effect was very erratic, the maximum pressures being in some cases as high as, and in other cases much lower than, when the valve was closed. The question as to how far the results may be considered to apply in the case of wave impact on a sea-wall, is of some interest. Probably the average sea-wall will be comparable, as regards rigidity, with the experimental pipe line, and, except as regards porosity, pressures of the same order of magnitude may be anticipated with the same velocity of wave impact. Any such porosity will, however, considerably reduce the maximum pressures obtained, whether due to water-hammer or to air compression, while the presence of any cavity forming an air chamber at the inner end of any open joint will effectively prevent hammer action. Still, since the magnitude of the hammer pressures are directly proportional to the velocity of impact of water surface on water surface, and since this is probably approximately proportional to the velocity of wave impact in all cases, it is evident that with velocities in the neighbourhood of 80 feet per sec. (6!3 times those obtaining in these experiments), internal pressures of the order of 40 tons per square foot may, under favourable circumstances, be developed. Fortunately each application of such a pressure only lasts for an almost infinitely small interval of time, and the shorter the effective length of the joint, and the less in consequence the modifying effect of the entrapped air and the greater the pressures attained, the less is the time over which the pressure is exerted. Still, even so, its effect, in gradually breaking down the adhesion of block to block, is likely to be extremely serious. ENGINEERING STRUCTURES 151 Conclusions — The main conclusions to be drawn from the investigations are that while, on the assumption of simple hydrostatic transmission of pressure, the effective internal pressure due to wave impact cannot exceed that exerted by wave impact on the sea face of a breakwater, the pressures produced, if the energy of the wave is devoted to compression of air in the open joints, may amount to approximately twice this magnitude. If, however, conditions are favourable to the production of water-hammer, considerably greater pressures, up to some fifteen times the face pressure with very high velocities of impact, are to be regarded as possible. The results suggest the desirability of providing a free outlet for such water as may percolate to the interior of a sea- wall or breakwater, by means of a series of weep holes or drains opening on its sheltered face. Such drains, preventing the accumulation of internal water, would be an effective guard against the production of internal pressures of sufficient magnitude to affect the stability of the structure, whether due to water-hammer or to air compression. ARNOLD HARTLEY GIBSON WILLIAM NELSON ELGOOD CHEMISTRY THE PREPARATION OF PARTIALLY METHY- LATED SUGARS AND POLYHYDRIC ALCOHOLS MUCH of the work which has appeared from the Chemical Research Laboratory of St. Andrews during the past ten years has been concerned with the preparation and properties of methylated sugars in which only the reducing group remains unsubstituted. As examples of such compounds we have the tetramethyl derivatives of glucose, fructose, mannose, and galactose, trimethyl arabinose and trimethyl rhamnose. The study of alkylated sugars of this nature has yielded results of theoretical interest which are referred to in detail in another communication to this volume.1 It will be seen, however, by inspection of the formula of a typical example, such as tetramethyl glucose : CH . OH / CH . OCH3 9 I \ CH.OCH3 CH CH . OCH3 I CH2 . OCH3 that the reactivity of the molecule is practically confined to 1 Young, A General Review ofPurdie's Reaction. 165 156 PREPARATION OF METHYLATED SUGARS one position — the unmethylated reducing group. Considering that methylation increases the stability of a sugar and also confers a greater range of solubility on the product, it is evident that partially methylated derivatives of the sugar group would be compounds possessing a special interest. The characteristic properties of the parent sugar would be more closely preserved than in the case of a fully substituted compound, a richer variety of reactions would be available, and, at the same time, the more obscure decompositions under- gone by sugars in virtue of the high hydroxyl content of the molecule, would, to a large extent, be precluded. As examples of the many possible applications of such compounds we have : (1) their use in testing the various theories of alcoholic fermentation ; (2) their capacity to be converted into mixed ethers of sugars or polyhydric alcohols, compounds which possess a special interest for the physiologist; (3) the accumulation of optical data showing the effect of successive substitution in the sugar group, and (4) the oppor- tunity afforded by the study of the mode of formation of these compounds for obtaining experimental evidence of configura- tion. Several examples of definite mono-, di-, and trimethylated hexoses have been prepared by workers in St. Andrews during the past three years, and, although the different lines of research indicated above are not yet completely developed, the work already done has furnished interesting results which are now submitted, along with an estimate of the direction which research in this field is likely to take. EXPERIMENTAL METHODS EMPLOYED The general method devised by Purdie and Irvine for the preparation of fully methylated sugars does not permit of AND POLYHYDRIC ALCOHOLS 157 the isolation of incompletely substituted derivatives other than compounds of the nature of trimethyl glucose. In the first paper of the series, however, the authors pointed out that, in order to protect a sugar from oxidation during the silver oxide reaction, all that is necessary is to substitute the reducing group by a condensed residue capable of subsequent removal by hydrolysis. For obvious reasons, derivatives of the nature of methylglucoside have been largely made use of for this purpose, but other types of sugar derivatives may also be employed. Of these, the compounds produced by the condensation of sugars with ketones or aldehydes are best adapted for the purpose required. For example, a glucosidic monoacetone derivative of a hexose must, irrespective of the linkage of the acetone residue, contain three hydroxyl groups capable of methylation, so that hydrolysis of the alkylated product should give a trimethyl hexose ; similar treatment of a diacetone derivative would result in a monomethylated aldose or ketose. The remaining type of a partially alkylated hexose would be represented by a dimethyl glucose, and this has now been prepared by alkylation of monobenzylidene methylglucoside, and removal of benzaldehyde and methyl alcohol by hydrolysis. The principles sketched above have been found to be capable of general application, and it is possible, by the introduction of hydrolysable residues into the sugar molecule, to protect selected hydroxyl groups from alkylation, so that the preparation of a large number of partially methylated sugars is 'thus rendered available. The process is also applic- able to the preparation of similar derivatives from polyhydric alcohols. The following table shows the methods adopted in the formation of the more important compounds of this class which have so far been obtained : — 158 PREPARATION OF METHYLATED SUGARS Condensed Sugar Derivative. No. of Methoxyl Groups intro- duced. Groups removed by Hydrolysis. Methylated Sugar Derivative. Glucosediacetone . . 1 Two mols. acetone Monomethyl glucose Benzylidene-a-methyl- glucoside 2 Methyl alcohol and benzaldehyde Dimethyl glucose Glucosemonoacetone . [ 3 One mol. acetone Trimethyl glucose Fructosediacetone . . 1 Two mols. acetone Monomethyl fructose Rhamnosemonoacetone 2 One mol. acetone Dimethyl rhamnose Mannitol monoacetone 4 One mol. acetone Tetramethyl mannita Mannitol diacetone 2 Two mols. acetone Dimethyl mannitol Glycerol monoacetone 1 One mol. acetone Monomethyl glycerol MONOMETHYLATED REDUCING SUGARS It has hitherto proved impossible to obtain definite monosubstituted sugars other than the glucosides or metallic derivatives. As the compounds now described are reducing sugars they are thus examples of a new class. MONOMETHYL GLUCOSE The constitution assigned to this compound will be dis- cussed later, and it will be shown that the methoxyl group is AND POLYHYDRIC ALCOHOLS 159 attached to the terminal carbon atom in the sugar chain, i.e. in the position marked No. 6 in the following formula : — (6) (5) (4) (3) (2) (1) CH2(OCH3) . CH(OH) . CH . CH(OH) . CH(OH) . CH(OH) 0 The substance may therefore be termed 6-monomethylglucose. The method of preparation adopted was to condense glucose with acetone under conditions which result in the formation of glucosediacetone (Fischer, Ber., 28, 1165, 2496). The reaction is troublesome and uncertain in its results, on account of the necessity to convert glucose in the first instance into a dimethyl acetal, and, moreover, as the acetone residues are exceedingly liable to undergo hydrolysis, precautions have to be taken throughout the preparation to avoid the access of either moisture or traces of acid. We have, however, been able to improve on Fischer's process, and have succeeded in increasing the yield of product considerably. The methyla- tion of the substance was conducted with the precautions found advisable in the case of the corresponding fructose derivative, and the same proportion of the alkylating mixture was used. The product was purified by fractional distillation (b.p. 139-140°/12mm.) and the yield was almost quantitative. Analysis showed the compound to be pure monomethyl glucosediacetone, the properties and solubilities of which resembled those of other methylated sugar derivatives of a glucosidic nature. The compound showed [a]f-32-2° in alcohol and -31-8° in acetone solution, the concentration in each case being 5 per cent. The removal of the acetone residues took place with extreme ease on heating an aqueous-alcoholic solution, containing 0'4 per cent, of hydrogen chloride, for 100 minutes in boiling water. The course of the reaction was followed polarimentric- ally, and the results showed that both acetone groups were 160 PREPARATION OF METHYLATED SUGARS removed simultaneously and apparently at the same rate. After neutralisation with silver carbonate, shaking with animal charcoal and concentration at 40°/15 mm., a syrup was ob- tained. The product was dissolved in methyl alcohol and precipitated in the crystalline form by the addition of acetone. Analysis showed the compound to be monomethyl glucose in a state of purity. The substance shows all the characteristic properties of a reducing sugar, melts sharply at 157-158° and is more soluble in organic solvents than the parent glucose. When crystallised as described the compound showed muta- rotation : c=2-46, Solvent= methyl alcohol. [a]2D0>+98'6° — ^68-0°. This form is accordingly regarded as the a-isomeride. The jS-form of the sugar has also been isolated by the method subsequently described under dimethyl glucose. This form showed the upward mutarotation +28° — > +68°, and the optical values are thus in fair agreement with those calculated by the method recently described by Hudson. The position of the methyl group in monomethyl glucose was readily established as, on treatment with phenyl hydra- zine and acetic acid, the sugar gave a monomethyl glucosazone identical with that previously obtained from monomethyl fructose. In the case of the latter sugar, direct experimental evidence exists which indicates that the methoxyl group is terminal. MONOMETHYL FBUCTOSE No detailed account of the isolation of monomethyl fructose need be given here, as a description of the compound is included in Mr. Young's contribution to this volume. The method adopted was substantially the same as that followed AND POLYHYDRIC ALCOHOLS 161 in the preparation of the corresponding glucose compound, and thus included the intermediate formation of monomethyl fructosediacetone. This compound crystallises in square plates melting at 115°, showed [a]D— 136'4° in methyl alcoholic solution, and was hydrolysed by heating with O'l per cent, hydrogen chloride. The rotatory changes then observed indicate that the two acetone residues are removed in suc- cessive stages, an observation which is of importance in establishing the constitution of the compound. The subse- quent isolation of the free sugar was carried out in the usual manner and yielded monomethyl fructose as a readily crystal- lisable sugar melting at 122-123° and showing all the properties of a reducing ketose. The following observations of mutarotation were made : Solvent. Initial [a],, Permanent Initial [a],, of a-form. [o]D. after fusion. Water . . . -70'5° -53'1° -41'9° Methyl Alcohol -74- 1° -•> -22-1° -12-5° The constitution of the sugar, deduced from its behaviour towards phenyl hydrazine, oxidising agents, and in condensa- tion reactions, is represented by the structure : CH2(OCH3) . CH . CH(OH) . CH(OH) . C(OH) . CH2(OH) 0 DIMETHYLATED REDUCING SUGARS DIMETHYL GLUCOSE For the preparation of this compound two of the hydroxyl groups in glucose, in addition to the reducing group, must be protected from methylation by the introduction of hydro- lysable residues, and we thus made use of the benzylidene a-methylglucoside described by Van Eckenstein. 162 PREPARATION OF METHYLATED SUGARS It will afterwards be shown that the most probable structure for the latter compound is that given below : CH2 . CH . CH . CH(OH) . CH(OH) . CH(OCH)3 o o I o V A C6H H According to this view of the constitution a new asymmetric carbon atom is introduced into the molecule, and thus benzal- dehyde should condense with a-methylglucoside so as to pro- duce two isomeric products. This possibility seems to have been overlooked by Van Eckenstein, but we have succeeded in isolating the hitherto unknown isomeride by cautious crystallisation of the accumulated mother liquors obtained in a large scale preparation of the condensation compound. The new stereoisomeride crystallises in short prisms melting at 148-149° and shows [ajjf+96'00 in aqueous solution. This compound, which is only produced in small amount, is therefore d-benzylidene-a-methyl-rf-glucoside. The methylation of Z-benzylidene-a-methylglucoside pro- ceeded with unusual smoothness as, although acetone was required to promote the solution of the compound in methyl iodide, one treatment with the alkylating mixture was sufficient to effect complete methylation. The product was readily obtained in the crystalline state in nearly quantitative amount, and, after recrystallisation from ligroin, melted at 122-123°. The specific rotation in acetone solution for c=l*64 was +97'03°. It was found possible, by heating the compound for one hour at 95° with one per cent, hydrochloric acid, to remove the aromatic residue without affecting the glucosidic group. The product of this reaction was therefore dimethyl a-methylglucoside, which was isolated in the usual manner. The compound, when crystallised from hot benzene, AND POLYHYDRIC ALCOHOLS 163 melted at 80-82° and behaved as a glucoside towards Fehling's solution. The specific rotation in aqueous solution was + 142'6° a value which is practically identical with that found for a solution in acetone. When boiled for 30 minutes with 10 per cent, hydrochloric acid, the glucosidic group was removed, and, on working up the product, dimethyl glucose was obtained in the form of a syrup which gradually crystallised on standing. After crystal- lisation from ethyl acetate, the sugar was obtained in the form of well-developed prisms which gave satisfactory analytical figures but which melted very indefinitely. This behaviour was shown to be due to the presence of the stereo- isomeric a- and /3-forms, both of which were ultimately separated. When the crystallisation from ethyl acetate is carried out with solutions containing not more than five per cent, of the solute, the sugar separates in clusters of delicate pointed prisms. This is the pure /3-form (m.p. 108-110°) and thus shows upward mutarotation when dissolved. Solvent. c. Change in Specific Rotation. Water . . . 5'00 +10-6° -- »- +64-4° Alcohol ... 5-02 5-7° 49" 4° Acetone. . . 3'84 6'5° — > 50'9° The compound showed the phenomenon of suspended mutarotation in acetone solution, and it was thus possible to obtain an exact record of the whole range of the optical change in this solvent. The a-form of the sugar separates, along with the /3-isomeride, from solutions in ethyl acetate containing from 5 to 10 per cent, of the solute. Separation of the two varieties was, however, effected by cautious precipitation from the equi- librium solution in alcohol by the gradual addition of ether. The first crops to separate consisted as before of the /8-form, but from the mother liquors the a-isomeride (m.p. 85-87°) was 164 PREPARATION OF METHYLATED SUGARS deposited in warty aggregates of microscopic prisms. These showed the reverse mutarotation in acetone solution. Initial Specific Rotation. Permanent Specific Rotation. +81-9° +48-3° Dimethyl glucose is thus well adapted for the study of mutaro- tation in that both stereoisomeric forms are available, and the ready solubility of the sugar in solvents in which the change is slow permits of the initial values being exactly determined. The optical study of the compound is now complete, but the results obtained are beyond the scope of this paper. DIMETHYL RHAMNOSE [Purdie and Young (Trans., 1906, 89, 1196).] The preparation of this compound resembled that just de- scribed, but differed in one essential, in that only two hydroxyl groups in rhamnose require to be protected from methylation. This was effected by the use of Fischer's rhamnosemonoacetone. The properties of dimethyl rhamnosemonoacetone, and of the alkylated sugar obtained from it by hydrolysis, were normal, but the detailed study of the substituted rhamnose was restricted by the fact that it could not be obtained in the crystalline state. It was however shown for the first time, in the investigation to which reference is made above, that the condensation of acetone with reducing sugars involves two hydroxyl groups in the latter compounds. This result is naturally of special importance in devising methods for preparing partially methylated sugars. AND POLYHYDRIC ALCOHOLS 165 TRIMETHYLATED GLUCOSES 2:3:5 TRIMETHYL GLUCOSE 1 Up to the present, three isomeric trimethyl glucoses have been prepared. One of these is the dimethyl methylglucoside already described : the remaining two compounds are reducing sugars which differ in the distribution of the methoxyl groups. It has already been shown by Purdie and Irvine that the methylation of methyl glucoside gives trimethyl methyl- glucoside as the main product when the reaction is carried out in methyl alcoholic solution. Considering the mode of preparation of this compound and the reactions of the tri- methyl glucose obtained from it on hydrolysis, it is evident that the primary alcoholic group present in the parent gluco- side escapes methylation when the reaction is carried out in presence of excess of alcohol. In solubility, reducing power, optical activity, and general chemical character, including its oxidation to a lactone, trimethyl glucose resembles tetra- methyl glucose closely. Substitution of methoxyl for hydroxyl in the three positions specified has therefore little effect on the properties of glucose, and this result is important as it will be afterwards shown that a change in the position of the alkyloxy-groups results in the complete alteration of the optical relationships. No doubt the method just described is capable of general application as a means of obtaining trimethylated aldoses containing an unsubstituted primary hydroxyl group, but the risk of the product being contaminated with other derivatives, and the experimental difficulties encountered in purifying viscous syrups by vacuum distillation, have not induced us to extend the method to other examples. 1 This nomenclature for derivatives of sugars, although not yet adopted in standard works of reference, seems specially advisable for the compounds described in this paper. Inspection of the formula for monomethyl glucose will indicate the carbon atoms to which the numbers refer. — J. C. I. 166 PREPARATION OF METHYLATED SUGARS 3:5:6 TRIMETHYL GLUCOSE CH2(OCH3) . CH(OCH3) . CH . CH(OCH3) . CH(OH) . CH(OH) 0 In the preparation of the above compound glucosemono- acetone was alkylated, first in acetone solution and afterwards in methyl iodide solution. The liquid product, isolated from the reaction, boiled at 138-139°/12 mm. and had apparently undergone partial hydrolysis as it possessed a decided action upon Fehling's solution, and gave, on analysis, figures inter- mediate between those required for trimethyl glucosemono- acetone and trimethyl glucose. This result is not surprising considering the ease with which glucosemonoacetone is hydrolysed. The optical values observed for the methylated acetone compound have in consequence little significance, but the fact that the compound is laevo-rotatory ([a]^0'— 27 '2° in methyl alcohol) is nevertheless remarkable. The hydrolysis was carried out exactly as in the prepara- tion of monomethyl glucose, but as the sugar could not be obtained in the crystalline state the syrup examined would consist of the equilibrium mixture of a- and /3-forms. The proportion of the latter variety was therefore increased by heating the compound at 70° for half an hour, and, on solution in alcohol, the following optical values were obtained :— Initial Specific Rotation. Permanent Specific Rotation. -6-2° — -> —8-3° As already indicated, the rotatory power of this sugar is in every respect abnormal. Whereas both the a- and /3-forms of glucose, and also of all the other known methylated glucoses, are dextrorotatory, the equilibrium value for 3:5:6: tri- methyl glucose is laevo. Not only so, but the mutarotation recorded above indicates that the /3-form is either dextro- or feebly laevo-rotatory, as the change ft —> a results in an increase AND POLYHYDRIC ALCOHOLS 167 of laevo-rotation. This is at variance with all other optical values obtained for the glucoses and admits of no simple explanation. The abnormality can hardly be due to a highly pronounced laevo-rotatory effect attending methylation of the primary alcohol group in the sugar chain, as inspection of the available optical data will show. APPLICATIONS OF PARTIALLY METHYLATED SUGARS The various applications of these compounds have already been briefly referred to in the introduction, and it will be seen that they afford considerable scope for investigation. An additional question of theoretical interest was raised in the course of the work, this being the varying capacity of members of the sugar group to enter into condensation with aldehydes or ketones. Thus methylmannoside gives both mono- and dibenzylidene derivatives, while methylglucoside and methyl- galactoside only condense with one molecule of benzaldehyde. Several other examples are known in which, although the necessary hydroxyl groups are present in a sugar derivative, and are moreover situated in the spatial proximity apparently favourable to condensation, are nevertheless incapable of participating in condensation reactions. This behaviour seems to be due to the stereochemical arrangement of the hydroxyl groups, and consideration of the available data points to the idea that condensation proceeds readily when the reacting hydroxyl groups are in the cis position with reference to the internal ring of the sugar molecule, but not when they are in the trans position. These considerations permit of the configuration of the a- and /8-forms of reducing sugars being definitely established, and obviously the study of partially methylated sugars will prove useful in developing this line of research, as compounds of this nature may be obtained in which only the trans positions in the molecule are methylated. i68 PREPARATION OF METHYLATED SUGARS Again, in view of their convenient solubilities and the tendency of partially methylated sugars to display suspended mutarotation, it is evident that these compounds are suit- able substances for the study of the rotatory powers of a- and /J-forms and of the rotatory changes shown in the forma- tion of equilibrium mixtures, as it is possible to obtain an accurate polarimetric record of the tautomeric changes without calculation of the true initial values. In this way we have shown that both monomethyl and dimethyl glucose conform to the generalisation established recently by C. S. Hudson regarding rotatory power in the sugar group. The general question of the optical effect of methylation on the rotatory powers of glucose has also been studied in the course of the work. The results will shortly be published, but the discus- sion is beyond the scope of this paper. With regard to the application of the new compounds in testing the theories of alcoholic fermentation it is evident that the use of monomethyl glucose offers special advantages. The current theories are chiefly based on the analogy of other reactions of sugars, and can thus be tested by the isolation of the intermediate products of the change. Although research in this direction has been highly profitable, the results obtained are in many cases conflicting or even contradictory. It will, however, be seen that, using monomethyl glucose as a sub- strate, the destiny of the methyl group at once gives a clue to the mechanism of alcoholic fermentation, as the position of the alkyloxy group in the molecule is known. Thus, according to Baeyer's dehydration theory, the fermentation products should be (1) methyl alcohol, (2) ethyl alcohol, (3) lactic acid, and (4) carbon dioxide. Wohl's theory, on the other hand, admits of a greater number of possibilities, but the most feasible should result in the formation of n-propyl alcohol, ethyl alcohol, a-hydroxybutyric acid, and carbon dioxide. Similarly the dimethyl glucose now described should give either (1) dimethyl ether and ethyl alcohol, or (2) methyl AND POLYHYDRIC ALCOHOLS 169 alcohol and ?i-propyl alcohol as the essential fermentation products. As the more fully alkylated sugars are not fermentable, work of this nature could not be extended beyond the limits specified. PARTIAL ALKYLATION OF POLYHYDRIC ALCOHOLS Generally speaking, the same principles are utilised in the preparation of partially methylated polyhydric alcohols as apply to the formation of the corresponding substituted sugars, i.e. the methylation of a derivative which still contains hydroxyl groups, and the removal of the substituting residue by hydrolysis. In the case of the alcohols most closely related to the sugars, suitable derivatives for this purpose are practically unknown, in fact the mechanism of the formation of condensation pro- ducts is probably dissimilar in the two classes referred to, in virtue of the presence of an acidic reducing group in the sugars and the absence of such a group in the alcohols. It will be seen that, in the case of alcohols where an even number of hydroxyl groups are available for condensation with a ketone, a completely substituted derivative will in most cases result. It is only when the alcohol under examination contains an odd number of hydroxyl groups that the methylating reaction may be directly applied to the acetone derivative. Thus, arabitol can only condense with two molecules of acetone as a maximum, and consequently one hydroxyl group is thus left available for methylation. On the other hand, in the case of mannitol, condensation with acetone gives a triacetone derivative which is naturally unaffected by alkylation, and, moreover, it is impossible to control the condensation so as to stop the reaction when only one or two acetone residues have entered the molecule. This difficulty may be overcome by taking advantage of the fact that the acetone residues in mannitol triacetone may 170 PREPARATION OF METHYLATED SUGARS be removed in definite steps. This behaviour is doubtless controlled by causes similar to those which influence the partial hydrolysis of a-fructose-diacetone. Irvine and Garrett have shown that in the formation of this compound the addition of the non-glucosidic acetone residue involves torsion of the internal ring of the sugar molecule, but that the glucosidic acetone group is differently linked and occasions less molecular strain. The expressions trans and cis may be used to dis- tinguish between the two types of linkage, and it has also been shown that carefully regulated hydrolysis removes the more unstable trans group, and thus a glucosidic monoacetone derivative remains. These considerations are illustrated in the formulae shown below : — ,CH20 C'(Me), . CIS CHoOH.CH C(Me), AND POLYHYDRIC ALCOHOLS 171 Before developing the argument further, it may be men- tioned that the above speculations assume the ketonic re- sidues to be attached so as to form five-membered rings. So far there is no rigid proof of this, but the results obtained in the study of the fructose-acetones are most easily explained on this assumption. Further, if we accept Fischer's view that ketones condense with ft rather than with a carbon atoms a series of contradictions arises. Thus the arrangement is impossible in the case of glucosediacetone, and similarly the ketonic residues in mannitol triacetone cannot be arranged in accordance with this idea, as at least one pair of a carbon atoms must participate in the condensation. The simpler view is that this type of condensation is symmetrical, wherever possible, so that mdnnitol triacetone becomes : 20 !HO O A CH< CHO\ I >' /ITT (\/ OlloU If Alt A2, A3 represent the acetone residues, it is seen that, considering the configuration of mannitol, A2 must of necessity be a trans grouping, applying this expression in the sense already indicated. On the other hand, A^ and A3 may be either cis or trans. It does not necessarily follow that in each of these cases the more stable cis linking will result, as the relative positions of the terminal hydroxyl groups may conceivably be affected by the configuration of the remaining asymmetric systems, and they may thus react preferentially in one or other of the two alternative positions. Our results are in fact in agreement with this view. i;2 PREPARATION OF METHYLATED SUGARS Presumably then a difference in stability is to be expected in the three acetone residues in mannitol triacetone in virtue of the different effect of cis and trans grouping on the molecular balance. A further difference in stability might also be expected to arise from the fact that one acetone residue (A2) substitutes the hydrogen atoms of two secondary hydroxyl groups, while the linkage of the remaining ketonic residues (Ai and ^3) involves one primary alcohol group in each case. This factor, although no doubt present, does not seem to exercise any marked effect on the stability of the different groups as glucose diacetone, a compound in which the same conditions prevail, undergoes hydrolysis in one stage only. This at once points to the fact that the stability of the acetone residues is controlled by their stereochemical arrangement. It is of course inadmissible to claim that results obtained with ring structures such as glucose and fructose must find exact analogies in the case of an open chain compound such as mannitol, but a close parallel has nevertheless been estab- lished in that carefully regulated hydrolysis of mannitol triacetone removes the ketonic residues simultaneously but at different rates. It is thus possible to isolate the inter- mediate compounds indicated in the following scheme :— Mannitol triacetone — > mannitol diacetone — > mannitol monoacetone — > mannitol. The reaction is naturally a troublesome one to control, but was effected by heating a 2*5 per cent, solution of the triace- tone compound in 70 per cent, alcohol, containing O'l per cent, of hydrogen chloride, to 40° for about three hours. A con- tinuous polarimetric record of the optical changes gave figures which, when plotted in a curve, showed two indefinite maxima followed by a regular fall to a constant value. On stopping the reaction at intermediate stages, it was found that, in the neighbourhood of the first maximum, the main product was a mannitol diacetone mixed with varying amounts of mannitol AND POLYHYDRIC ALCOHOLS 173 and unhydrolysed material. Similarly, when the reaction was arrested at a time corresponding to the second maximum on the curve, the product was essentially mannitol monoacetone. The formation of a definite mannitol diacetone, as the first step in the production of the monoacetone derivative, is obviously of great importance in tracing the course of this interesting hydrolysis, as it eliminates the possibility of the hydrolytic change being confined, in the first instance, to the acetone residues coupled in the primary alcoholic positions. Moreover, it indicates that there is a difference in the stability with which the terminal residues At and A3 are attached to the molecule, although the general symmetry of the structure would not justify this conclusion except on the assumption that there is a difference in the linkage of the groups (A1 and As). The study of mannitol monoacetone lends support to this view. Methylation of the compound by the silver oxide reaction resulted in the formation of tetramethyl mannitol monoacetone, from which tetramethyl mannitol was obtained on hydrolysis. The fact that this compound may be oxidised by Fenton's reagent to give an alkylated reducing aldose, shows that the stable acetone residue was originally attached to a terminal primary alcohol group. This series of reactions may therefore be interpreted in the following way : — CH20N 3(Me)2 Trans. CHO I CHC CHO - )>C(Me)o Trans. ' C HO\ >C(Me)2 Cis. 0X CH2OH CH2OCH3 CHjjOCHjj CHOH CHOCH3 1 CHOCH3 i CHOH --> CHOCH3 -> CHOCH3 1 1 i 1 i CHOH CHOCH3 CHOCH, CHOV 1 >C(Me)2 CH20/ CHO v 1 >C(Me)2 CH20/ CHOH CH2OH (A) . . . (B) . . . (G) 174 PREPARATION OF METHYLATED SUGARS (A) Mannitol monoacetone : — Crystallises in prisms melting at 85°. Specific rotation in alcohol +26'4° (c=2-7). (B) Tetramethyl mannitol monoacetone : — Liquid boiling at 137-140°/11 mm. Specific rotation in alcohol + 39'0° (c=2*8). (C) Tetramethyl mannitol : — Liquid boiling at 167-169°/13 mm. Obtained crystalline on standing. Specific rotation in alcohol — 12'5°. The experimental evidence bearing on the constitution of mannitol diacetone is in the meantime somewhat incomplete. Alkylation of the compound gave dimethyl mannitol diacetone (b.p. 140-141°/13 mm.), and, on hydrolysis, a crystalline dimethyl mannitol was obtained (m.p. 93°; [a]D— 8'8° in alcohol). The position of the methyl groups in the latter compound has not yet been determined, but there can be little doubt that they occupy positions 3 and 4 in the carbon chain. Inspection of the above results will show that the behaviour of mannitol triacetone on hydrolysis may also be explained to some extent by the assumption that the com- pound exists in two, hitherto unrecognised, isomeric forms, the arrangement of the substituent groups A19 A2, and A3 being respectively cis, trans, cis, and trans, trans, cis. Until further work has been done on the constitution of dimethyl mannitol, it is perhaps premature to speculate too freely on the validity of this assumption, which certainly adds a new feature of complexity to the condensation reactions of opti- cally active compounds, but our unsuccessful attempts to isolate a second form of mannitol triacetone render this alternative unlikely, and the existence of a monoacetone derivative is also opposed to it. In the meantime, therefore, the bulk of the evidence points to the idea that the terminal alcoholic groups in AND POLYHYDRIC ALCOHOLS 175 mannitol, although unconnected with an asymmetric carbon atom, assume preferentially different positions which affect the stability of the condensation derivatives. According to this view, mannitol triacetone exists in only one form, the linkage of the ketonic residues being trans, trans, cis. APPLICATIONS OF PARTIALLY METHYLATED POLYHYDRIC ALCOHOLS The most important application of these compounds will doubtless be found in the opportunity they afford for con- firming configuration. From their mode of formation, the compounds contain hydroxyl and methoxyl groups, the position of the former alone being favourable to condensation with acetone. In other words, these groups represent cis and trans positions respectively in the original compound so that determination, by standard methods, of the distribution of the alkyloxy groups gives the configuration. Another possible development of this line of research is the preparation of methylated aldoses, isomeric with alkylated sugars prepared from glucosides, but containing the methoxyl group in the y-position and thus capable of functioning as aldehydes only. The examination of compounds of this class, with respect to their capacity to display mutarotation and to participate in glucoside formation, should afford definite evidence as to the structural causes of these phenomena in the reducing sugars. Again, as in the case of partially methylated sugars, the behaviour of the corresponding alcohols towards catalytic fermenting agents should be capable of yielding results bear- ing directly on the mechanism of these changes. Finally, the compounds are readily converted into mixed 176 PREPARATION OF METHYLATED SUGARS ethers. Thus, ethylation of tetramethylmannitol results in the formation of a diethyl-tetramethyl-mannitol, and a tetraeihyl- dimethyl-mannitol has also been prepared from dimethyl- mannitol. Mixed ethers of this type are now receiving attention in view of their powerful narcotic properties, and, as the methods available for their preparation are not numerous and are of limited application, several processes for their production have recently been protected (D.R.P., 226454). By the pro- cess now described it should be possible to prepare mixed ethers of considerable complexity such as dimethyl-diethyl- dipropyl hexitols, in which the relative positions of the three types of alkyloxy groups could be varied, according to the order in which the alkylations were effected. Although several workers have contributed the experi- mental data necessary for this paper, special acknowledgement is due to Mr. J. P. Scott, M.A., B.Sc. (Carnegie Fellow), who investigated the partially methylated glucoses now described for the first time. The section of the paper dealing with ethers of polyhydric alcohols is based on results obtained by Miss B. M. Paterson, B.Sc. (Carnegie Scholar). It should also be stated that all necessary references to original papers on the alkylation of the sugars will be found in the bibliography appended to Mr. Young's contribution to this volume. JAMES COLQUHOUN IRVINE A GENERAL REVIEW OF PURDIE'S REACTION: ALKYLATION BY MEANS OF DRY SILVER OXIDE AND ALKYL IODIDES INTBODTTCTOBY NOTE THE process of alkylation by means of silver oxide and alkyl iodides was first employed by Purdie and Pitkeathly in 1899 (I).1 Prior to that time the original work which had appeared from the St. Andrews Chemical Laboratory had been con- cerned chiefly with the preparation, resolution, and examina- tion of active acids and their derivatives. Several racemic alkyloxy-acids had already been obtained and resolved into their active forms. It was realised that these substances were of much importance, since their activity was found to be considerably greater than that of the parent hydroxy-com- pounds, and further, they were free from the disturbing effect on optical activity which is exercised by the hydroxyl group. The discovery of the ' silver oxide reaction,' as it may be called, rendered possible the direct conversion of esters of active hydroxy-acids into active esters of alkyloxy- acids, and thus greatly facilitated the work in hand. The application of the process to the synthetical introduction of alkyl groups has been extended in various other directions, which are indicated in this paper. In the course of the preparation of ethereal salts by the 1 The figures in brackets are the reference numbers to the original papers, the titles of which are arranged in chronological order in the bibliography appended to this communi- cation. The abbreviations used are those adopted by the Chemical Society of London. Z 178 A GENERAL REVIEW OF action of alkyl iodides on silver salts of hydroxy-acids, certain anomalous results were obtained. The ethereal tartrates prepared in this way, for instance, displayed abnormally high rotations, and it was ascertained that alkyloxy-derivatives were produced during the reaction. In endeavouring to ob- tain evidence of the production of these alkyloxy-derivatives, Purdie and Pitkeathly (1) found that the reaction between silver malate and isobutyl iodide gave rise to very little ethereal salt but to free malic and isobutoxy-succinic acids, and further, that isopropyl isopropoxysuccinate was formed during the interaction of isopropyl iodide and silver tartrate. Consideration of these results led to further experiments. Ethyl malate, ethyl iodide, and litharge, when heated together, showed no interaction, but on substituting mercuric oxide for litharge, a product which was more active than ethyl malate was obtained. Finally, when silver ox^de was added to a solution of ethyl malate in isopropyl iodide, a vigorous reaction was found to ensue. A closer study of the reaction was now made ; the materials used were ethyl malate, ethyl iodide, and silver oxide in the proportions calculated on the assumption that the reaction proceeds according to the following equation :— OH . C2H3(COOEt)2+2EtI+Ag20 = OEt . C2H3(COOEt)2+EtOH+2AgI. From the reaction mixture pure ethyl Z-ethoxysuccinate was isolated and its optical activity was found to agree with that of the ester prepared from the active acid previously obtained by resolution of the racemic acid (Trans., 1895, 67, 972). The value of the reaction was further exemplified by ethylating ethyl tartrate by treating it with silver oxide and ethyl iodide in a similar manner ; the product of the reaction was ethyl rf-diethoxysuccinate. All previous at- tempts to alkylate the alcoholic hydroxyl groups of tartaric acid had been unsuccessful. It is thus seen that the discovery PURDIE'S REACTION 179 of this method of alkylation was not a chance result, but was due to a careful and systematic tracing of an effect to its cause. The reaction was immediately utilised for the preparation of the methyl methoxy- and ethyl ethoxy-propionates (Purdie and Irvine (2) ), and the optical activity of the corresponding acids and their salts was determined and compared with the values obtained for the same compounds which had previously been prepared by resolution of the racemic acids by morphine (Purdie and Lander, Trans., 1898, 73, 862). The comparison indicated that the alkylation process proceeded without any racemisation occurring. At the same time McKenzie (3) was able to prepare inactive and active phenylalkyloxyacetic acids by the alkylation of i- and Z-mandelic acids and their esters. In the following year (1900) Lander (4) published the results of an elaborate investigation of the general applica- bility of this method of alkylation. He was able to show that, by treatment with silver oxide and alkyl iodides, it is possible to substitute alkyl groups for the hydrogen in the hydroxyl groups of compounds of such widely different types as /-menthol, i-benzoin, ethyl acetoacetate and salicylic acid. The reaction of benzaldehyde with silver oxide and ethyl iodide leads to the oxidation of the substance and sub- sequent esterification of the resultant acid, ethyl benzoate being obtained. It was further shown that silver oxide and alkyl iodides react with amides and substituted amides, a hydrogen atom being replaced by an alkyl group. Acet- anilide, for instance, yielded N-phenylacetiminoethyl ether C6H5.N : C(OC2H5).CH3. The production of imino-ethers is therefore much facilitated by this reaction, since the prepara- tion and isolation of the intermediate silver compounds are obviated. The results which have been mentioned rendered it evident that the use of dry silver oxide and alkyl iodides constituted i8o A GENERAL REVIEW OF a general means of syiithesising alkyl derivatives of compounds of more or less well-defined acid character, possessing hydrogen atoms replaceable by alkyl radicles. ADVANTAGES OF THE PROCESS There are many advantages attending the use of this method of alkylation. The products are generally obtained pure, in good yield, and there is no difficulty in separating the inorganic by-products. If an alkyl iodide has been em- ployed in the alkylation, these by-products are silver iodide and water. The latter can in some cases be removed by employing a sufficient excess of the alkylating materials ; in any case the insolubility of water in alkyl halides renders it of little effect on the course of the main reaction if no extraneous solvent is being used. The insolubility and stability of the remaining inorganic by-product, silver iodide, obviate any difficulty in the extraction of the organic product. The reaction is generally smooth and rapid in its course, and it can be very readily controlled. Alkyl iodides are, for the purpose of this reaction, more conveniently used than other halides. This is, possibly, simply a manifestation of the well-known fact that the elimination of silver iodide in organic synthesis occurs with great readiness. It is further possible that the iodides are more suitable than other halides by reason of their inferior stability. Alkylation proceeds most readily in the case of those compounds which are soluble in alkyl iodides, and in such cases no further addition of a solvent is necessary. It should be noted that the reaction is carried out in a neutral medium, and is therefore advantageously employed in the etherification of substances which readily undergo hydrolysis. A further advantage of this method of alkylation is that it can be applied to optically active compounds without the occurrence of racemisation or inversion. In fact, up to the PURDIE'S REACTION 181 present time, the reaction has been almost exclusively em- ployed in the preparation of derivatives of active compounds, and in no case has an optical change of the nature of the Walden inversion been noticed. Purdie and Barbour (9) definitely proved that no inversion of this kind occurred on methylating methyl d-tartrate, as on hydrolising the methyl di- methoxysuccinate produced they obtained a dimethoxysuccinic acid, which, when reduced with hydriodic acid, gave d-tartaric acid. The proof has been extended by Irvine (33) to the methyl Z-methoxy-propionate obtained by the methylation of methyl Wactate. The substance was reduced with hydri- odic acid and yielded Mactic acid, which was identified by conversion into its zinc salt. The silver oxide method of alkylation has therefore no disturbing effect on the configura- tion of an active lactate. In some few cases (McKenzie (3) ) racemisation has been observed, but it is not certain whether this occurred during the actual reaction or in the isolation of the product. A modification of the reaction has been devised by Forster (44 and 45) for the purpose of esterifying acids without risk of racemisation. Silver oxide, in slight excess of the calculated amount, was mixed with an ethereal solution of Z-a-triazo- propionic acid, and, without separating the silver salt from the unchanged oxide, excess of ethyl iodide was added and left in contact with the mixture for thirty-six hours. The filtered solution yielded the desired ethyl Z-a-triazopropionate CH3.CHN3.C02C2H5. Ethyl Z-a-triazobutyrate was prepared in a similar manner. By the action of silver oxide and methyl iodide on salicylic acid, Lander (4) obtained pure methyl o-methoxybenzoate. Similarly McKenzie (3) prepared i-ethyl phenylethoxyacetate from i-mandelic acid, and Denham converted glyceric acid into methyl o/J-dimethoxypropionate (private communica- tion). We have here instances of esterification of the acids and alkylation of alcoholic hydroxyl groups proceeding 182 A GENERAL REVIEW OF simultaneously, and from these results it would appear that the method might prove to be useful for the esterification of acids which give unstable silver salts. The possibility of utilising the reaction as a means of determining constitution and configuration will be discussed later. RANGE OF APPLICATION OF THE REACTION It may at once be stated that all stable compounds con- taining primary or secondary alcoholic hydroxyl groups or carboxyl groups invariably give positive results with the reaction if the compounds are soluble in the alkyl iodides or other neutral solvent ; no cases have yet been encountered in which substances of the nature described have escaped alkylation by this process. Tertiary alcoholic groups, exclud- ing those formed in tautomeric compounds by the keto-enol change, appear to entirely resist the action of the alkylating mixture. Lander (4) found, for instance, that triphenyl carbinol showed no change either when boiled for twelve hours with silver oxide and ethyl or isopropyl iodides, or even when heated with silver oxide and ethyl iodide in a sealed tube at 160° C. for four hours. The point is well illustrated by the experience of McKenzie and Wren (53), who subjected i-tri- phenylethylene glycol to two alkylations with silver oxide and methyl iodide, and obtained a monomethyl ether as the sole product. This product they proved to be ^8-hydroxy-a- methoxy-a^-triphenylethaneC6H5.CH(OCH3).C(OH).(C6H5)2 and not the isomeric a-hydroxy-/3-methoxy-a/3/8-triphenyl- ethane, since they obtained a substance identical with this product by the action of magnesium phenyl bromide on either i-benzoin methyl ether or i-methyl phenylmethoxy- acetate. It is clear, therefore, that only the secondary alcoholic hydroxyl group of triphenylethylene glycol is methylated by the alkylating mixture ; the tertiary hydroxyl group remains unattacked despite the excess of alkylating PURDIE'S REACTION 183 materials used. A further instance was noted by Purdie and the writer (58), who found that the two hydroxyl groups of yS - dimethoxy - fie - dimethylhexane - /3e - diol HO. C (CH3)2.CH (OCH3).CH(OCH3).C(CH3)2.OH resisted alkylation by this method. As regards phenolic hydroxyl groups data are lacking, since experiments do not appear to have been made on the simple phenols. With certain substituted phenols, however, alkyla- tion of the phenolic group occurred readily. Thus Irvine (5 and 11) prepared salicylaldehyde methyl ether in 90 per cent, yield by the action of silver oxide and methyl iodide on salicylaldehyde. Practically no oxidation of the aldehyde occurred, since only a trace of methyl o-methoxybenzoate was found in the product. Similarly Lander (4) prepared the latter ester directly from salicylic acid. The phenolic ethers so obtained were free from resinous matter, such as is generally produced during the preparation of these substances by the aid of alkyl iodides and caustic potash. On the other hand, Meldola and Kuntzer (52) obtained a negative result with a substituted phenol. Although the silver oxide reaction finds its chief applica- tion in the alkylation of hydroxyl groups, yet, as Lander has shown (4, 6, 7, 13, 21, and 24), it is possible by its aid to substitute alkyl radicles for hydrogen in the molecules of amides and substituted amides. In this way imino-ethers may be prepared. An account of the results which have been obtained in this direction is given later. METHODS OF APPLYING THE REACTION It is important that the silver oxide used should be freshly prepared and carefully dried. The oxide is most conveniently prepared by adding hot, filtered barium hydroxide solution to a hot solution of silver nitrate, washing the precipitated oxide with boiling water until all excess of barium hydroxide has 184 A GENERAL REVIEW OF been removed, and drying the filtered substance first on a porous plate and afterwards in a steam-oven, the door of which remains open. The oxide should be finely powdered to facili- tate the drying, and kept in a desiccator until required for use. Freshly prepared silver oxide appears to act with greater readiness than that which has been kept for some time. As mentioned above, those substances are most readily alkylated which are soluble in alkyl iodides. If, however, the compound to be alkylated is insoluble in the alkyl halide, it may be dissolved in a neutral solvent such as acetone or benzene. In such cases the alkylation is slower, and is fre- quently incomplete after one treatment. In this connection it is to be noted that alkylation has the general effect of increasing the solubility of a substance in organic solvents. Hence if the substance under treatment is insoluble in the alkyl iodide used, and is one into which it is possible to intro- duce more than one alkyl group, then the partially alkylated product may be, and frequently is, soluble in the halide. This being so, an extraneous solvent is not required in the subsequent alkylation which is necessary to complete the reaction. The sugars furnish extreme cases of this kind. Cane sugar is so insoluble in organic solvents that in order to methylate it (19), it was found necessary by Purdie and Irvine to convert the sugar into a syrup by adding its own weight of water, to dissolve this in methyl alcohol, and then to add silver oxide and methyl iodide in small quantities at a time in order to prevent precipitation of the sugar by the iodide. Water, alcohol, and sugar were doubtless attacked simultane- ously by the alkylating materials. The product from the first alkylation was soluble in methyl alcohol, and after three alkylations, in methyl iodide. The fourth and final alkylation was therefore conducted in methyl iodide solution, and a completely methylated product was thereby obtained. If it is necessary to employ an extraneous solvent, the best method of procedure is to dissolve the substance in the PURDIE'S REACTION 185 minimum amount of the hot solvent and to add the silver oxide and alkyl iodide in small, aliquot quantities. The preparation of dimethyl 0 CH . CPh/ and a triphenylbutyrolactone respectively. The same furan derivative is obtained by the action of magnesium phenyl bromide on methyl maleate. PREPARATION OF DIALKYLOXY-DERIVATIVES OF SUCCINIC ACID The first compound of this type to be synthesised by means of the silver oxide reaction was ethyl d-diethoxysuccinate, prepared by Purdie and Pitkeathly (1). The production of methyl rf-dimethoxysuccinate from methyl tartrate by Purdie and Irvine (8) has already been described ; from ethyl tartrate, by treatment with silver oxide and methyl iodide, the same authors prepared ethyl rf-dimethoxysuccinate. The crystalline rf-dimethoxysuccinic acid is obtained from either of the methoxy-esters by hydrolysis with barium hydroxide, and subsequent decomposition of the barium salt by sulphuric acid. A number of normal and hydrogen-metallic salts of d-dimethoxysuccinic acid were prepared, and the determina- tion of the rotation of these, and of the esters mentioned above, furnished material from which certain generalisations on the optical activity of this series of compounds could be drawn. d-Dimethoxysuccinamide was also prepared, but attempts to convert this into the corresponding imide, like other attempts made at a later date, were fruitless. The work on the optical activity of methyl-, ethyl-, and propyl-dimethoxysuccinates was continued by Purdie and Barbour (9), who traced the influence of solvents on the IQ2 A GENERAL REVIEW OF rotatory powers of these esters and of the corresponding tartaric esters. Purdie and the writer (58) have prepared a series of dimethoxy-compounds similar to those derived from mono- methoxysuccinic acid. It was hoped that information bearing on the tautomeric behaviour of succinyl chloride might be obtained by the study of its active analogue d-dimethoxy- succinyl chloride. Neither this compound nor Z-methoxy- succinyl chloride, however, exhibited mutarotation when dissolved in indifferent solvents, and from these and other experiments it must be concluded that the substances do not consist of mixtures of dynamic isomerides. Treatment of methyl . CH(OMe) . CMe/ A corresponding compound is directly obtained when mag- nesium phenyl bromide is substituted for magnesium methyl iodide in the above reaction ; this product is 3 : 4-dimethoxy- 2:2:5: 5-tetraphenyltetrahydro-furan : — CH(OMe) . CPh2 1 CH(OMe).CPh THE ALKYLATION OF MONOSACCHARIDES Although benzoin and salicylaldehyde can be directly alkylated by the silver oxide method, the process is not directly PURDIE'S REACTION 193 applicable to the simple hexoses. Glucose dissolved in methyl alcohol and treated with silver oxide and methyl iodide yields an acid syrup which undergoes decomposition when subjected to distillation (Purdie and Irvine (19) ). The only product of the change which can be identified is methyl oxalate, evidently formed by the disruptive oxidation of the sugar molecule. In order to prepare etheric derivatives of aldoses and ketoses, it was found necessary to first mask the aldehydic or ketonic group of the sugar by conversion into methyl glucosides. The first of what was to prove a long and important series of communications on the sugar group was made by Purdie and Irvine in 1902 (19 and 22). Starting with a-methyl- glucoside and submitting this, in methyl alcoholic solution, to the action of silver oxide and methyl iodide, trimethyl a-methylglucoside was obtained as a syrup, and was purified by vacuum distillation. The substance was readily hydrolised by dilute aqueous hydrochloric acid, whereby only the glucosidic methyl group was removed and the corresponding sugar, trimethyl glucose, thus obtained. This sugar proved to be a viscous syrup which did not lend itself to further work. Trimethyl a-methylglucoside was, however, found to be soluble in methyl iodide, and, on treating this solution with silver oxide, the remaining hydroxyl group underwent methylation and tetramethyl a-methylglucoside was formed. This substance was a mobile, colourless liquid, readily purified by vacuum distillation. The product of its hydrolysis with aqueous hydrochloric acid was found to be the crystalline sugar tetramethyl glucose, a body which afterwards played an important role in the chemistry of alkylated sugars. Finally, when tetramethyl glucose was itself subjected to alkylation, no oxidation occurred, but there resulted a mixture containing tetramethyl a-methylglucoside and a large excess of a crystalline isomeric substance. This crystalline substance was later proved to be, not a pentamethylated aldehydic 2s A GENERAL REVIEW OF glucose, but tetramethyl /3-methylglucoside (25). The changes described are represented as follows : — CHOH CHOCH3 CHOCH3 ,CHOCH3 / CHOH /I / CHOH /I / CHOCH3 / CHOCH3 0 | \ CHOH — > \ CHOH 0 | > \ CHOCH3— (< 1 > \ CHOCH3 — \CH \CH VH \CH 1 | 1 1 CHOH CHOH CHOCH3 CHOCH3 CH2OH CH2OH CH2OH CH2OCH3 Glucose a-Methylglucoside. Trimethyl Tetramethyl (•y-oxidic). a-methylglucoside. a-methylglucoside. \ ;CHOH CHOCH3 CHOH /I / CHOCH3 CHOCH3 / CHOCHj o o 1 0 | — > \ CHOCH3 — > CHOCH3 \ CHOCH3 \ \ CH CH CHOCH3 CH.OCH, CHOCH, CH2OCH3 (•y-oxidic) Tetramethyl Tetramethylglucose. a- and /S-methylglucosides. CHOCH3 CH2OH (y-oxidic) Trimethylglucose. It has not yet been ascertained which of the hydroxyl groups in a-methylglucoside remains unalkylated when it is converted into trimethyl a-methylglucoside, and hence the formula given for the latter substance may not be strictly accurate as regards the positions of the substituent methyl groups. Work at present being conducted by Professor Irvine tends to show, however, that the view adopted is correct. On oxidation by Kiliani's method, tetramethyl glucose PURDIE'S REACTION 195 yielded tetramethyl gluconolactone. Bearing in mind the tendency of gluconic and similar acids to form y-lactones, it will be seen that the unmethylated carbinol group in tetra- methyl gluconic acid and in tetramethyl a-methylglucoside is that in the y-position. It is therefore the y-carbon atom in the methylated and in the parent glucosides which is united with the oxygen atom of the ring. These facts are of great importance, since they constitute what is probably the strongest experimental evidence in favour of Fischer's formula for glucosides. It has been pointed out that, excluding a few doubtful exceptions, the silver oxide reaction does not cause race- misation of reacting substances ; the sugars furnish a striking illustration of this point. Not only do the methylated compounds mentioned above retain optical activity, but they are chemically and optically more stable (if such a term may be used) than the corresponding unmethylated substances. It is beyond the scope of this paper to discuss, in a general manner, the results which have been obtained by the study of the optical activity of the methylated sugars and their compounds. These results are of importance, and are likely to prove of further value in any generalisations on the activity of the sugar group, since in the methylated com- pounds the constitutive influence of the hydroxyl groups is much modified, if not entirely eliminated. Tetramethyl glucose, as was to be expected, is very soluble in organic solvents. It is therefore possible to examine the substance polarimetrically in solutions from which water is absent and in which no ionisation can occur. As is well known, the reducing sugars exhibit mutarotation in aqueous and in alcoholic solutions, and this mutarotation is now believed to be due to the interconversion, in such solutions, of the a- and /8-forms of the dissolved sugar, these two forms being dynamic isomerides, differing in the position in space of the hydrogen and hydroxyl radicles attached to the terminal 196 A GENERAL REVIEW OF carbon atom. The view prevails that the interchange of positions of the radicles cannot occur directly, but is due to rupture of the oxidic ring, and it has been suggested that the rupture is accompanied by union of the molecule with a molecule of water. The fact that tetramethyl glucose shows marked mutarotation in such solvents as benzene and carbon tetrachloride from which all traces of water have been care- fully removed, at once disposes of all theories of the mechanism of the isodynamic change, which assume the addition of water or of alcohol to the molecule of the sugar (Purdie and Irvine (25) ). Tetramethyl glucose can be obtained in two forms, the a- and /3-isomerides, and these are similar, in their nature and mutarotation, to the corresponding varieties of glucose itself. Tetramethyl a-glucose, when dissolved in water, alcohol, benzene, toluene, or carbon tetrachloride, has initially a high dextro-rotation falling until a constant value is reached ; the /3-isomeride shows a low dextro-rotation initially, and this rises to the same equilibrium value. The values for the initial and equilibrium rotations are little affected by the nature of the solvent, a fact which seems to point to the elimination of the constitutive effect of the hydroxyl groups by methylation. This study of the alkylated glucoses and glucosides was continued by the correlation of the tetramethyl a-glucose with tetramethyl a-methylglucoside and of the /S-sugar with the corresponding glucoside. Irvine and Cameron (28) completed the identification of tetramethyl /3-methylglucoside by preparing it by direct alkylation of /8-methylglucoside. The ^-glucosides can therefore be alkylated by means of the silver oxide reaction in the same manner as their a-isomerides. The production of tetramethyl /8-methylglucoside by alkylation of tetramethyl glucose in the usual manner has already been mentioned. Curiously enough, this alkylation proceeds at -10° C., and takes exactly the same course as when carried out at higher temperatures (Irvine and Moodie (36) ). Further experiments by the same authors PURDIE'S REACTION 197 supply an explanation of this apparent anomaly, and afford an insight into the mechanism of this particular case of alkylation. Such methylation might be supposed to proceed either by the intermediate formation of an additive compound of sugar and alkyl iodide and subsequent removal of hydrogen iodide, or by the intermediate formation of a silver derivative in which silver replaces the glucosidic hydrogen atom. In the latter alternative, if the /3-sugar formed a silver derivative in this way and methyl iodide reacted with the resultant substance, the presence of a large excess of the /3-glucoside in the product would be accounted for. This supposition is rendered unlikely, however, by the failure, after repeated attempts, to obtain any evidence of the formation of a silver derivative of tetramethyl glucose. On the other hand, solutions of tetramethyl glucose in alkyl halides showed remarkable abnormalities in optical activity at low tempera- tures, and these abnormalities (the details of which cannot be given here) can be explained by association between the sugar and the alkyl iodide. It appears probable, therefore, in view of the existing evidence of the occasional quadrivalency of the oxygen atom, that in this case alkylation occurs by intermediate formation of a methyl iodide additive compound from which hydrogen iodide is subsequently removed by the agency of silver oxide. The change may be represented schematically as follows : — C— C C— C C— C ,OH | I ,OH ' \/ \OCH3 Y y/ \ \ C— C /\ I X I I /OCH3 CH3 I CH3 C G/ X/ \H 0 198 A GENERAL REVIEW OF Mention should be made of the preparation of a number of alkylated glucose derivatives which have been prepared with the object of determining the constitution of the parent glucose compounds ; these are referred to in a later section. The methods which have been applied to the preparation of alkylated derivatives of glucose serve also for the production of similar compounds of other sugars. a-Methylgalactoside, when fully methylated, gives a colourless, liquid tetramethyl a-methylgalactoside, and on hydrolysis with dilute hydro- chloric acid, the latter compound yields the corresponding sugar, tetramethyl galactose (Irvine and Cameron (26) ). This substance differs from tetramethyl glucose in being an uncrystallisable syrup, and experiments on the mutarotation of the compound are therefore restricted, but not so far as to conceal the analogy between the two alkylated sugars. Further alkylation of tetramethyl galactose results in the production of a mixture of the stereoisomeric tetramethyl a- and /3-methyl- galactosides in which the yS-isomeride is in large excess. The analogy is further exemplified by the fact that tetramethyl /3-methylgalactoside is, like the corresponding glucoside, a crystalline compound. Again, by processes similar to those already described, Irvine and Moodie (31) from a-methyl- mannoside, have prepared tetramethyl a- and /3-methyl- mannosides and tetramethyl mannose. In this case the methylated a-mannoside is a crystalline solid, the isomeric ^6-mannoside a liquid, and the mannose a colourless syrup. With respect to their optical activity and other attributes, the compounds fall into line with the corresponding glucose and galactose derivatives. The aldo-pentoses and methylpentoses behave like the aldo-hexoses mentioned, in that they yield alkylated deriva- tives when their methylglucosides are subjected to the alkylation process. Purdie and Rose (35), starting with a-methylarabinoside, obtained trimethyl a-methylarabinoside, a compound which forms extraordinarily large and beautiful PURDIE'S REACTION 199 crystals. Hydrolysis of the latter gave the syrupy sugar trimethyl Z-arabinose, which, when alkylated in its turn, yielded a mixture of trimethyl a- and /3-methylarabinosides containing a large excess of the latter isomeride. In the case of the methylpentose rhamnose, Purdie and the writer (34) obtained very similar results, with the exception that mixtures of the stereoisomeric a- and ^-rhamnosides were dealt with throughout. The sugar trimethylrhamnose was identified by the formation of a crystalline hydrazone. The only ketose to which the silver oxide reaction has so far been applied is fructose, and here much difficulty was encountered (Purdie and Paul (38) ; Irvine and Hynd (46) ). As was the case with rhamnose, the glucosidic derivative used (methyl fructoside) could not be obtained in the crystalline form. The material initially subjected to alkylation was a syrupy mixture of the isomeric a- and j8-methylfructosides possibly contaminated with other substances, and this, after treatment with silver oxide and methyl iodide and purification of the product by vacuum distillation and otherwise, yielded a liquid mixture of tetramethyl a- and ^8-methylfructosides. Considerable difficulty was experienced in isolating the pro- duets ; this may have been due to the susceptibility of the ketoses and their derivatives to oxidation. The sugar ob- tained by hydrolysis of the alkylated fructoside mixture was syrupy, but by realkylating this and again hydrolising the product, a small quantity of crystalline tetramethyl fructose was eventually isolated. It was found impossible to prepare either of the pure tetramethyl a- or /3-methylfructosides ; mixtures of these, however, were obtained, in one of which the a-isomeride, and in the other the /3-variety, predominated. Irvine and Hynd subsequently obtained a definite mono- methyl fructose which is described in another communication to this volume. 200 A GENERAL REVIEW OF THE ALKYLATION OF DISACCHARIDES The preparation of derivatives of disaccharides is a matter of difficulty on account of the readiness with which they undergo hydrolysis and of the insolubility of these substances in solvents other than water. The former difficulty does not apply to alkylation by means of the silver oxide reaction, and in the case of cane sugar, the solubility difficulty has been overcome by Purdie and Irvine (19, 22, and 30) by a method described in the Introduction. The materials were used in the following proportions : cane sugar (1 mol.), methyl iodide (20 mols. ), silver oxide ( 10 mols. ). Four alkylations with these proportions were necessary for the production, from ten grams of sucrose, of an equal weight of a neutral, syrupy liquid. This product received no purification beyond drying in a vacuum, but nevertheless gave analytical figures approximat- ing to those required for an octamethylated sucrose. Hydro- lysis of this substance by means of dilute, aqueous hydrochloric acid gave a syrup which reduced Fehling's solution and from which crystalline tetramethyl glucose has been obtained by simple nucleation. The uncrystallised portion of the last- mentioned syrup doubtless contained tetramethyl fructose, but the difficulties encountered in this work have so far pre- vented its isolation in a pure state. The results described are of considerable significance, since they afford direct experi- mental evidence of the correctness of Fischer's formula for cane sugar. According to this formula, cane sugar possesses a structure analogous to that of the alkylglucosides : — 0 CH2(OH) . C . CH(OH) . CH(OH) . CH . CH2OH A L. CH(OH) . CH(OH) . CH . CH(OH) . CH2OH 0 PURDIE'S REACTION 201 Now it has been shown that methylglucoside and sucrose yield methylated derivatives when alkylated, and these two methylated substances yield the same tetramethyl glucose upon hydrolysis. It follows, therefore, that the constitution and linkage of the glucose group in sucrose must be the same as in the simple glucoside. In view of the experimental evidence, obtained in this work, bearing on the constitution of methylglucoside, it will be seen that the above formula is verified so far as it represents the glucose half of the molecule. The only other disaccharide which has been alkylated by the silver oxide method is maltose. The sugar, in methyl alcoholic solution, was twice subjected to the action of silver oxide and methyl iodide, and the product being now soluble in methyl iodide, received two further treatments without the addition of alcohol. Methylation was here accompanied by oxidation of the free aldehydic group and subsequent esteri- fication of the resultant carboxyl group. The viscid product was hydrolysed by dilute hydrochloric acid, and eventually yielded a syrup from which, after distillation in a vacuum, crystalline tetramethyl glucose was isolated. Evidence as to the mode of linkage of the glucose residues in maltose has been lacking, but was furnished by these experiments. Fischer suggested that such linkage might be either of an acetal or of a glucosidic type. In the latter case the molecule of maltose may be represented thus : — CHO . (CHOH)4 . CH2 . O . CH . (CHOH)2 . CH . CHOH . CH2OH (or by the corresponding y-oxidic formula). Whether the link- age be of the acetal or of the glucosidic nature, the glucose residue containing the free aldehydic group would undergo oxidation during alkylation. The remaining half of the maltose molecule would yield a pentamethyl glucose on hydrolysis of the alkylation product if the linkage were of the acetal form. But such is not the case ; maltose cannot 2 G 202 A GENERAL REVIEW OF therefore possess an acetal structure, and inspection of the above formula shows that it explains the production of tetramethyl glucose, under the conditions mentioned. The glucosidic linkage in maltose is thus confirmed. The methylated sugars are more stable than the parent compounds, and so offer greater resistance to the destructive action of condensing agents than the latter. The alkylated sugars too are soluble in chemically indifferent solvents, and would thus appear to be available for the synthesis of methy- lated disaccharides. Tetramethyl glucose has, in fact, been used for such a purpose (Purdie and Irvine (30) ). The sugar was dissolved in benzene containing 0'33 per cent, of hydrogen chloride, and the solution heated in a sealed tube at 105°-115° C. for ten hours. The product was a syrup which was subjected to two further treatments similar to the above. The resultant viscid liquid was purified by distillation in a vacuum, and was found to have no action on Fehling's solution. Condensation had evidently occurred, and the product proved to be an octamethyl glucosidoglucoside of the structure : vxj-j-g L. CH2 . OCH3 CH2 . OCH3 .OCH, (CH . OCH3)2 CH The formula represents three stereoisomerides, the aa-, and aj8-varieties, and the substance obtained was probably a mixture of all three modifications. This was the first recorded instance of the synthesis of a derivative of a non- reducing disaccharide, and a similar method of self-condensa- tion has recently been adopted by Fischer for the preparation of disaccharides (Ber., 1909, 42, 2776). PURDIE'S REACTION 203 DETERMINATION OF THE CONSTITUTION OF SUGAR DERIVATIVES BY MEANS OF THE ALKYLATION PROCESS Reference has already been made to the evidence, obtained by means of the silver oxide reaction, bearing on the con- stitution of methylglucoside, sucrose, and maltose. These and other similar applications of the reaction have been collated by Irvine in a paper in the Biochemische Zeitschrift (50), which gives a useful summary and bibliography of such results of this nature as had been obtained prior to September 1909. Of the natural glucosides, only salicin and gynocardin have so far been investigated by means of the reaction. The results obtained by Irvine and Rose (32) in the former case are of great interest, and show that salicin is constituted similarly to the artificial glucosides. By alkylation of the glucoside in the usual manner, a crystalline pentamethyl salicin was obtained. The hydrolysis of this compound by dilute aqueous acid led to the production of resinous substances which could not be further examined ; it was therefore impos- sible to obtain evidence of constitution in this way. The following synthetic evidence, however, demonstrated the presence of the y-oxidic linkage in salicin. Saligenin and tetramethyl glucose were dissolved in benzene containing 0*25 per cent, of hydrogen chloride, and the solution was heated in sealed tubes at 120° C. A syrupy, glucosidic condensation product resulted, which in all probability was a mixture of the a- and /3-forms of tetramethyl salicin : — CH2(OH) . C6H4 . 0 . CH . (CH . OCH3)2 . CH . CH . (OCH3) . CH2OCH3. 0 ' The hydroxyl group remaining in this product was now alkylated, when a crystalline pentamethyl salicin resulted and 204 A GENERAL REVIEW OF was found to be identical with the pentamethyl salicin obtained by direct alkylation of the glucoside. Thus salicin, like methylglucoside, possesses the y-oxidic linkage as do also the related glucosides helicin and populin, which can be obtained from salicin by reactions which do not interfere with the gluco- sidic linkage. The alkylation process has also been applied by Moore and Tutin (55) to the natural glucoside gynocardin, or rather to the gynocardinic acid derived from it by the action of barium hydroxide and subsequent decomposition of the barium salt by sulphuric acid. The acid was methylated in the usual manner, first in methyl alcohol and afterwards in methyl iodide solution, and yielded methyl pentamethyl- gynocardinate ; the two remaining hydroxyl groups resisted the action of the alkylating agents, and are therefore probably phenolic. The substance, like pentamethyl salicin, gave resin- ous products on hydrolysis by dilute acids, and no attempt was made to overcome this difficulty or to isolate the methy- lated sugar. The hexoses readily form condensation compounds with acetone. The monoacetone derivatives are glucosidic, and, in their formation, one molecule of sugar unites with a molecule of acetone with the elimination of a molecule of water. Methylation of these compounds by means of the silver oxide reaction affords an insight into their structure. Acetone- rhamnoside treated in this way yields a dimethylated deriva- tive, and hence the formula suggested by Fischer for the parent compound, Me . CH(OH) . CH . (CHOH)2 . CH . 0 . CMe : CHe, I o_ is excluded, since it contains three secondary hydroxyl groups. Dimethyl acetonerhamnoside is readily hydrolysed, yielding PURDIE'S REACTION 205 the corresponding sugar dimethyl rhamnose, which forms a hydrazone but no osazone. Fischer's alternative formula for acetonerhamnoside -0- Me . CH(OH) . CH . CH(OH) . CH . CH i A V /\ Me Me is therefore also excluded, since, in methylating the sub- stance, of the two C-atoms attacked one is evidently in the a-position. It would appear that the second linkage of the acetone residue is attached to the ft- or 8-carbon atom (Purdie and Young (34) ). By methylating fructose diacetone by means of silver oxide and methyl iodide, Irvine and Hynd (46) have obtained clear evidence of the structure of that compound, which evidently contains one hydroxyl group, since it yields only a monomethylated derivative. The latter substance is easily hydrolysed by dilute hydrochloric acid, giving a crystalline monomethyl fructose possessing all the properties of a reducing sugar. The sugar, when heated with phenyl- hydrazine, gives monomethylglucosazone, and when oxidised with bromine water, it is converted into a dihydroxymethoxy- butyric acid which is incapable of forming a lactone. Con- sideration of these results will show that the formula for monomethyl fructose must be 0-- HO . CH2 . C(OH) . CH(OH) . CH(OH) . CH . CH2OCH3, 206 A GENERAL REVIEW OF and the following formula for fructose diacetone is thus probable : — --- 0 --- - CH2— C— CH— CH— OH . CH2OH O O 0 V V C C x\ A Me Me Me Me The various compounds obtained by condensation of fructose with acetone have been fully examined by Irvine and Garrett (54). In the course of this work a syrupy sub- stance was obtained which was considered to be a mixture of the a- and y8-f orms of 2 : 3-f ructosemonoacetone. The com- pound could not be sufficiently purified for analysis, but on methylating the substance by the same method as that employed for a-methylglucoside a liquid was obtained which could be purified by distillation in a vacuum. The product proved to be a trimethyl fructosemonacetone, and hence the substance from which it was prepared was shown to be a fructose monoacetone containing three hydroxyl groups. Certain condensation compounds of glucose, such as the anilide and oxime, have hitherto been regarded as being derived from the aldehydic form of the sugar. Recent work, however (Irvine and Moodie (41) ), in which the silver oxide reaction has been utilised, shows these compounds to possess the y-oxidic linkage in the sugar residue, and they must there- fore be considered to be derived from the a- and /8-forms of the sugar and not from the aldehydic isomeride. Tetra- methyl glucoseanilide is readily prepared by boiling an alcoholic solution of tetramethyl glucose and the base, and in this way a mutarotatory form is obtained. The compound PURDIE'S REACTION 207 remains unaltered after several treatments with silver oxide and methyl iodide, and therefore has the formula— CH30 . CH2 . CH(OCH3) . CH . CH(OCH3) . CH(OCH3) . CH . NHC6H5. 0 ' Glucoseanilide is prepared in the same way as its tetramethyl analogue, and in its preparation and mutarotation Irvine and Gilmour (42) have obtained evidence of the existence of two stereoisomeric forms of the substance. Methylation of the compound (a matter of great difficulty by reason of its susceptibility to oxidation) led to the formation of a crystal- line tetramethyl glucoseanilide, identical with that prepared as above. The constitution of glucoseanilide is thus estab- lished, and shown to be represented by the formula — (HO) . CH2 . CH(OH) . CH . CH(OH) . CH(OH) . CH . NH . C6H5. 0 (a- and /3-fonns) It may be mentioned that the optical values for the two anilides obey Hudson's Law. Irvine and McNicoll (56) have extended this work to the anilides of tetramethyl mannose, tetramethyl galactose, and trimethyl rhamnose, and further, to the anilides of the parent unmethylated sugars, mannose, galactose, and rhamnose. Their results are in every way comparable with those described above. Similar methods and reasoning have been adopted by Irvine and Moodie (41) and Irvine and Gilmour (42) in their work on glucoseoxime, which appears to be capable of reacting according to the structure — HO . CH2 . CH(OH) . CH . CH(OH) . CH(OH) . CH . NH . OH. 0 (a- and /9-forms) 208 A GENERAL REVIEW OF This compound is converted, by treatment with silver oxide and methyl iodide, into tetramethyl glucoseoxime methyl ether. The oximido-group is evidently methylated, along with the other hydroxyl groups, by the silver oxide reaction, and that this is generally the case is shown by the application of the reaction to other oximes. Thus cenanthaldoxime yields a monomethyl ether, while salicylaldoxime and benzoin oxime give dimethyl ethers under these conditions (Irvine and Moodie, loc. cit.). Attempts have been made, by means of the silver oxide reaction, to obtain evidence bearing on the constitution of tetramethyl glucosephenylhydrazone, glucose phenylhydra- zone, -p-toluidide, -p-phenetide, -/3-napthylamide, and -o-carboxyanilide (Irvine and pupils (41, 42, and 56) ). The compounds mentioned proved, however, to be so unstable that molecular rupture resulted, and the method had neces- sarily to be abandoned. The o-carboxyanilides of galactose, rhamnose, mannose, and maltose were found to be similarly unstable and to undergo decomposition when treated with silver oxide and methyl iodide (Irvine and Hynd (63) ). PREPARATION OF BENZOIN DERIVATIVES By interaction of benzoin, silver oxide, and ethyl iodide, Lander (4) obtained benzoin ethyl ether, C6H5.CH(OC2H6).CO.C6H5. The reaction was carried out at the boiling-point of the halide, and was seen to be accompanied by the formation of water and of very dark silver residues. This last fact was indicative of oxidation, and in addition to the benzoin ether, benzaldehyde and ethyl benzoate were formed during the reaction, probably as follows : — fC6H6.CH(OH).CO.C8HB+Ag20 = C6H5 . CHO+C6H8CO2H+2Ag. 2C6H6C02H+Ag20 = 2C6H5C02Ag+H20. I C6H6C02Ag+C2H5I = 2C6H5C02C2H5+AgI. PURDIE'S REACTION 209 By substituting isopropyl iodide for ethyl iodide in the above reaction, and using benzene as a solvent, a mixture of un- changed benzoin, benzaldehyde, isopropyl benzoate, and the isopropyl ether of benzoin resulted. Under improved conditions, however, it is possible by this process to convert benzoin practically quantitatively into an ether. Thus Irvine and Weir (40) effected the com- plete solution of the benzoin by adding acetone to the methyl iodide used, and by then proceeding with the alkylation in the usual manner obtained benzoin methyl ether in a purer state than the material prepared by Fischer's hydrochloric acid method. Similarly Irvine and McNicoll (43) have succeeded in eliminating those secondary reactions which Lander found to accompany the formation of benzoin ethyl ether. The materials here used were benzoin (1 mol.), silver oxide (3 mols.), and ethyl iodide (9 mols.), dry ether being added until the boil- ing-point of the solution was reduced to 50° C. During the heating at this temperature the benzoin slowly passed into solution, and, after further heating at a slightly higher temperature, was converted into the benzoin ether. The yield was somewhat greater than 70 per cent, of the benzoin used. The methylation of o-dimethoxybenzoin has been shown by Irvine (5 and 11) to proceed with perfect smoothness. The benzoin (1 mol.) is dissolved in excess of methyl iodide, and silver oxide (3 mols.) is gradually added. Gentle reaction takes place and is completed by 30 minutes' heating on the steam-bath. The product is extracted with ether and recrystallised from carbon bisulphide. In this way pure o-dimethoxybenzoin methyl ether is obtained in almost quantitative yield. The corresponding hydrobenzoin is in- soluble in methyl iodide, and, in the absence of a solvent, resists alkylation by the silver oxide method. If hydro- benzoin itself, however, be brought into solution by the addition of acetone to the methyl iodide, vigorous action occurs 2 D 210 A GENERAL REVIEW OF on the addition of silver oxide, and by repeating the alkylation, the crystalline hydrobenzoin dimethyl ether is obtained in good yield (Irvine and Weir (40) ). No alkylation occurs in the case of deoxybenzoin treated with silver oxide and ethyl iodide, since from such a reaction mixture Lander (10) was only able to separate unchanged deoxybenzoin and a little bidesyl. This result is confirmed by Irvine and Weir (40), who dissolved deoxybenzoin in methyl iodide and boiled the solution with silver oxide for twenty hours. The greater part of the substance was thereafter recovered unchanged ; the remainder had been converted into bidesyl. Both these attempts to obtain derivatives of deoxybenzoin of an enolic character were therefore unsuccessful. Irvine and Moodie (39) have made a detailed study of the reduction products of o- and p-dimethoxybenzoin. In the course of this work it was necessary to prepare anisoin methyl ether, and the preparation of this substance was found to proceed just as readily as that of the corresponding ortho- compound (vide supra). Prior to alkylation the anisoin was brought into solution by the addition of a small quantity of dry acetone to the methyl iodide required in the reaction. As already indicated, two methods are available for the preparation of methyl- and ethyl-derivatives of such sub- stances as benzoin, anisoin, o-dimethoxybenzoin, and furoin ; these are the silver oxide method and Fischer's process, which consists in passing dry hydrogen chloride into methyl- or ethyl-alcoholic solutions of the substances in question. A comparative study of the two reactions has been made by Irvine and McNicoll (43), who find that the former reaction gives practically quantitative yields of the methyl ethers of the substances mentioned above, and that the ethyl ethers are likewise obtained in good yield and in such a condition that they readily crystallise. The hydrochloric acid method does not give very uniform results. In methyl alcoholic solu- PURDIE'S REACTION 211 tion, furan derivatives are produced, and oxidation products further contaminate the alkyl ethers formed and render purification difficult. For further details illustrating the advantage of the employment of the silver oxide reaction in this direction, the original paper should be consulted. A modification of the usual procedure of alkylation was used by these authors in the methylation of furoin by the silver oxide method, and should be noted. The sparing solubility of this compound in methyl iodide and its ready oxidation to furil rendered the usual methods unavailable. Furoin methyl ether was, however, obtained by adding silver oxide to a solution of furoin in ethyl acetate to which had been added the requisite quantity of methyl iodide and also sufficient dry ether to reduce the boiling-point of the solution to 50° C. Optically active benzoin has now been obtained by McKenzie and Wren (Trans., 1908, 93, 310), who prepared Z-benzoin by the action of magnesium phenyl bromide on Z-mandelamide. Wren (48) has since prepared a number of derivatives of Z-benzoin, among them being the methyl ether. The method of alkylation by silver oxide and methyl iodide was utilised for this preparation in preference to the other method as being less likely to yield a racemised product ; further, Fischer's method has been shown to yield a quantity of by-products (vide supra). Z-Benzoin methyl ether was readily prepared by the method indicated, and is remarkable for the extraordinary influence of solvents on the rotation of the substance ; the specific rotation varies from -88*2° (chloroform) to 147'8° (heptane). In attempting to prepare Z-benzoin ethyl ether by Fischer's method, Wren (49) found that complete racemisation occurred. Z-Benzoin and alcoholic hydrogen chloride therefore interact to form r-benzoin ethyl ether. On ethylating Z-benzoin by the silver oxide process, a partially racemised product was obtained, and it is very probable that the racemisation did not occur during the alkylation process but during the vacuum 212 A GENERAL REVIEW OF distillation at a high temperature which was necessary to purify the substance. In the course of an investigation of the racemisation phenomena observed in Z-benzoin and its derivatives, McKenzie and Wren had occasion to prepare (53) the monomethyl ethers of both i- and Z-triphenylethylene glycols. The alkylation of each of the parent glycols proceeded slowly, three treat- ments being necessary for completion. In each case alkyla- tion of the secondary hydroxyl group alone occurred ; the proof that this group was alkylated, and not the tertiary hydroxyl group, has already been referred to. PREPARATION OF IMINO-ETHERS By the action of silver oxide and alkyl iodides on amides and substituted amides it is possible to prepare imino-ethers : a number of such preparations have been carried out by Lander, the sole worker in this field. Preliminary experi- ments (4) showed that benzamide was converted, by treatment with excess of silver oxide and ethyl iodide, into benzimino- ethyl ether, which was identified by conversion into the crystalline hydrochloride C6H5 . C(OC2H5) : NH . HC1. Under similar conditions, acetanilide yielded N-phenylacetiminoethyl ether C6H5 . N : C(OC2H5) . CH3 or ethyl isoacetanilide, no trace of the isomeric N-ethyl ether being detected in the reaction product. When, however, methyl iodide was substituted for ethyl iodide in the reaction, the isomeric N-phenylacetimino- methyl ether C6H5. N: C(OCH3) . CH8 and N-methylacetanilide C6H5. N(CH3) . COCH3 were obtained in almost equal quantities (6). Analogous results were obtained by alkylation of aceto- o-toluidide, which is converted by silver oxide and ethyl iodide into N-o-tolylacetiminoethyl ether CH3. C6H4. N: C (OC2H5) . CH3 exclusively, while with silver oxide and methyl iodide it yields a mixture of the corresponding iminomethyl ether and N-methylaceto-o-toluidide CH3 . C6H4 . N(CH3) . CO . CH8. PURDIE'S REACTION 213 Aceto-p-toluidide differs from its o-isomer in being con- verted by silver oxide and methyl iodide into N-methyl- aceto-p-toluidide only. By similar methods, N-a- and N-/?-napthylacetiminoethyl ethers were prepared from aceto-a- and aceto-/8-napthalides, and N-phenylbenzimino- ethyl ether C6H5 . N : C(OC2H6) . C6H5 from benzanilide. The latter substance gives rise to the corresponding methyl ether mixed with a little benzoylmethylaniline when subjected to the action of silver oxide and methyl iodide (13) and the benz-o- and benz-p-toluidides behave similarly. Under parallel conditions derivatives of oxalic acid yielded results in agreement with those already mentioned. Thus oxanilide was converted into di-N-phenylimino-oxalic diethyl ether(C6H5. N: COEt)2and ethyl oxanilate into semi-N-phenyl- imino-oxalic diethyl ether. On the other hand, methyl oxanilate in benzene solution, when alkylated by means of silver oxide and methyl iodide, gave semi-N-phenylimino- oxalic dimethyl ether C02Me . C : (NPh) . OMe along with some isomeric methyl phenylmethyloxamate COgMe . CO . NMePh (24). The effect of ethylating methyl oxanilate in the usual manner was somewhat curious. A product intermediate in com- position betweenC02Me.C:(NPh)OEt and C02Et.C:(NPh)OEt resulted, and interchange of alkyls in the carboxylic ester group must therefore have occurred during the ethylation of the CO . NH residue. A preliminary attempt to prepare a toluimino-ether from o-toluamide resulted only in the conversion of the latter into o-toluonitrile. This attempted alkylation was carried out in boiling alcoholic solution, and it was afterwards found that different amides (21), under the same conditions, give rise to nitriles together with some imino-ether. Thus whilst o-tolu- amide gives a 13'6 per cent, yield of the imino-ether, p-tolu- amide gives a 70 per cent, yield. This result is explained by supposing that the o-compound loses alcohol more readily than its p-isomeride : C7H7 . C : NH . OEt — > C7H7 . CN+EtOH. 214 A GENERAL REVIEW OF The silver oxide reaction has also proved to be of use in the preparation of certain of the aliphatic imino-ethers, which, as a class, are difficult to isolate and identify, by reason of their fugitive nature. If ethyl oxamate is treated with ethyl iodide and silver oxide a vigorous reaction ensues, and Nef's semi- imino-oxalic diethyl ether C : NH(OEt) . C02Et is produced (7). Acetamide undergoes decomposition under the same condi- tions. Alkylation of urethane takes place at the ordinary temperature if the substance is dissolved in a mixture of ethyl iodide and dry ether and the solution allowed to stand in contact with silver oxide for ten days. The product, imino- carbonic diethyl ether (C2H50)2 : C : NH, is isolated in the form of the broniimino-ether. It is seen that ethylation of substituted amides by silver oxide and ethyl iodide under the usual conditions gives imino- ethers . C(OEt) : N almost exclusively, while on substituting methyl for ethyl iodide imino-ethers and isomeric substituted amides are produced at the same time. It is however possible, as Lander (17) has shown, by conducting the alkylation at 100° C., to obtain both ethyl homologues PhN : C(OEt) . Me and PhNEt . CO . Me simultaneously, by the action of silver oxide and ethyl iodide on acetanilide. The results thus resemble those obtained by the methylation of the substance in an open vessel at 40°-50° C. For comparative purposes, Lander investigated the action of silver oxide and methyl iodide on formanilide, and obtained N-phenylformiminomethyl ether H . C(OMe) : N . C6 H5 mixed with a small quantity of the isomeric amide H . CO . NMe . C6H5 and some diphenylformamidine. When silver formanilide was boiled with ether and methyl iodide, a small quantity of imino-ether was formed, more of the amide and considerably more of the amidine than in the former experiment. These results are of considerable theoretical significance. PURDIE'S REACTION 215 ALKYLATION OF OTHER TAUTOMERIC COMPOUNDS In his general investigation of the applicability of the silver oxide reaction (4) Lander tested the action of silver oxide and ethyl iodide on ethyl acetoacetate for the purpose of preparing, if possible, the isomeric ether in place of the alkyl compound formed by the action of sodium ethoxide and ethyl iodide. That is to say, it was hoped that ethyl aceto- acetate would react in the enolic form as ethyl /3-hydroxy- crotonate and so be converted into ethyl ^-ethoxycrotonate CH3 . C(OC2H5) : CH . COOC2H5. The product of the reaction was fractionally distilled, and thus separated into the two substances ethyl ethylacetoacetate and ethyl /8-ethoxy- crotonate. As might perhaps have been expected, the yield of the latter was very small (5 per cent.). The simultaneous pro- duction of . OC2H5 and • C . C2H5 derivatives of ethyl aceto- acetate may be regarded as evidence of the existence of both ketonic and enolic forms in the original substance. The fact that the product is largely composed of the ketonic derivative agrees with the physical evidence that ethyl acetoacetate exists, under ordinary conditions, mainly in the ketonic form. When benzoylacetic ester is treated with the same reagents, the course of the reaction is entirely similar to that shown by acetoacetic ester, the alkylated product consisting very largely of the C-ethyl homologue C6H6.CO.CHEt.COOEt mixed with very small quantities of the isomeric /3-ethoxycinnamic ester, C6H5 . C(OEt) : CH . COOEt (Lander (10) ). It would appear from these results that the silver oxide reaction, unlike the usual method of alkylating tautomeric substances of this type by which C-ethers only are obtained, does not interfere with the equilibrium between the dynamic isomerides of which the parent substance is composed. Alkylation of ethyl ethylacetoacetate only occurred to a very slight extent (4) in an experiment carried out under 2i6 A GENERAL REVIEW OF similar conditions to the above, and apparently the homologue of ethyl $-ethoxycrotonate was not one of the products. A modification of the usual alkylation process was tried by adding silver oxide to a mixture of ethyl acetoacetate and ethyliodoacetate. It was found necessary to cool the flask containing the reaction mixture and latterly to add benzene for the purpose of lowering the temperature. The product was purified by distillation in a vacuum and ethyl aceto- succinate was thus obtained. Silver oxide can therefore be substituted for sodium ethoxide in the synthesis of acyl- substituted succinic esters. In contact with silver oxide and ethyl iodide, ethyl malon- ate does not react in the hydroxy-form to give /S-diethoxy- acrylate, but yields ethyl ethylmalonate as the sole product (Zoc. cit.). Ethyl oxaloacetate and its silver derivative show similar be- haviour in being completely converted by silver oxide and ethyl iodide into ethyl ethoxyfumarate EtOOC. C(OEt) : CH. COOEt, which is also exclusively formed by interaction of the silver derivative and ethyl iodide. No C-ester is formed in either of the above reactions (Lander (17) ). The unsuccessful attempts to obtain alkylation derivatives of deoxybenzoin have already been referred to in the section dealing with benzoin derivatives. THE MECHANISM OF THE REACTION Prior to the discovery of the silver oxide reaction, dry silver oxide had been used as a synthetical reagent by Wurtz in the formation of ethyl ether from ethyl iodide (Ann. Chem. PTiys., 1856, iii. 46, 222), by Erlenmeyer for a similar purpose (Annalen, 1863, 126, 306), and by Linneman (Annalen, 1872, 161, 37). In preparing isopropyl ether, Erlenmeyer used moist silver oxide. It is stated by Wurtz that dry silver PURDIE'S REACTION 217 oxide and methyl or ethyl iodides react energetically immedi- ately on coming into contact with one another, with evolution of much heat. Such, however, is not the case ; Lander (4) found that with either iodide the reaction is slow, and he considers that interaction with the formation of ethers depends upon the presence, or initial formation, of small quantities of alcohol or water : — Ag20+2C2H5OH = 2C2H5OAg+H20 C2H5OAg+C2H5I = (C2H5)20+AgI. Purdie and Bridgett (20) record similar observations. Dry silver oxide shows no apparent change when heated with excess of methyl iodide, but when dry methyl alcohol is present the reaction starts immediately on warming and then proceeds spontaneously. In the course of two hours' heating on the water-bath, 35'5 per cent, of the silver oxide was converted into silver iodide in the former case, while in the latter 88 per cent, underwent the same change. The authors are of the opinion that in the absence of every trace of moisture and alcoholic substance, silver oxide and methyl iodide would not interact, and they share Lander's view that the Wurtz syn- thesis of ethers can only occur in the presence of traces of moisture which act catalytically in producing alcohols, these being afterwards alkylated. The statements which have been given are necessary in order to explain the apparent anomaly that silver oxide and an alkyl iodide should etherify a hydroxyl group in a compound rather than interact with one another with the formation of simple ethers. The anomaly is non- existent, since the statement of Wurtz is inaccurate. It must be remembered, however, that water is formed during alkyla- tion by means of silver oxide and alkyl iodides, and that loss of the alkylating materials will therefore occur owing to simple ether formation in the manner mentioned above. If an alcoholic solvent is employed, the loss of the alkylating reagents is considerable. The necessity for the employment of a large 2E 2i8 A GENERAL REVIEW OF excess of the alkylating mixture in every case is therefore evident. Certain of the higher alkyl iodides react more readily with silver oxide to form alkyl ethers than do methyl and ethyl iodides. It is possibly for this reason that the latter give better results in the alkylation process than, for instance, isopropyl iodide (Lander (4) ). There is little direct experimental evidence bearing on the mechanism of the silver oxide reaction, so that any con- ception of the course of the reaction must, for the present, be largely speculative. It is highly improbable that, in the alkylation of hydroxy-compounds, the silver oxide acts simply by removing hydrogen iodide, since no reaction occurs if litharge, zinc oxide, cupric oxide, or magnesium oxide are sub- stituted for silver oxide. McKenzie (3) suggests that the most plausible hypothesis is that, by replacement of alcoholic H by Ag, an unstable silver derivative is formed and subse- quently undergoes double decomposition with the alkyl halide. The suggestion is endorsed by Lander (4 and 17) and by Purdie and Irvine (8). In accordance with this view, cuprous oxide might be expected to behave similarly to silver oxide. Alkylation of methyl tartrate by means of cuprous oxide and methyl or ethyl iodides has not been effected, but in the case of isopropyl iodide there is reason to believe that alkylation does take place, but imperfectly. (Private communication from Professors Purdie and Irvine.) The remarkable series of colour changes which are sometimes noticed during alkyla- tion by this method might possibly be advanced as a further argument in support of the silver derivative hypothesis. It must, however, be admitted that, as yet, there is no positive evidence of the formation of a definite derivative of this kind during alkylation. Apparently the only attempt that has been made to isolate such an intermediate compound is that of Irvine and Moodie (36), already mentioned in connection with tetramethyl glucose. It is unlikely that alkylation of PURDIE'S REACTION 219 hydroxy-compounds of the ordinary type proceeds by the formation of an oxonium additive compound with methyl iodide, as methyl tartrate, a substance which is most readily alkylated, shows no tendency to form such derivatives (Irvine and Moodie, loc. cit.). The methylation of tetramethyl glucose is undoubtedly brought about in this way by pre- liminary addition of methyl iodide and subsequent removal of the elements of hydrogen iodide from the oxonium com- pound by silver oxide, but this is to be regarded as an altogether exceptional instance. Lander (4 and 17) takes the view that the first step in the alkylation of tautomeric compounds of the keto-enol character is the formation of silver derivatives of both isomerides, that is to say, OAg and CAg compounds ; these subsequently undergo double decomposition with alkyl iodides, yielding the corresponding 0- and C-alkyl derivatives. The alkylation of amides and substituted amides has been supposed (4 and 17) to take place in a similar manner, by the intermediate formation of silver derivatives, but the recently published work of Matsui (64) appears to negative this view. It is shown that in the alkylation of amides such as acetamide and benzamide, silver oxide can be replaced by cuprous oxide, lead oxide, or even anhydrous potassium carbonate. These substances, and silver oxide also would therefore seem to act simply by removing hydrogen iodide. It will thus be seen that the discovery of the silver oxide reaction has opened many lines of research, and it has proved to be of more immediate profit to pursue these lines of work rather than to closely scrutinise the reaction itself. Further discussion of the mechanism of the reaction must therefore be postponed until the present evidence has been considerably amplified. CHARLES ROBERT YOUNG 220 A GENERAL REVIEW OF BIBLIOGRAPHY Arranged in Chronological Order 1. PufcDlE and PITKEATHLY. Trans., 1899, 75, 153. Production of Mono- and Di-Alkyloxysuccinic Acid, etc. 2. PURDIE and IRVINK. Trans., 1899, 75, 483. The Rotatory Powers of Optically Active Methoxy- and Ethoxy-Propionic Acids. 3. McKENZiE. Trans., 1899, 75, 753. Active and Inactive Phenylalkyloxy- acetic Acids. 4. LANDER. Trans., 1900, 77, 729. Alkylation by Means of Dry Silver Oxide and Alkyl Halides. 5. IRVINE. Trans., 1901, 79, 668. Preparation of o-Dimethoxybenzoin and a New Method of Preparing Salicylaldehyde Methyl Ether. 6. LANDER. Trans., 1901, 79, 690. Alkylation of Acylarylamines. 7. LANDER. Trans., 1901, 79, 701. Preparation of Aliphatic Imino-ethers from Amides. 8. PURDIE and IRVINE. Trans., 1901, 79, 957. Optically Active Dimethoxy- succinic Acid and its Derivatives. 9. PURDIE and BARBOUR. Trans., 1901, 79, 971. The Influence of Solvents on the Rotatory Powers of Ethereal Dimethoxysuccinates and Tartrates. 10. LANDER. Proc., 1901, 17, 59. Action of Dry Silver Oxide and Ethyl Iodide on Benzoylacetic Ester, Deoxybenzoin, and Benzyl Cyanide. 11. IRVINE. Inaug. Diss., Leipzig, 1901. Ueber einige Derivate des Ortho- methoxybenzaldehydes. 12. FORSTER. Trans., 1902, 81, 264. Studies in the Camphane Series. Part Tii. 13. LANDER. Trans., 1902, 81, 591. Synthesis of Imino-ethers. N-Arylben- ziminoethers. 14. PURDIE and IRVINE. Brit. Assoc. Reports, Belfast, 1902. The Alkylation of Sugars. 15. FORSTER. Trans., 1903, 83, 98. Studies in the Camphane Series. Part x. 16. LANDER. Trans., 1903, 83, 320. Synthesis of Imino-ethers. N-Ethyl-, N-Methyl-, and N-Benzyl-Benziminoethers. PURDIE'S REACTION 221 17. LANDER. Trans., 1903, 83, 414. The Nature and Probable Mechanism of the Replacement of Metallic by Organic Radicles in Tautomeric Compounds. 18. McKENZiK and HARDEN. Trans., 1903, 83, 424. The Biological Method for Resolving Inactive Acids into their Optically Active Components. 19. PURDIE and IRVINE. Trans., 1903, 83, 1021. The Alkylation of Sugars. 20. PURDIE and BRIDGETT. Trans., 1903, 83, 1037. Trimethyl o-Methyl- glucoside and Trimethyl Glucose. 21. LANDER and JKWSON. Proc., 1903, 19, 160. Imino-ethers corresponding with Ortho-substituted Benzenoid Amides. 22. PURDIE and IRVINE. Brit. Assoc. Reports, Southport, 1903. A Contri- bution to the Constitution of Disaccharides. Methylation of Cane Sugar and Maltose. 23. FORSTER. Trans., 1904, 85, 892. Studies in the Camphane Series. Part. xiv. 24. LANDER. Trans., 1904, 85, 984. Imino-ethers and allied Compounds corresponding with the Substituted Oxamic Esters. 25. PURDIE and IRVINE. Trans., 1904, 85, 1049. The Stereoisomeric Tetra- methyl Methylglucosides and Tetramethyl Glucose. 26. IRVINE and CAMERON. Trans., 1904, 85, 1071. The Alkylation of Galactose. 27. FRANKLAND and GKBHARD. Trans., 1905, 87, 864. The Ethereal Salt* and Amide of Dimethoxypropionic Acid derived from rf-Glyceric Acid. 28. IRVINE and CAMERON. Trans., 1905, 87, 900. A Contribution to the Study of Alkylated Glucosides. 29. McKENZiE and THOMPSON. Trans., 1905, 87, 1004. Racemisation Pheno- mena during the Hydrolysis of Optically Active Menthyl- and Bornyl- Esters by Alkali. 30. PURDIK and IRVINE. Trans., 1905, 87, 1022. Synthesis from Glucose of an Octamethylated Disaccharide. Methylation of Sucrose and Maltose. 31. IRVINE and MOODIE. Trans., 1905, 87, 1462. The Alkylation of Man- nose. 32. IRVINE and ROSE. Trans., 1906, 89, 814. The Constitution of Salicin. Synthesis of Pentamethyl Salicin. 33. IRVINE. Trans., 1906, 89, 935. The Resolution of Lactic Acid by Morphine. 34. PURDIE and YOUNG. Trans., 1906, 89, 1194. The Alkylation of Rham- nose. 222 A GENERAL REVIEW OF 35. PURDIE and ROSE. Trans., 1906, 89, 1204. The Alkylation of Arabinose. 36. IRVINE and MOODIE. Trans., 1906, 89, 1578. The Addition of Alkyl Halides to Alkylated Sugars and Glucosides. 37. MARCKWALD and PAUL. Ber., 1906, 39, 3654. Ueber die Umwandlung von Racemkorpern in die optisch-activen Verbindungen. 38. PURDIE and PAUL. Trans., 1907, 91, 289. The Alkylation of d-Fructose. 39. IRVINE and MOODIE. Trans., 1907, 91, 536. The Reduction Products of -o and ^-Dimethorybenzoin. 40. IRVINE and WEIR. Trans., 1907, 91, 1384. The Application of Baeyer's Reduction to Benzoin and its Derivatives. 41. IRVINE and MOODIE. Trans., 1908, 93, 95. Derivatives of Tetramethyl Glucose. 42. IRVINE and GILMOUR. Trans., 1908, 93, 1429. The Constitution of Glucose Derivatives. Part i. Glucose-Anilide, -Oxime, and -Hydrazone. 43. IRVINE and McNicOLL. Trans., 1908, 93, 1601 The Formation of Ethers from Compounds of the Benzoin Type. 44. FORSTER and FIERZ. Trans., 1908, 93, 1859. The Triazo-group. Part v. Resolution of a-Triazopropionic Acid. 45. FORSTER and MULLER. Trans., 1909, 95, 191. The Triazo-group. Part viii. 46. IRVINE and HYND. Trans., 1909, 95, 1220. Monomethyl Laevulose and its Derivatives. 47. IRVINE and GILMOUR. Trans., 1909, 95, 1545. The Constitution of Glucose Derivatives. Part ii. Condensation Derivatives of Glucose with Aromatic Ammo-Compounds. 48. WREN. Trans., 1909, 95, 1583. Some Derivatives of /-Benzoin. 49. WREN. Trans., 1909, 95, 1593. Racemisation Phenomena observed in the Study of /-Benzoin and its Derivatives. 50. IRVINE. Siochem. Zeit., 1909, 22, 357. Ueber die Verwendung alkylierter Zucker zur Bestimmung der Konstitution von Disacchariden und Glucosiden. 51. FORSTER and JUDD. Trans., 1910, 97, 254. The Triazo-group. Part xii. 52. MELDOLA and KUNTZER. Trans., 1910, 97, 455. Salts and Ethers of 2:3:5 Trinitro, 4-Acetylamino Phenol. 53. McKENZiE and WREN. Trans., 1910, 97, 473. Optically Active Glycols derived from /-Benzoin and from Methyl 2-Mandelate. 54. IRVINE and GARRETT. Trans., 1910, 97, 1277. Acetone Derivatives of d-Fructose. PURDIE'S REACTION 223 55. MOORE and TUTIN. Trans., 1910, 97, 1285. Note on Gynocardin and Gynocardase. 56. IRVINE and McNicoLL. Trans., 1910, 97, 1449. The Constitution and Mutarotation of Sugar Anilides. 57. PURDIE and NEAVE. Trans., 1910, 97, 1517. Optically Active Methoxy- succinic Acid from Malic Acid. 58. PURDIE and YOUNG. Trans., 1910, 97, 1524. Optically Active Deriva- tives of Z-Methoxy- and d-Dimethoxy-Succinic Acids. 59. PURDIE and ARUP. Trans., 1910, 97, 1537. Action of Grignard Re- agents on Methyl Z-Methoxysuccinate, Methyl Maleate and Maleic Anhydride. 60. FORSTER and NEWMAN. Trans., 1910, 97, 2570. The Triazo-group. Part xv. 61. H. BILTZ Ber., 1910, 43, 1600. Ueber den Abbau der Tetramethyl- harnsaure und ueber das Allokaffein. 62. H. BILTZ. Ber., 1910, 43, 1999. Methylierung und Konstitution von Allantoin. 63. IRVINE and HYND. Trans., 1911, 99, 161. o-Carboxyanilides of the Sugars. 64. MOTOOKI MATSUI. Abstr., 1911, (i) 185. (Mem. Coll. Sci. Eng. Kyoto, 1910, 2, 397.) C. R. Y. THE PREPARATION OF ANHYDRIDES OF ORGANIC ACIDS THE reagents most frequently used for the preparation of anhydrides and chlorides of organic acids are the chlorine derivatives of phosphorous and phosphoric acids, and this notwithstanding the fact that these compounds are difficult to handle, and give rise in some cases to products contaminated with phosphorus compounds which are not easily removed. It thus appears remarkable that chlorides of other inorganic acids have not come into general use for similar purposes. The inorganic chlorides generally, however, are no less troublesome to work with than those of phosphorus, but, apart from this, the number of such compounds available for practical purposes is limited. In the third group of the periodic system we have boron trichloride, but the cost of this compound at once puts it out of court as a reagent. In the fourth group occurs carbonyl chloride, the use of which as a means of preparing organic anhydrides has been suggested by Hentschel (Ber., 1884, 17, 1285). Carbon tetrachloride does not appear to have been used, but the application of silicon tetrachloride to this purpose has been patented (U.S. Pat. 944372), as has that of silicon tetrafluoride (D. R. P. 171146). In the fifth group none of the chlorides except those of phos- phorus have been employed : the action of nitrosyl chloride on silver salts is referred to below. In the sixth group we have sulphur with the numerous acids derived from its oxides. The most commonly occurring of the chloroanhydrides of these acids is sulphuryl chloride, and this compound has been used for anhydride formation, as has a mixture of sulphur dioxide and chlorine (Abstr., 1906, i. 3, 621 ; D. R. P. 167304). 2F 226 THE PREPARATION OF Chlorosulphonic acid and chlorosulphonates have also been employed (D. E. P. 146690; Abstr., 1904, i. 282). Sulphur tetrachloride, as is well known, does not exist under ordinary conditions, but thionyl chloride, which may be regarded as being related to sulphur tetrachloride in the same manner as phosphorus oxychloride is to phosphorus pentachloride, is a familiar reagent. This compound has been applied to the preparation of acid chlorides by H. Meyer, who prepared a number of acid chlorides by heating the free acid with five or six times its weight of thionyl chloride. He noticed that the anhydrides were formed in some cases (Monatsheft., 1901, 22, 777), and later (Chem. Zeit., 1909, 1036) the same author outlines a method for the preparation of the anhydride of the sulphonic acids from thionyl chloride and the free acids of their potassium salts, apparently under similar conditions. In the methods considered above, the formation of an anhydride is usually regarded as being due to the primary formation of an acid chloride, which then, on reaction with more anhydrous salt, gives rise to the anhydride and metallic salt. Thus, in the familiar example of phosphorus oxy- chloride, we have the following equations to represent the two stages in the reaction : — 2CH3 . COONa+POCl3 = 2CH3 . CO . Cl+NaP03+NaCl CH3 . COONa+CH3 . CO . 01 = (CH3 . CO)20+NaCl. Similar reactions occur when other non-metallic chlorides, such as the chlorides of sulphur, are used. The reaction which takes place between excess of the so-called sulphur dichloride and a dry salt of an organic acid was investigated by Heintz, who found that the chloro- anhydride is formed among other products (Jahresb. Chem., 1856, 569). Thus, using sodium benzoate, the reaction is represented by the equation : — 4C6H5 . COONa+3SCl2 = 2S+Na2S04+2NaCl+4C6H5CO.Cl. ANHYDRIDES OF ORGANIC ACIDS 227 With excess of sodium benzoate the benzoyl chloride first formed yields benzoic anhydride. Technical processes based on the above reactions have been patented by H. Kessler (D. R. P. 132605 ; Abstr., 1903, i. 309) and by T. Goldschmidt (Eng. Pat. 25433; J. Soc. Chem. Ind., 1910, 112, 592). Kessler specifies the use of sul- phur dichloride SC12> while Goldschmidt causes excess of chlorine to react with the dry sodium salt in presence of sulphur. The acid chloride is thus formed, and the mixture on being heated gives the anhydride. Carius (Annalen, 1858, 106, 291) investigated the reaction between equimolecular proportions of sodium chloride and sulphur monochloride S2C12, both in the presence and in the absence of solvent, and found that although the reaction is not altogether smooth, the initial and final stages are repre- sented essentially by the equation : — 2C6H6 . COONa+2S2Cl2 = 2C6H5 . CO . Cl-f 2NaCl+S02+3S Gerhardt proposed another method for the preparation of organic anhydrides, in which the mechanism of the reaction is somewhat different from that described above for cases in which inorganic chlorides are used. This method is based on the observation that an organic acid chloride, such as benzoyl chloride, gives a mixed anhydride on reaction with a metallic salt of a different organic acid, according to the equation : — CH3 . COONa+C6H5 . CO . Cl = CH3 . CO . 0 . CO . C6H6+NaCl. Such mixed anhydrides are unstable, so that on being heated they decompose with formation of the simple anhydrides : — 2CH3 . CO . 0 . CO . C6H5 = (CH3 . CO)2O+(C6H6 . C0)20. Mixed anhydrides can also be prepared by heating excess of an anhydride such as acetic anhydride with another acid (Autehrieth, Ber., 1887, 20, 3188). 228 THE PREPARATION OF Consideration of the above reactions suggests at once the possibility of preparing organic anhydrides by formation and subsequent decomposition of mixed anhydrides of organic and inorganic acids ; such mixed anhydrides are known, and, like those which contain organic radicals only, they are unstable and readily undergo a similar decomposition. Thus Pictet and his colleagues (Abstr., 1903, i. 309, 456, 675 ; 1906, i. 3) have prepared mixed anhydrides of acetic acid with boric, arsenious, phosphoric, sulphuric, and nitric acids respectively by the interaction of glacial acetic acid or of acetic anhydride with the inorganic anhydride, while Francis (Ber., 1906, 39, 3798) has prepared benzoyl nitrate from benzoyl chloride and silver nitrate, and Francesconi and Cialdea (Abstr., 1903, i. 788 ; 1904, i. 707) have made mixed anhydrides of nitrous acid and organic acids by the interaction of nitrosyl chloride and silver salts of organic acids. In the methods already described, in which sulphur dichloride is used, the reactions are carried out in absence of solvent, and the primary product is the acid chloride; while Carius, using equimolecular proportions of sulphur monochloride and sodium benzoate in equimolecular pro- portions, obtained benzoyl chloride even when he used carbon disulphide as solvent. By a modification of the conditions under which Carius worked, the writer has succeeded in pre- paring a series of acyl derivatives of the unknown acid, thio- sulphurous S2(OH)2, of which sulphur monochloride may be regarded as the chloride. Thus, if dry sodium benzoate (2 mols.) and sulphur monochloride (1 mol.) are boiled together in presence of carefully dried ether or other indifferent solvent, the yellow colour of the sulphur chloride quickly disappears, and, after filtration from the sodium chloride formed in the reaction and concentration, the solution deposits crystals of benzoyl thiosulphite. The reaction is represented by the equation : — 2C6H6 . COONa+S2Cl2 = (C6H5.COO)2S2+2NaCl. ANHYDRIDES OF ORGANIC ACIDS 229 Benzoyl thiosulphite crystallises . in well-defined colourless crystals. Sulphur. Molecular Weight (in Benzene F.P. Method). Found 21-32 per cent. 281 (C6HSCOO)2S2 requires . 20'92 per cent. 306 The compound is extremely unstable and soon turns yellow owing to spontaneous decomposition, the products of which are benzoic anhydride, sulphur dioxide and free sulphur, as shown by the equation :— 2(C6H5COO)2S2 = 2(C6H5CO)20+S02+3S. That the formation and decomposition of this substance take place quantitatively is shown by the following Table, where the weights are given of the crude products from 3*4 grams of sulphur chloride and 12'5 grams (instead of 11 '5 grams) of silver benzoate. The theoretical quantities are calculated by means of the above equations from the weight of sulphur chloride used : — Calculated. Found. Weight of benzoyl thiosulphite (after filtration from silver chloride and dis- tillation of ether), . . . . 7'6 g. 7'6 g. Weight of sulphur (residue after extraction of the benzoic anhydride with ether), 0-95 g. 0'90 g. Weight of anhydride (after distillation of ether from the ethereal extract), . 5' 65 g. 5-60 g. The method is widely general, and has been applied to a large number of unsubstituted fatty and aromatic acids and to acids containing halogen and nitro groups, the reactions in all cases being exactly similar to those just described : with acids of high molecular weight such as stearic and naphthoic acids the last traces of sulphur cannot be completely removed. In general, after the decomposition of the intermediate compound is complete, a change which is accelerated, as would be expected, by the application of heat, the anhydride can be 230 THE PREPARATION OF extracted by means of ether and can be freed from traces of sulphur by recrystallisation. An even simpler way to prepare acid anhydrides is afforded by the employment of thionyl chloride in the same manner. If a solution of this reagent in ether is shaken with an organic silver salt, immediate reaction occurs, but usually no inter- mediate compound can be isolated, so that after filtration from the silver chloride the anhydride can be crystallised in a pure condition from the filtrate :— 2C6H5 . COOAg+SOCl2 = (C6H5 . CO)20+S02+2AgCl. The yield of anhydride by this method is nearly quanti- tative, even although only small quantities are prepared. Thus, in one experiment, 2'1 grams of crude but nearly pure benzoic anhydride were obtained instead of the calculated quantity, 2'3 grams. The method is applicable to the prepara- tion of anhydrides of fatty and aromatic acids, both of small and large molecular weights, to the preparation of anhydrides of halogen and nitro acids and of some dibasic acids. The use of a solvent is an essential feature of the new methods, and it is important that the solvent be carefully dried if successful results are to be obtained. The low temperature at which these reactions occur, and the smooth manner in which they proceed, suggested their applica- tion to the preparation of normal anhydrides of hydroxy- acids, a type of compound which cannot be obtained by the methods hitherto in use, since by the employment of chlorides of phosphorus chlorination occurs, while, by the influence of heat, anhydro-acids and similar compounds are produced. It was found, however, that with sulphur chloride and salts of glycollic, mandelic and malic acids the reaction is abnormally slow, while with salts of the hydroxybenzoic acids it is irregular, so that it is improbable that the normal cnhy- drides of the hydroxy-acids can be prepared by the use of this reagent. Experiments with thionyl chloride gave more promising ANHYDRIDES OF ORGANIC ACIDS 231 results. Reaction takes place immediately, and proceeds apparently as smoothly as with acids of other types. Silver glycollate gave an intermediate compound which can be isolated in the pure state, while similar derivatives of mandelic and malic acids were obtained, but not pure. If to silver glycollate suspended in ether, an ethereal solution of thionyl chloride be added, immediate reaction occurs, and, after filtration, there is obtained, on addition of light petroleum, a white crystalline substance, the composition of which is in agreement with the formula [CH2(OH)COO]2SO. It loses sulphur dioxide when allowed to stand, but only slowly. When nearly all the sulphur dioxide has been evolved there is left, after washing with ether, a white powder the composition of which agrees with that required by the formula [CHa(OH)CO]20. It is inadvisable to hasten the decom- position by the application of heat, as a syrup is then formed which cannot be caused to crystallise, and elevation of tem- perature favours the formation of anhydro-compounds other than the normal anhydride. The anhydride melts at about 100°, is almost but not completely soluble in water, and appears to be different from the anhydride obtained by Fahlberg by the action of sulphur trioxide on glycollic acid (J. Pract. Chem., [2], 7, 336). The manner of formation of this compound would point to its having the normal structure, were it not for the comparative stability of the intermediate compound. It is possible that in this reaction silver glycollate behaves in the abnormal manner sometimes displayed by silver salts of hydroxy-acids, and the participation of the hydroxyl group in the reaction is therefore not excluded. With silver mandelate the tendency to form chlorinated products is pronounced, and the consequent formation of water has made it impossible to isolate the intermediate compound free from admixture with acid. The chlorine- containing impurities may be removed by repeated precipita- tion of the substance from its ethereal solution by means of 232 THE PREPARATION OF light petroleum ; the relatively large quantity of even care- fully dried solvent thus necessary tends, however, to the introduction of traces of water and consequent formation of more free acid. In absence of water the decomposition of the intermediate compound proceeds very slowly, and it is necessary to assist the decomposition by application of heat. After decomposition is complete a glass is obtained on extrac- tion with benzene and subsequent removal of the solvent, the analytical figures for which agree nearly with those required for a compound [C6H5.CHOH.CO]2O, while the figures obtained on titration point also to this formula. From this substance, which still contained a trace of sulphur, mandelic acid was regenerated by treatment with water or by dissolving in sodium hydroxide solution and acidifying, though a small portion remained undissolved by the solvents. The intermediate compound formed from silver malate is more easily decomposed, and most of the sulphur dioxide can be expelled by passing carbon dioxide through the ethereal solution for a day or two. After distillation of the ether, soft feathery crystals separate from the residual syrup, and these, after washing with ether and recrystallisation, give analytical figures which agree nearly with those required for CH(OH)COv malic anhydride )>0. The substance is soluble CH2.CO / in water, and malic acid is formed in the aqueous solution on standing. This substance is probably not the same as the porcelain-like anhydride of malic acid obtained by Walden (Ber., 1899, 2706, 2819). It appears to be opticaUy active. A fuller investigation of these interesting compounds must be deferred until they can be prepared more easily and in larger quantity. The experimental difficulties met with in their preparation are at present considerable. Attempts to prepare normal anhydrides of the amino- benzoic acids were unsuccessful by both methods. ANHYDRIDES OF ORGANIC ACIDS 233 SUMMARY OF RESULTS Type of Acid Behaviour of Silver Salt towards Sulphur Chloride Behaviour of Silver Salt towards Thionyl Chloride Fatty Acids Reaction normal. Intermediate compounds : Oils. Reaction normal. No intermediate com- pounds. Aromatic Acids Reaction normal. Intermediate compounds : Crystalline Solids. Reaction normal. No intermediate com- pounds. Dibasic Acids Anhydrides already known formed. No intermediate com- pounds isolated. Anhydrides already known formed. No intermediate com- pounds. Acids which con- tain halogen (fatty and aromatic) Reaction normal. Intermediate compounds less stable than those from the unsubstituted acids. Reaction normal. No intermediate com- pounds. Nitro-aromatic Acids Reaction normal. Intermediate compounds sparingly soluble in the usual solvents. Reaction normal. No intermediate com- pounds. Aromatic Amino-Acids Reaction abnormal. Reaction abnormal. Hydroxy- Acids Reaction varies. Intermediate compounds formed. Reaction varies. Intermediate compounds formed. 2 G 234 THE PREPARATION OF THE CONSTITUTION OF SULPHUR MONOCHLORIDE If we may look upon the glycollyl intermediate compound described above as a mixed anhydride of glycollic and sulphurous acids, that is, as a representative of the inter- mediate compounds whose existence in the case of non- hydroxy acids is too fugitive to be observed, the analogy between the course of the reactions when thionyl chloride is used, and those with sulphur chloride, is pronounced. The resemblance becomes complete when it is remarked that the decomposition of the acyl thiosulphites proceeds as a reaction of the first order. That such is the case has been ascertained by absorbing the sulphur dioxide evolved on its decomposition in iodine solution. The decomposition is conveniently carried out in toluene, maintained at its boiling-point, in a flask provided with a reflux condenser and gas delivery tube ; the latter is branched, and each branch connected through a stopcock with absorption bulbs. In this way, if a regular current of carbon dioxide is passed through the flask, the quantity of iodine used from time to time, and thus the amount of decomposition, is readily determined by titration with sodium thiosulphate. One such set of observations is here given :— Time in Minutes x=cc. Iodine a—x 1 . a - log — t 6a— x 16 9-08 62-14 0-0059 26 21-38 49-84 0-0060 45 32-97 38-25 0-0060 71 43-71 27-51 0-0058 85-5 48-22 23-00 0-0057 Total quantity of iodine used after some hours=a=71-22cc. The fact that the decomposition of benzoyl thiosulphite is monomolecular may be taken as pointing to the transient ANHYDRIDES OF ORGANIC ACIDS 235 existence of thiosulphurous anhydride S20, and, if this is so, we should then have the following schemes to represent the course of the reactions with sulphur chloride and thionyl chloride respectively : — Sulphur Chloride 2C6H6 . COOAg+S2Cl2 = (C9H5 . COO)2S2+2AgCl, (C6H5.COO)2S2 = (C6H6.CO)20+S20, 2S20 = S02+3S. Thionyl Chloride 2C6H5.COOAg+SOCl2 = (C6H5COO)2SO+2AgCl, (C6H5 . COO)2SO = (C8H5 . CO)20+SO2. These methods of anhydride formation would thus be classified with that in which an unstable mixed anhydride is first formed by the action of benzoyl chloride on a salt of an organic acid. The analogous behaviour of thionyl chloride and sulphur chloride may be further developed. If the reaction between thionyl chloride and silver benzoate be carried out, as already described in presence of ether, but with equimolecular pro- portions of the reagents, benzoyl chloride is obtained. With sulphur chloride the further reaction does not take place so readily, but it may be brought about by boiling benzoic anhydride with the chloride. Sulphur dioxide is then evolved and sulphur is liberated. The reactions may be represented by the equations : — (C6H5.CO)20+S2C1, = 2C6H5.CO.C1+S20, 2S20 = SO2+3S. (C6H5 . CO)20+SOC12 = 2C6H5 . CO . C1+S02. This manner of viewing sulphur chloride as a chloro- anhydride the parent acid of which is extremely unstable, allows of some of its reactions being expressed very simply. 236 THE PREPARATION OF For example, its decomposition by means of water to give hydrochloric acid, sulphur dioxide, and free sulphur : — S2C12+2HOH = S2(OH)2+2HC1 2S2(OH)2 = (2S20+2H20) = 3S+So2+2H20. The general resemblance between sulphur and oxygen compounds led Carius (Annalen, 1858, 106, 291) to regard sulphur chloride as being sulpho-thionyl chloride, so that its formula should be — CxCl corresponding to O : S< and not a . S . S . a. \C1 He looked upon the reaction between sodium benzoate and sulphur chloride, in which benzoyl chloride is formed, as taking place in two stages, in the first of which benzoyl sulphide and thionyl chloride are produced :— 2C6H5 . COONa+2S2Cl2 = (C6H8CO)2S+2SOCl2+Na2S = 2C6H5 . CO . Cl+2NaCl+S02+3SJ while he formulated the reaction between sulphur chloride and water similarly : — 2H20+S2C12 = H2S+2HC1+S02 2H2S+S02 = 2H20+3S. It is thus evident that Carius, having in mind the possible analogy between thionyl chloride and sulphur chloride, looked upon reactions in which the latter substance and oxygen-containing compounds take part as consisting in the primary formation of thionyl chloride, which then may react further. As already shown, however, it is possible to view these reactions more simply, and to express, perhaps more clearly, the analogy between the two chlorides, although at the same time it should be noted that the analysis of the mechanism of these reactions previously detailed holds good for either constitutional formula. The similarity in the reaction of ANHYDRIDES OF ORGANIC ACIDS 237 the two compounds, though general, are similarities in behaviour common to acid chlorides, but if one takes into account the far-reaching analogy between oxygen and sulphur compounds, they may be held as pointing to similarity in constitution. The amides corresponding to thionyl chloride and sulphur chloride are unknown, but Michaelis has prepared the tetra- alkylated derivatives of these compounds by the action of thionyl chloride (Ber., 1895, 28, 1016) and of sulphur chloride (foe. cit., 165) respectively on dialkylamines in presence of ether, the reactions in each case being quite parallel : — SOC12+4NH(C2H6)2 = SO[N(C2H5)2]2+2NH(C2HS)2 . HC1. S2C12+4NH(C2H5)2 = S2[N(G'2H5)2]2+2NH(C2H5)2 . HC1. The existence of amides and acyl derivatives of thio- sulphurous acid makes it appear probable that corresponding esters might also be obtained, and such compounds have been described by Lengfeld (Ber., 28, 449), who investigated the reactions which take place between sulphur chloride and the methoxide and ethoxide of sodium respectively in presence of light petroleum. The writer has not, however, succeeded in preparing these compounds either by Lengf eld's method or by modifications of it. What appears to be direct evidence in favour of the sulphothionyl constitution of sulphur chloride is afforded by its formation from thionyl chloride by the action of phos- phorus pentasulphide : — P2S5+5 0 : SC12 = P205+5 S : SC12 (Carius, loc. cit.) but according to Prinz (Annalen, 223, 355) the change does not occur directly. The converse transformation, that of sulphur chloride into thionyl chloride, can be effected by the action of sulphur trioxide at a temperature of 75°-80°. S03+S2C12 = SOC12+S02+S. 238 THE PREPARATION OF This reaction forms indeed the basis of a method for the preparation of thionyl chloride, for if a current of chlorine be passed into the mixture the liberated sulphur is recon- verted into sulphur chloride, which then reacts with a further quantity of trioxide (D. R. P. 139455; Abstr., 1902, ii. 420). The behaviour of sulphur chloride towards hydrocarbons and phenols may be noticed. Boeseken (Eec. Trav. Chim., 1905, 24, 209) found that benzene and sulphur chloride react together in presence of aluminium chloride, so that diphenyl sulphide and free sulphur are formed quantitatively according to the equation : — 2C6H6+S2C12 = (C6H5)2S+S+2HC1, while Cohen and Skirrow (Trans., 1899, 75, 887) obtained diphenylene disulphide C6H4< >C6H4 and free sulphur from the same substances when aluminium chloride was employed as catalyst. The reaction between phenol and sulphur chloride is vigorous, and among the products are dihydroxy-diphenyl sulphide S(C6H4OH)2, free sulphur, and probably a disulphide S2(C6H4OH)2. Somewhat similar products were obtained by Henriquez (Ber., 1894, 27, 2992) from sulphur chloride and naphthol. If we accept the unsymmetrical formula S : SC12 we can readily understand that the divalent sulphur atom may easily be split off in all the above cases with formation of a monosulphide : — R2S : S ^ R2S+S, while with the symmetrical formula Cl. S. S. Cl the formation of a monosulphide is less intelligible. The salt-like metallic derivatives of imides react with sulphur chloride though less readily than ordinary metallic ANHYDRIDES OF ORGANIC ACIDS 239 salts. Thus if silver succinimide (2 mols.) is shaken for some time with sulphur chloride (1 mol.) in presence of dry benzene, there is obtained, after filtration and evaporation of the benzene in vacuo, a quantitative yield of sulphur succinimide as a white crystalline powder which is stable when dry but decomposes fairly readily when in solution in such solvents as acetone. It is at once decomposed by sodium hydroxide solution. The molecular weight of this compound, as deter- mined experimentally, points to the formula [C2H4(CO)2.N]2S2. Silver phthalimide does not react smoothly with sulphur chloride, but, from potassium phthalimide, the sulphur derivative can be obtained though the yield is not very good. It is remarkable that the values found for the molecular weight of sulphur phthalimide agree with the simple formula C6H4.(CO)2NS, a fact which, taken in conjunction with the quantitative yield of the bimolecular succinimide derivative, points to the unsymmetrical formula for sulphur chloride, as is indicated by the formulae : — >N-S >N\ i >N-S a consideration of which shows that a theoretical yield of the monomolecular substance might be obtained by the breaking down of the double molecules of symmetrical structure, but less probably by the breaking down of those of unsymmetrical structure. Sulphur phthalimide can be prepared in good yield by acting on phthalimide with excess of sulphur chloride and pyridine in presence of an indifferent solvent. It is a stable substance which crystallises from chloroform in colourless crystals which contain chloroform of crystallisation. Like the succinimide compound, it is at once decomposed by sodium hydroxide solution. Thionyl chloride and silver succinimide when shaken together in presence of benzene appear to give the thionyl 240 ANHYDRIDES OF ORGANIC ACIDS compound corresponding to sulphur succinimide, but this substance has not been obtained in the pure state owing to its extreme sensitiveness to moisture, with which it at once gives sulphur dioxide and succinimide : — [C2H4(CO)2N]2SO+H20 = 2C6H4(CO)2NH+S02. It is seen from the foregoing that while neither formula for sulphur chloride is in conflict with the behaviour of this compound towards water, metallic salts and amines, the unsymmetrical one S : SC12 affords a better explanation of its reactions with hydrocarbons, phenols, and metallic derivatives of imines, and is therefore to be preferred, quite independently of any presumption in its favour from the standpoint of analogy. WILLIAM SMITH DENHAM INDIUM AND THALLIUM IN CRYSTALLO- GRAPHICAL RELATIONSHIP INTRODUCTORY EXCEPT in the case of the double sulphates (1), isomorphous relationships between salts of indium and thallium have not been made the subject of investigation. The literature dealing with the crystallography of indium salts is in fact remarkably scanty. Thiel and Koelsch (2) have described an oxide of indium — presumably In2O3 — which crystallises in octahedra, with spinel habit, like Fe3O4. The compound In F33H20 crystallises, according to Thiel (3), in four-sided, probably rhombic, prisms. No similar compound of other trivalent metal has been investigated crystallographically, and no compound of the type MF33H20, where M=A1, Tl, Ga, has hitherto been isolated. Some interesting cases of isomorphism among the silicotungstates of the trivalent metals, including indium, have been studied by Wyrouboff (4) . Thus in the series of salts R4(W12Si040)3 60H20, where R=A1, Fe, Cr, Ga, Bi, the results furnished by Wyrouboff are interpreted by Groth (5) to indicate isomorphism in the case of the salts of Al, Fe, Cr, Ga, and similarity of axial angle, though considerable diver- gence in axial constants, in the salt of Bi, and in In4(W12SiO40)3 63H20, which differs from the other salts by three molecules of water of crystallisation. The series R4(W12Si040)3 87H20, where R=A1, Cr, Ga, is undoubtedly isomorphous. The members of the two series R4(Mo12Si040)3 93H20, where R=A1, Cr, Fe, and R4(W12SiO40) 93H2O, where R=A1, Cr, Fe, Ga, In, crystallise in octahedra. Lastly, the salt K3InCl6 2n 242 INDIUM AND THALLIUM IN 1 £H20, which was first prepared by Meyer (6), and re-examined by Fock (7), was found to be isomorphous with K3T1C16 2H2O and with (NH4)3T1C16 2H2O, and mixed crystals were obtained by Fock of K3InCl6 IJH20 and K3T1C16 2H20, the crystal constants of which closely resembled those of its components. Pratt (8) found that K3TlBr6 H20 was isomorphous with the above three salts, so that the unusual case is here presented of salts of the same chemical type, but varying in the number of molecules of water of crystallisation, showing isomorphous relationships. In this connection Groth (9) remarks : ' The explanation of these remarkable relationships can naturally only be obtained by again completely and systematically examining these and all analogous compounds.' So far then as the crystallography of the salts of indium has been determined, it is evident that the results show only indefinite isomorphous relationships between indium and a wide class of trivalent metals, including Fe, Cr, Al, Ga, and Tl, and do not point to any specially close relationship within the sub-group Ga, In, Tl. It seemed advisable to compare the crystallographical character of corresponding salts of thallium and indium in particular, especially as the work on the double sulphates had brought to light no isomorphous relationships between indium and thallium compounds. As the complex alkali halides gave promise of the best results, and as the work on these salts might at the same time lead to the elucida- tion of the unusual case of isomorphism referred to above, attention was confined to the chlorides and bromides of indium and thallium with K, NH4, Rb, Cs, and Tl. As already mentioned, the work of Rammelsberg (10) and Fock (7) on K3TlCla 2H20, of Meyer and Groth (6), and of Fock(7),on K3InCl6 liH20,of Rammelsberg (10) on (NH4)3T1CI6 2H20, and of Pratt (8) on Rb3TlBr6 H20, seemed to indicate iso- morphism in this rather remarkable group. Pratt also found that the salts Rb2TlCl5 H20 and Cs2TlCl5 H20 were isomorphous in the rhombic bipyramidal system, and Meyer (11) prepared CRYSTALLOGRAPHICAL RELATIONSHIP 243 an indium salt which corresponded in formula to (NH4)2InCl5 H20, but which he did not examine crystallographically. A review of previous work seemed then to indicate the possibility of obtaining two fairly long isomorphous series, of the types R3MX6?H20 and R2MX5H2O respectively, where R=K, NH4, Rb, Cs, and possibly Tl ; M=T1, In ; and X=C1, Br. Still another series — of the type RMX4 ?H2O — might prove of interest in this connection, more especially with X=Br or I. Nickles (12) obtained the salts KTlBr4 2H20 and NH4TlBr4 2H20, and considered them to be rhombic, but gave no crystallographical details ; while Pratt (8) found that the salts RbTlBr4 H2O and CsTlBr4 crystallised in the cubic system. As part of the proposed investigation it was considered necessary to redetermine the crystallography of such members of isomorphous series as had already been investigated. A certain amount of the published data was too indefinite in detail to be made use of in an investigation on iosmorphism, where it is imperative that the data be the most accurate that can be obtained. It was also necessary to analyse the com- pounds of the type R3MX6 ?H20, with reference especially to the amount of water which they contain. If the formulae given for these salts is correct, it would seem that the water of crystallisation is in solid solution in the crystals, and does not play any fundamental part in determining the crystal structure. METHODS OF PREPARATION AND ANALYSIS The general method adopted for the preparation of the crystals was as follows. Thallic or indie oxide was dissolved in rather more than the requisite amount of halogen acid, and the alkali added in the proportion required by the formula. The salt was then allowed to crystallise on evaporating the solution either at room temperature or by gradual cooling in a crystallisation apparatus. As the salt sought for might be 244 INDIUM AND THALLIUM IN in equilibrium at room temperature with a solution widely differing in composition from that represented by the formula of the salt, the concentration of the solution had to be varied till the required salt was found to crystallise. On dissolving thallous oxide in the halogen acid, any thallous ions present go to form insoluble thallous halide ; consequently the difficulties which attend investigations on the thallic sulphates on account of the continuous reduction in the solution of thallic ions to thallous ions, were obviated here. So long as an excess of acid is present, practically no thallous ions can remain in solution. Excess of acid also prevents hydrolysis from taking place ; in the case of the indium solutions, precipitation of the hydroxide was only prevented by main- taining the solutions decidedly acid, during the process of crystallisation. The analyses were carried out as follows. After being carefully crushed, and dried at room temperature, the salt was gradually heated in a drying oven, and weighed at intervals, all precautions being taken to avoid overheating and conse- quent disintegration. On constant weight being attained, the water of crystallisation was estimated. The salt was dissolved in water, and the thallic or indie hydroxide precipitated by addition of ammonia. Meyer (13) found that only two-thirds of the chloride in a thallic chloride solution is precipitated by silver nitrate in strong nitric acid, a fact which points to the formation of complex ions in the solution. It is therefore necessary to remove the thallium or indium from the solution before estimating the halogen. Indium was estimated as In2O3, precautions being taken to ensure that no sublimation took place during the heating of the hydroxide. Thiel and Koelsch (2), on investigating this method of estimating indium, found that at a temperature of 850° C. no loss of weight took place through sublimation, while at 1000° C. the sublimation was considerable. Unless in the case where the solution con- tained a large amount of ammonium nitrate, it was found that CRYSTALLOGRAPHICAL RELATIONSHIP 245 a temperature of 850° C. was sufficiently high to transform In(OH)3 into In2O3. When, however, there is excess of NH4N03, a certain amount of nitric oxide remains absorbed in the oxide at 850°. Here the amount of ammonia used for precipitating the hydroxide was the actual minimum sufficient to ensure complete precipitation, since indium hydroxide passes to some extent into colloidal solution in presence of much ammonia. Consequently there was small likelihood of error due to absorption of gas by In203 ; the hydroxide was, however, heated to 900°, to ensure the elimina- tion of any residual gas. There was no evidence of sublima- tion at this temperature. To estimate thallium, the precipitated thallic hydroxide was redissolved in sulphuric acid, and reduced to the thallous state by passing a current of sulphurous acid through the solu- tion. This was then evaporated to dryness to drive off the last trace of sulphurous acid, and the thallium was now estimated by the bromine method (14). As any sulphurous acid in the solution would reduce the bromate and lead to too high a result for the thallium-content, it was essential that all traces of the gas be first removed. Although the bromine method involved some rather troublesome processes, it was, on the whole, more trustworthy than the peroxide method (1), since it took into account any thallium that might be reduced to the thallous state during the precipitation of the hydroxide. The solution from which the hydroxide had been pre- cipitated was then acidified with nitric acid, and the halide was precipitated and estimated as silver halide. In the case of bromides excess of silver nitrate was added and the solution boiled. The excess of silver was then got rid of by addition of hydrochloric acid ; the filtrate was evaporated to dryness with sulphuric acid, and the alkali metal estimated as sul- phate. 246 INDIUM AND THALLIUM IN DETAILED CRYSTALLOGRAPHY OF THE VARIOUS SALTS K3T1C16 2H20 I : The series R3MX6 2H20 This salt crystallises equally well in two distinct habits, one of which has already been described by Fock (7). In the one case the crystals are elongated along the c axis (Fig. 1), and the prism faces jllOj are well developed. The other A: •^k '' o "V- r * a m a ***• 2HaO (6) FIG. 2 KjTlCl, 2H20 (a) FIG. 1 habit, which was not observed by Fock, is illustrated in Fig. 2. The crystals are here tabular on JOOlj, and the faces of jllOj are reduced to very narrow bands. jlOlj is in all cases small, but was represented in all the crystals examined, jlllj is well developed, and gives particularly good reflections. The following are the crystallographical data : — System : ditetragonal bipyramidal. Axial constants : a : c=l : '7941. Angle No. of Measure- ments Limits Average Calcu- lated Diff. Fock Rammeli- berg {001} :{!!!} 56 48° 9'— 48°27' 48*19' 48° 13' 48°30' {001}: {101} 18 3815'— 38°36' 38*27' 38°27' 0' {111}:{111} 48 63°29'— 63*54' 63*42' 63°45' 3' 63°38' Forms present : J100} J110} JOOlj jlOlj Jill}. S. G.= 2-859 at 20°. Cleavage indistinct,// jlOOj. CRYSTALLOGRAPHICAL RELATIONSHIP 247 Double refraction very weak. For this and other thal- lium salts, the refractive index is so high that no suitable liquid can be found of sufficiently high refractive index to be made use of in the total reflection method of measuring the double refraction. Consequently no refractive index data are given in this investigation. The following are the results of the analyses of the water of crystallisation : — Average of Analyses K3T1C1«2H2O K^TICI, 1$H,O H20 6-43 % 6-31 % 4-82 % This salt undoubtedly crystallises with two molecules of water of crystallisation. (NH4)3T1C16 2H20 This salt crystallises tabular on jOOlj (Fig. 3). Large, well-formed crystals were obtained, which in no case showed (NH4),TlCl62HaO Fio. 3 indications of prismatic growth parallel to the c axis. The faces of jlOOj were well developed, while those of {110! were small. jlll| was always represented, and J113J occurred as small faces giving quite good reflections. Faint indications of |103| were also observed, but the reflections were not good enough to give reliable measurements. The best reflections were obtained from the faces of jlllj. Rammelsberg (10) mentions the forms jlOOj, jllOj, {101 j, jlllj, {OOlj, but he evidently found no indication of {113J or J103J. 248 INDIUM AND THALLIUM IN System : Ditetragonal bipyramidal. Axial constants : a : c=l : '8097. Angle No. of Measure- ments Limits Average Calcu- lated Diff. Rammele- berg {001} :{!!!}* {001}: {101} {lllhhll} 43 17 34 48°37'— 49" 3' 38°44'— 39°11' 64°12'— 64°33' 48"52' 39° 6' 64°22' 39° 0' 64'22' 6' 0' 48°22' 38°40' 64° 2' {001} : {113} 5 20°29' — 21°29' 21° 9' 20°54' 15' Forms present: J100J, jllOj, jlOlj, jlll(, jOOlj, J113J, with indications of J103J. Cleavage very poor, // J100S. S. G.= 2-389 at 20°. It will be observed that the crystallographical details differ somewhat widely from the value given by Rammelsberg. Rammelsberg gave no details with regard to his measure- ments, which were made some thirty years ago, and would consequently require in any case to be revised to-day. The estimations of the water of crystallisation gave the following results : — Average of Analyses H20 7-15% (NH4)3TlCle2H20 7-11 % (NH4)3T1C1« 5-43 % Considerable caution had to be exercised in heating this salt. At temperatures above 150° dissociation took place, the salt losing continuously in weight. Heating was con- tinued at a temperature slightly under 150° until a practically constant weight was reached, and presumably the whole of the water of crystallisation had been liberated. The ammonium salt, like the potassium salt, crystallises with two molecules of water. K3InCl6 2H20 This salt had been obtained by Meyer (6), and examined CRYSTALLOGRAPHICAL RELATIONSHIP 249 crystallographically by Groth. Fock (7) re-examined the salt, and both Meyer and Fock gave the formula as K3InCl6 1 JH20. The salt is very soluble in water, and forms small, slightly yellowish crystals. From solutions containing potassium chloride and InCl3 in the proportions 3:1, potassium chloride was precipitated on evaporation at room temperature till very little solution remained. Precipitation of the complex salt then took place in a solution in which the concentration of indium ions was very high. The crystals were either tabular on jOOlj (Fig. 4), or elongated along the C axis, and K3InCl0 2HaO FIG. 4 showed large faces of jlllj. On only one of all the crystals examined did the form jlOlj appear, and in that case it was very poorly developed. The faces of jlOOj were more pro- nounced than those of \110\, and the reflections were very good. System : Ditetragonal bipyramidal. Axial constants : a : c=l : -8173. Angle No. of Measure- ments Limits Average Calcu- lated Diff. Fock Groth {001}:{111}» 47 48°55'— 49°25' 49° 8' 49°ir 49°13' {111}:{111} 46 64°15'— 64°52' 64°40' 64°40' 0' 64°47' {001}: {101} 1 39°15' 39°15' 0' 1 Forms present: JlOOj, jllOj, J001|, \lll\, jlOlj on one crystal. S. G.= 2-483 at 20°. 2 i 250 INDIUM AND THALLIUM IN The analyses gave the following results :— H20 K. In. Cl. 6-87% 25-00% 23-26% 43'92 % Calculated for K,InCls 2HSO 7-50 % 24-40 % 23-87 % 44-24 % „ K3InCl6lJH,0 5-74% 24-86% 24-04% 45-25% With the exception of the potassium value, which is necessarily, owing to the method of analysis, less reliable than the other values, the averages all distinctly favour the formula K3InCl6 2H2O. There is in this case no possibility of dissocia- tion at temperatures between 150° and 200°, so that the values obtained for the water of crystallisation would be abnormally high for the salt K3InCl6 1|H2O. Fock directed his attention mainly to investigating whether the formula K3T1C16 2H20 was correct ; although for the indium salt he obtained the analysis H.,0 Cl. In. 5-52 45-25 23-23 Calculated 5-74 45-25 24-04 it seems probable that the salt was not heated sufficiently to drive off the last traces of the water of crystallisation, which is liberated at a temperature considerably above 150°. At any rate, it seems conclusively proved that this salt con- tains two molecules of water, and consequently agrees in formula with the two salts already described, with which it is isomorphous. Rb3TlBr6 2H2O From solutions in which the relative proportions of RbBr and InBr3 were those represented by the above formula, there were deposited on evaporation cubic crystals of RbTlBr4 H20, and on further evaporation these were replaced by tetragonal crystals of Rb3TlBr6 2H20. This salt crystallises in honey- yellow crystals, developed on JOOlj, but not so pronouncedly tabular as in the case of (NH4)3T1C16 2H20 (Fig. 5). The faces CRYSTALLOGRAPHICAL RELATIONSHIP 251 of Jlllj and jlOlj are well developed ; and |110j occurs as fairly broad faces. The reflections are very good. Pratt (8) described this salt as crystallising in the tetragonal system, but with only one molecule of water of crystallisation. Rb3TlBr6 2H20 FIG. 5 The results of his investigations are given in a separate column in the following statement of the crystallography of the salt. System : Ditetragonal bipyramidal. Axial constants : a : c=l : '8038. Angle No of Measure- Limits Average Calou- latf-H Diff. Pratt ments {111}:{101 44 31°57'— 32°15' 32° 6' 32° 8i' {101}: {001 12 38°46'— 38°56' 38°53' 38°48' 5' 38054|' {001}: {111 10 48°44'— 48°48' 48°46' 48°40' 6' 48'53' {110}:{101 28 63°26'— 63°47' 63°40' 63°40' 0' iiiiMoiij 6 79-20'— 79°54' 79036' 79°30' 6' Forms present : JlOOj, jllOj, J001|, }101j, {lllj. S. G. -4-077 at 20°. The following are the results of analysis : — Rb. Tl. Br. 25-69 Calculated for Rb,TlBr.2H,O 26-25 „ Rb,TlBr, HS0 26-76 20-46 20-90 21-30 48-85 49-15 50-06 HaO 3-60 3-69 1-88 The analysis shows that the salt crystallises with two molecules of water. All the water of crystallisation is driven off by heating the salt to 120° C., and it is rather difficult to 252 INDIUM AND THALLIUM IN account for the low value for the water of crystallisation obtained by Pratt. K3InBr6 2H20 From solutions containing potassium bromide and indium bromide in the proportions 3:1, precipitation of KBr takes place till the solutions are almost completely evaporated. Very small reddish-brown crystals then begin to appear. Although on varying the concentration of the solution more favourable conditions for the deposition of this salt are obtained, it crystallised invariably as a very fine powder, and good crystals could not be isolated. The angular measure- ments obtained in the goniometer showed that the crystals were isomorphous with the compounds already described. This salt effloresces, however, so readily on exposure to the air that it was found impossible to obtain exact crystallo- graphical data. The specific gravity of the crystalline powder was found to be — as accurately as possible under the circum- stances— 3-140 at 20°. Analyses gave the following results : — K. In. BT. 16-89 15-24 63-97 Calculated 15-90 15-56 65-00. Owing to the very efflorescent character of the salt, no attempt was made to obtain the value of the water of crystal- lisation. The salt is, however, undoubtedly isomorphous with compounds of the type R3MX6 2H2O, and one is justified in assuming that it also contains two molecules of water of crystallisation. No other salt isomorphous with those already described has been obtained in this investigation. It would appear that the only compounds of the series R3MX6 2H2O, where R=K, (NH4), Rb, Cs or Tl; M=In or Tl; X=C1 or Br, that are in equilibrium with their solutions under ordinary conditions of temperature and pressure are (1) K2T1C16 2H20 ; (2) (NH4)3 CRYSTALLOGRAPHICAL RELATIONSHIP 253 T1C16 2H20 ; (3) Rb3TlBr6 2H20 ; (4) K3InCl6 2H20 ; and (5) K3InBr6 2H20. It is rather remarkable that while Rb3TlBr6 2H2O has been isolated, neither K3TlBr$ 2H20 and (NH4)3TlBr6 2H20 on the one hand, nor Cs3TlBr6 2H20 on the other, can be obtained from their solutions. Although the order of stability in a series of salts of similar formula — dependent as it is in the main on the varying solubilities of the different salts which may be precipitated — seems as a rule to have a definite relationship to the order of atomic weights, there are evidently markedly exceptional cases. II : The series R2MX5 H20 Crystallographical data have already been published on two salts belonging to this series, i.e. on Rb2TlCl5 H20 and Cs2TlCl5 H20 (see Pratt, loc. cit.). An isomorphous salt where X =Fe has been examined by Johnson( 15) — namely (NH4)2FeCl5 H2O. In the investigation of this series it was found that the crystals should be placed in a different position from that adopted by Pratt and Johnson, and it may be well to explain here why an interchange of the crystallographical axes has been made. The crystal system of the isomorphous series is rhombic bipyramidal, but there is a close resemblance to tetragonal habit. In the case of Cs2InCl6 H2O, for instance, the relative values of the crystallographical axes are "9841 : 1. Unless there were some valid objection, the third axis, which differs markedly from these two, would be taken as the c axis, and the two specified axes as the a and b axes respectively. Now, on examining the crystals of Rb2InCl5 H2O, it was found that practically every individual was twinned on the unit prism face between the two almost equal axes as twinning plane, so that the twinned individual was turned almost through a right angle round the axis parallel to the prism face, thus accentuating the pseudo-tetragonal symmetry. It was 254 INDIUM AND THALLIUM IN therefore decided to take this prism face as jllOj, and the shorter of the two almost equal axes as the axis a. This prism had been formerly taken to be J101{ ; so that the new position of the crystal simply involved an interchange of the 6 and c axes, the a axis remaining as before ; the change necessitated, in other words, a rotation of the crystal through 90° round the a axis. In the various crystals of the series there is a pronounced cleavage parallel to jOllj, which has the same indices for both positions of the crystal. There is then no a priori reason, from cleavage considerations, why the crystal should have been placed in a position which tends to hide, rather than accentuate, its pronounced pseudo-tetragonal character. Rb2TlCl5 H20 From the solution containing the chlorides RbCl and T1C13 in the proportion given in the salt formula, thin flakes of Rb3TlCl6 H20 (monoclinic) are first precipitated. On further evaporation these disappear, being replaced by large well- formed crystals of Rb2TlQ5 H20. These crystals are much distorted, being always tabular on J101J, the faces of which are extremely well developed. Truncating the edges of the pinacoid formed by two large faces of J101J are three smaller faces on the upper side, and three on the lower side of the crystal plate. These are the two remaining faces of J101(, and the four faces of jOOlj. Were it not for the pronouncedly tabular habit, the combination of the two forms jlOlj and 01 Ij would strongly suggest the octahedron ; as it is, each crystal appears as a triangular plate with the vertices cut away. The characteristic habit of the crystal is shown in Fig. 6. The faces of jlOOj occur very seldom, and are then represented by narrow threadlike bands. The reflections are very good. The interfacial angular measurements of Pratt are appended to the following table of crystallographical data : — CRYSTALLOGRAPHICAL RELATIONSHIP 255 Crystal system : Rhombic bipyramidal. Axial constants : a : b : c=-9770 : 1 : 1-4386. Angle No. of Measure- Limits Average Calcu- Diff. Pratt ments {Oil [={011} 11 69°31'— 69°41' 69°36' 69'36' {Oil :{101} 52 70°55'— 71°35' 71°18' 71°24£' {101}: {101} 12 67°58'— 68°40' 68°20' 68°22' 2' 68° 7£' {100 :{120} 68*53}' {Oil :{120} 6319' Forms present: jOllj, jlOlj. J100J occurs seldom and very poorly developed. J120J not found, though observed by Pratt. S. G. =3-513 at 20°. No analyses were made in the case of this salt, as there was no question as to its identity, and sufficiently full analyses had already been made by Pratt to establish its formula. Rb2TlCl5 H20 Fie. 6 Cs2TlCl5 H20 PIG. 7 Cs2TlCl5 H20 The solution contained CsTl and T1C13 in the proportion 2:1. There crystallised on evaporation large hexagonal plates of CsgTlgClg, and these gradually gave place to prisms of Cs2TlCl5 H20. This salt is sparingly soluble in water, and crystals 5 — 6 mms. long were frequently obtained. The larger crystals were usually opaque. They are elongated along the b axis, with well-developed faces of jlOlj (Fig. 7). The end 256 INDIUM AND THALLIUM IN faces are comparatively small, but give good reflections, whereas the faces of jlOlj are badly striated. jlOOj occurs, but the faces are narrow and very poorly developed. J120J was not found on any crystal examined. Crystal system : Rhombic bipyramidal. Axial constants : a : b :c=*9690 : 1 : 1-4321. Angle No. of Measure- Limits Average Calcu- Intorl Diff. Pratt ments {Oil}: {Oil}* 18 69°17'— 70° 9' 69°51' 70° 0' {Oil}: {101}* 24 70°54'— 71°30' 71°17' 71°15' {101} :{101} 23 67°59'— 68°38' 68° 9' 68°10' 1' 68°22' {100}: {120} 62°51' {Oil}: {120} 43° 9' S. G.= 3-879 at 20°. Forms present : jOllj, jlOlj, JIOOJ. Here, too, Pratt observed the form J120J on some of the crystals he examined, although the form did not occur on the crystals from any of the crops examined by me. As in the case of the preceding salt, a quantitative analysis was deemed unnecessary here. (NH4)?InCl6 H20 This salt was obtained by Meyer( 16) on evaporating a solution containing (NH4)C1 and InCl3, in his attempt to isolate a compound similar to K3InCl6 1 £H20. He mentioned that the crystals seemed different in form from those of the potassium salt, but evidently no detailed examination was made. The habit of the crystals seems to vary with the composi- tion of the original solution. From a solution where the proportion of the NH4 ions to the In ions was that represented by the formula (NH4)3InCl6 2H20, crystals of (NH4)2InCl5 H20 were precipitated which were elongated along the 6 axis, had well-developed faces of \101\, fairly good faces of |120(, CRYSTALLOGRAPHICAL RELATIONSHIP 257 and small faces of JOllj (Fig. 8). From solutions, however, where the relative proportions of the metallic ions were those represented by (NH4)2InQ5 H20, the crystals always showed large faces of J100|. Such crystals were either developed along (NH4)2InCl6 H20(a) FIG. 8 (NH4)2InCl5 H20 (6) FIG. 9 the c axis, and carried fairly large faces of J120J, or were tabular on jlOOj (Fig. 9). jlOlj was always well represented. The crystals were slightly yellowish, and the cleavage parallel to JOllj was especially pronounced. System : Rhombic bipyramidal. Axial constants : a : b : c='9663 : 1 : 1-4005. No of Angle Measure- ments Limits Average Calcu- lated Diff. {101}:{I01}* 22 69' 6'— 69° 18' 6913' {101}: {120}* 36 67°40'— 68° 5' 67°47' {120}: {120} 16 54° l'_55° 2' 54°38' 54°42' 4' {011}:{01I} 20 70° 14'— 71° 0' 70°49' 71° 2' 13' {120}: {011} 24 43- 8'— 43°48' 43°33' 43°42' 9' loiHouj 16 70° 10'— 71° 4' 70°43' 70°43' 0' Forms present (on both types of crystal habit) : J101J, J120J, JOllj, jlOOj. S. G.=2-281 at 20C 2K 258 INDIUM AND THALLIUM IN The crystals give up their water of crystallisation very slowly on heating. The analyses of the salt gave the following result : — In. 01. H,O 32-70 49-63 5-30 Calculated 33-19 51-21 5-20. RbaInCl5 HaO A solution which contains RbCl and InCl3 in the proportion of two molecules of the first to one molecule of the second deposits on evaporation well-formed crystals of the above composition. This salt is sparingly soluble, and medium-sized crystals can be grown without difficulty. In habit the crystals are more or less tabular on JlOlj, and are usually elongated along the b axis. They show large faces of jlOlj, and fairly well-developed faces of J120J; jOllj and jlOOj are small, but are represented on all the crystals investigated. The general Cl,, H,O FIG. 10 habit is shown in Fig. 10. The reflections are good from all faces except from those of J101J, which frequently give multiple reflections. Dispersion is high. Practically all the crystals are twinned, the twinning plane being JllOj ; the two individuals of the interpenetrating twin cross at an angle of 88° 22'. The interpenetrating twin, looked at from above, bears a marked resemblance to the iron cross of pyrites, the |120| faces of the second individual appearing through the large faces of JlOlj on the first. CRYSTALLOGRAPHICAL RELATIONSHIP 259 The crystallographical data are as follows : — System : Rhombic bipyramidal. Axial constants : a : b: c=*9725 : 1 : 1-4065. Vn nf Angle Measure- Limits Average Calcu- Iat«H Diff. ments {101} :{10l}* 27 68°49'— 69°37' 69° 15' {120}: {120}* 63 53°59'— 54'45' 54°25' {011} :{011} 16 70°26'— 71° 8' 70°47' 70°45' 2' {120}:{011} 11 43°16'— 43°53' 43°32' 43'31' V {101}: {120} 8 67°43'— 68' 1' 67°46' 67°54' 8' {101}: {011} 14 70"34'— 71°14' 70°50' 70°48' 2' Forms present : jlOOj, JlOlj, J120J, jOllj. Twinning on jllOj. S. G.= 3-037 at 20°. The analyses of the carefully dried powder gave the follow- ing results : — Kb. In. 01. HS0 35-01 23-92 35-32 1-46 Calculated 35-52 23-86 36-87 2-75. The values for chlorine and for water are both low. The water of crystallisation is held firmly in this salt, and is probably not entirely driven off even at 200°. For some reason not fully understood, the method of analysis adopted for this series of isomorphous salts gave consistently low results for the halogen. CsjjInClg H2O The crystals of this salt, which were obtained on allowing a solution which contained CsCl and InCl8 in the proportion 2 : 1 to evaporate at laboratory temperatures, were very simple in type. They represented a combination of the two forms JlOlj and J011}, usually equally developed (Fig. 11), so that the crystals appeared cubic, developed on jlllj. 260 INDIUM AND THALLIUM IN Occasionally there was a tabular development on |101j. On only one crystal was the form jlOOj found. No cases of twinning were observed. (NH4)2InBr,s H20 FIG. 12 The crystals were colourless, and relatively insoluble in water. They were as a rule small, badly formed, and gave multiple, indefinite reflections. From the various crops which were grown, it was found difficult to obtain a sufficiently large number of crystals capable of yielding fairly reliable results. The crystals of this salt are distinctly poorer than those of any other member of the series. System : Rhombic bipyramidal. Axial constants : a : b : c='9841 : 1 : T4033. Angle No. of Measure- Limits Average Calcu- lated Diff. ments {Oil}: {Oil # 14 30 70°45' — 71°12' 70"20'— 70"44' 70°57' 70°32' {101}: {101 I 5 69°44'— 70°0' 69°56' 70'5' 11' Forms present : jlOlj, jOllj. JlOOj on one crystal. S. G.=3-350 at 20°. CRYSTALLOGRAPHICAL RELATIONSHIP 261 Analyses gave the following results : — Cs. In. 01. HSO 45-62 20-43 30-09 3-27 Calculated 46'13 19-94 30'80 3'13. (NH4)2InBr6 H2O From a solution containing NH4Br and InBr3 in the proportions of two to one, a fine-grained crystalline powder was precipitated, brownish in colour and very soluble in water. The crystals proved to be very deliquescent, and no exact crystallographical measurements could be made. The habit of the crystals is illustrated in Fig. 12. They are elongated along the c axis, slightly tabular on the well-developed faces of jlOOj, and show fairly large faces of J120J, jlOlj, and JOllj. The specific gravity is, as accurately as could be measured under the circumstances, 3' 167 at 20°. Analyses gave the following results : — In. Br. H5O 1970 70-11 5-92 Calculated 19-73 70-30 6-48. After standing in the solution for four days, the crystals became quite opaque. The slightest changes of temperature seem to affect the stability of this salt when in contact with the solution at ordinary room temperatures. Rb2InBrs H20 From solutions in which RbBr and InBr3 are in the pro- portions respectively of 1 : 1 or 2 : 1, crystals of the above salt are formed at room temperatures. The crystals are colourless, fairly insoluble, and are frequently rather cloudy. They are usually developed along the 6 axis, show broad faces of jlOOj, large faces of JlOlj, and comparatively small faces of [120J and 011J (Fig. 13). 262 INDIUM AND THALLIUM IN The following are the crystallographical data : — System : Rhombic bipyramidal. Axial constants : a : b: c :='9803 : 1 : 1*3951. Angle No. of Measure- Limits Average Calcu- l&ted Diff. ments {101} :{011}* 24 69°59'— 70°48' 70*26' {Oil}: {Oil}* 16 70°49'— 71*42' 71*16' {101}: {101} 18 69'26'— 70° 5' 70* 1' 70°11' 10' {120} :{011} 3 43°39'— 43°51' 43°44' 43*37' 7' {120}: {101} 9 67"28'— 68°30' 67°50' 68° 10' 20' {120}: {120} 54° 3' Forms present : jlOOj, J120J, jlOlj, jOllj. S. G.=3-409 at 20°. The results of the analyses are as follows :— Rb. In. Br. HS0 23-10 16-67 56-18 3-25 Calculated 24'29 16-64 56-32 2'61 Rb2InBr6 H2O Fio. 13 ., H2O Fio. 14 Cs2InBr6 H20 From solutions which contained CsBr and InBr3 in the proportion 2:1, crystals of this salt were very easily obtained on evaporation. They are very insoluble, colourless, and markedly lustrous. The crystals are small and well formed, and give good reflections. No evidence of twinning was found. The general habit of the crystals somewhat resembles that described for Rb2InBr6 H20 (Fig. 14). There is a marked CRYSTALLOGRAPHICAL RELATIONSHIP 263 elongation along the b axis : jlOOj is here very small, while JlOlj is particularly well developed. JOllj has fairly good faces, but J120J was found on only one of the crystals examined. The following are the crystal measurements : — System : Rhombic bipyramidal. Axial constants : a : b : c='9734 : 1 : 1-4180. Angle Measure- Limits Average Calcu- Diff. ments {101}: 0111* 35 70° 35'— 7T10' 70°58' {Oil}: 0111* 21 {70°}14'— 70°43' 70°23' 101}: 101} 11 68° 29'— 69° 4' 63°45' 68°55' 10' 120}: 120} 54°23' 120}: {011} 2 43° 16'— 43C38' 43°27' 43°22' 5' Forms present : jlOOi, J101|, J011| : J120J occurs only once. S. G.=3-776 at 20°. Analyses give the following results : — CB. In. Br. 32-80 14-45 49-42 Calculated 33-30 14-38 50'07 H2O 2-47 3-26. Ill : The Series RMx4 xH20 From the point of view of the present investigation this series is relatively unimportant, as each member of the series crystallises in the cubic system. The crystallographical values are therefore identical, and, as the values for the re- fractive indices could not be obtained for the thallium salts, there was nothing to indicate the change in crystalline structure due to the replacement of one element by another. Conse- quently no thorough investigation was made of the salts belonging to this series. Nickles (12) stated thatthe two salts of the composition KTlBr4 2H20 and NH4TlBr4 2H20 are iso- morphous, crystallising in the rhombic system ; and Meyer (17) also refers to the former of these two salts as rhombic. Pratt (8) 264 INDIUM AND THALLIUM IN has described RbTlBr4 H20 and CsTlBr4 as crystallising in cubes. These salts were all obtained in this investigation, but the crystalline form of KTlBr4 2H20 and NH4TlBr4 2H2O was found to be cubic, not rhombic. The crystals were tabular, showing only the face of jlOOj, with depressions in the form of inverted rectangular pyramids. Under the microscope the crystals were perfectly isotropic in all positions. No corresponding indium bromides were obtained, and no thallium or indium chloride of this general type was isolated. From solutions, however, which contained KBr and TlBr3 in the proportion 3 : 1, a salt was obtained, which, although not belonging to this series, may be described here. This salt was deposited in beautiful yellowish brown crystals of high lustre from solutions which had previously precipitated cubic crystals of KTlBr4 2H20. K3Tl2Br9 3H2O Rammelsberg (18) had obtained a salt of this composition from solutions containing TIBr, Br, KBr, and water. He described the crystals as yellowish and apparently regular, showing the faces of jlllj, *100|, and jllOj. Meyer (13) failed to obtain this salt, and considered that the salt which Rammelsberg obtained was probably KTlBr4 2H2O. I succeeded in obtaining both salts, crystallising together from various solutions of the composition K3Tl2Br9 xH2O. The stability conditions of the salt under consideration were not fully made out. Whenever it appeared, it crystallised subsequent to the precipitation of KTlBr4 2H2O, and the slightly reddish tinge of the crystals made them conspicuous among the pale yellow plates of the other salt. But in many cases — usually on slight rise of room temperature — the crystals disappeared shortly after formation ; and from several solutions no precipitation of the salt took place. Low room- temperatures and fairly acid solutions were distinctly favour- able conditions. CRYSTALLOGRAPHICAL RELATIONSHIP 265 When formed under the most suitable conditions, the crystals possessed a markedly high lustre ; but usually the lustre was dull and the reflections poor. This may probably be accounted for by the fact that at temperatures above the average room temperature, efflorescence was observed to take place. Fig. 15 shows the general habit of the crystals. They belong to the ditetragonal bipyramidal class, and are usually K3TljBr9 2H20 Fio. 15 slightly elongated along the c axis. The form jlOOj is well developed, as is also the form Slllj. jllOj is less pronounced, and jlOlj very small, occurring in only a few crystals. The crystals are capped by small faces of jOOlj. Under suitable conditions of growth, large well-formed crystals were always obtained. System : Ditetragonal bipyramidal. Axial constants : a : c=l : '7556. Angle No. of Measure- ments Limits Average Calcu- lated Diff. {Oio}:mi}» 46 58°38'— 59° 3' 58°55' {110} : {111} 24 43° _43°25' 43°12' 43° 6' 6' {110} :{011} 8 64°42'— 65°12' 64°49' 64°46' 3' ? 4 76°10'— 76"24' 76°17' Forms present : jOOlj, jlOOj, JllOj, jlllj, JlOlj. 266 INDIUM AND THALLIUM IN The analyses gave the following results : — K. 8-46 Calculated 9-03 Tl. Br. H,0 31-74 55-05 4-75 (diff.) 31-42 55-30 4-16. COMPARISON OF DATA Full crystallographical details have been given in the foregoing section for the members of two distinct series of isomorphous salts — (1) the ditetragonal bipyramidal series, consisting of the following salts : K3T1C16 2H20, (NH4)3T1C16 2H20, K3InCl6 2H2O, Rb3TlBr6 2H20, to which may be added K3InBr6 2H2O, for which incomplete details are given ; (2) the rhombic bipyramidal series, consisting of : Rb2TlCl5 H20, Cs2TlCl6 H2O, (NH4)2Ina6 H2O, Rb2InCl6 H20, Cs2InCl5 H2O, Rb2InBr5 H2O, Cs2InBr5 H2O, to which may be added (NH4)2InBr5 H20, for which incomplete details were obtained. On referring to the literature it would appear that only one other salt has been described which shows any marked similarity in its crystallography to the members of the first series, i.e., K3SbCl3Br3 1£H20. This salt was obtained by Atkinson and described by Solly (19), who gave the following details : — Class : Tetragonal bipyramidal. Crystal constants : a : c=l : '7629. Angle Calculated Observed {111}:{1T1}« {111}:{111} 85°39' 62°29' 85'40' Forms : jlllj, with occasionally small faces of jOOlj. Although the estimated water value differs by half a molecule from that given for the isomorphous series, and the CRYSTALLOGRAPHICAL RELATIONSHIP 267 crystals are markedly poorer in faces, the general similarity of chemical composition, and of angular values, justifies the inclusion of this salt in the isomorphous series. In comparing this salt with the others of the series, the values already quoted will be used. Isomorphous with the second series are two salts, K2FeCl5 H20 and (NH4)3FeCl5 H20. The former is the mineral Erythrosiderite (20), for which the crystal constants are (adopt- ing the same placing of the crystal as in the isomorphous series) a : b : c :=*9628 : 1 : 1*3931, and in which the combination JlOlj, |011j, jlOOj, and J120J occurs : on crystals formed in the laboratory the octehedral-like combination of J101| and jOllj, similar to that already described for Cs2InCl6 H20, is charac- teristic. The salt (NH4)2FeCl5 H20 has been described by Johnson (15). He found that the forms J101J and jOllj pre- dominate, jlOOj is sometimes large, and J120J small. Two twinning laws were observed — (1) twinning axis perpendicular to \lll\ ; (2) twinning plane {110J. System : Rhombic bipyramidal. Axial constants : a : b : c='9749 : 1 : 1'4239. Angle Calculated Observed {101}: {101 {100}:{120 ; 68°48' 62°51' {011}:{011 70° 10' 70° 17' {Oil}: {120} 43*16' 71° 3' 43°10' 71*11' {101}:{120} 68°28' 68°30' S. G.=l-99. These two compounds of iron are the only salts hitherto described which are undoubtedly isomorphous with the series of indium and thallium salts under discussion. The crystallo- graphy, as quoted above, will be used when comparison is made between the various members of the series. 268 INDIUM AND THALLIUM IN Within the limits of the present contribution it is impossible to enter into a full discussion of the practical results here described ; that must be reserved for subsequent publication elsewhere. Some general conclusions drawn from a detailed comparison of the various salts which have been examined is given, however, in the summary which follows. SUMMARY 1. The ditetragonal bipyramidal series, consisting of the salts (a) K,T1C1, 2H,0, (6) (NH«),T1C1, 2HaO, (c) KJnCl. 2H,0, (d) Rb.TlBr, 2HaO, («) K.In Br, 2H,0, was investigated, and full crystallographical details are given for all the salts except K3InBr, 2H,0, which effloresces so readily that exact measurements are impossible. It had formerly been considered were isomorphous. The investigation has shown that all the salts of this isomorphous series have two molecules of water of crystallisation. 2. The rhombic bipyramidal series, consisting of the salts (a) RbsTlCl6 H,0, (6) Cs.TlCl, H,0, (c) (NH4)JnCls H ,O, (d) RbJnCl, H ,0, («) Cs.InCl, H,0, (/) (NH4)aInBr6 H,0, (g) Rb,InBr5 H,0, (h) Cs,InBr6 H20, was investi- gated. Of these, the indium salts — with the possible exception of (NH4),IuCl, HaO — were prepared for the first time, and quantitative analyses are appended. The series is isomorphous, and crystallo- graphical details are given in full, except in the case of (NH4)aInBrs HaO, which is very deliquescent. 3. The following salts were found to crystallise in cubes in the regular system :— KTlBr, 2HaO, (NH4)TlBr4 2H,0, RbTlBr, H,0, andCsTlBr,. Of these the first two had formerly been taken as rhombic. Details of the crystallography of the salt K,TlaBr, 3HaO, which crystallises in the ditetragonal bipyramidal class, are also appended. 4. The results of the investigation on the isomorphous relationships in the above-mentioned series, and the additional salts KaFeCl5 HaO and (NH4),FeClt HjO (which are isomorphous with the second series), may be summarised as follows : — (a) Crystal Habit — The alkalies stand in the following order : — NH4, Rb, Cs. Cl and Br are very closely related. The salts of Fe, In, Tl differ widely from each other. CRYSTALLOGRAPHICAL RELATIONSHIP 269 (6) Interfacial Angles — For the alkalies the order is : — Rb, NH4) Cs, with NH, very near Rb. The interval Cl — Br is of the same order of magnitude as the interval Rb — Cs. The greatest change in interfacial angles is obtained by replacement within the group Fe, In, Tl; the effect is roughly proportional to the change in atomic weight. The effect of the replacement In > Tl is opposite in sense to that of NH4 > Cs, Cl > Br, Fe>In,orRb>NH4. (c) Molecular Volumes, Axial Constants, and Molecular Distance Ratios — In regard to molecular volume and molecular distance ratios, the alkalies stand in the following order: — K, NH1( Rb, Cs, with NH« near to Rb. Replacement affects mainly the x and $ values. Substi- tution of Br for Cl causes an especially large extension along the three axial directions, a fact which probably indicates a symmetrical disposi- tion of the halogen atoms in the molecule. Within the group Fe, In, Tl, replacement has only a small effect on the molecular volume and molecular distance ratios, the effect being seen mainly in the