A CATEGORY OF TRANSFORMATION GROUPS IN THREE AND FOUR DIMENSIONS. INAUGURAL DISSERTATION Submitted to the Philosophical Faculty of the University of Leipzig for the Degree of Doctor of Philosophy, BY JOHN VAN ETTEN WESTFALL. ITHACA, N. Y. ANDRUS & CHURCH 1899 VRE ‘Written under the guidance of Professor Sophu at the University of Leipzig. z Day of examination: July 26, 1898. 4 o) INTRODUCTION. Riemann, in his well known publication on the ‘“‘ Hypothesen, welche der Geometrie zu Grunde liegen,” started from his definition of an arc-element in a manifoldness of n dimensions, ds ==> ae i in which the quantity under the radical is always posi- tive and then proceeded to the development of the con- ception of constant measure of curvature, which is per- haps the substantial result of the publication. Helm- holtz on the contrary goes still further back and at- tempts to found Riemann’s assumption. He makes his own assumptions and, from these as a base, he attempts to prove that all the transformations possible under his conditions have the invariant Sod cx Clore, “ze in an article in the “ Leipziger Berichte,” for Oc- tober, 1886, notes inaccuracies in the development and points out that, under one interpretation of Helm- holtz’s axiom of unrestricted motion (freie Beweglichheit), his monodrom axiom is entirely superfluous, while un- der another interpretation, even all the axioms would be insufficient to determine the groups, which preserve the geometric qualities of a rigid body in space. Lie, in a later article in the same publication,’ proves rigidly the 1 Leipziger Berichte, Oct., 1890. isk yaaa truth of his statement made in his article of 1886. He proceeds from Helmholtz’s assumptions, with the excep- tion of one, namely, the monodrom axiom and deter- mines all the groups, satisfying the given conditions. Under the most general interpretation of the axiom of ‘unrestricted motion,” he gets, besides the groups of Euclidian and non-Euclidian motion, five others. Kow- alewski, a pupil of Lie, has in his inaugural disser- tation’ extended Lie’s investigations in space of three dimensions to that of four and five. Besides the Euclid- ian and non-Euclidian groups, he finds three others that satisfy the condition of unrestricted motion. The dis- cussion of these eight groups, or more particularly the one-parametric sub-groups, is the chief aim of this dis- sertation. ; The behavior of points under certain special con- ditions is in some cases most remarkable. Some- times all the points of a surface remain at rest and some- times not, depending upon the choice of the points we hold stationary. In some cases too, we find all the points of a surface invariant wherever we may choose our points. Then, too, in space of four dimensions, we find one group has in some cases closed path-curves and in others not. To give Lie’s development in its entirety is of course out of the question, but at the same time, in order that we may have a proper insight into the groups, a short sketch is necessary. 1 ‘Uber eine Kategorie von Transformationsgruppen einer vierdim- ensionalen Mannigfaltigkeit.’’ 1898. CHAPTERIL SKETCH OF LIE’S DEVELOPMENT OF THE GROUPS SATISFVING THE AXIOMS OF HELMHOLTZ, Lie sums up the Helmholtz axioms in the following words :' Bet Yi 6 OX) WV) oy aie ea Woe (EV Ay ene tec Zp VY NEO ee tne eee be a set of real transformations in space of three dimen- sions under the following conditions : (A) The functions -, ¢, w are analytical functions of the variables and the parameters. (B) Two points shall have one and only one invari- ant in the group. (C) There shall be unrestricted motion in space, that is: the point x, y, z, can be transformed into every other point in space. If we keep x, y, z, fixed, then a second point can take o” positions. Hold two points station- ary, then a third point can take o!' positions. Finally, if we hold three points stationary, then all the points in space remain stationary. (D) If we hold two points stationary and transform the remaining points in all possible ways, then the points, after traversing a finite distance, shall return at the same time to their initial positions, 1 Leipziger Berichte, Oct., 1890, slg sn That this set of transformations under the conditions A, B, C form a group with six parameters is evident. We have then the problem, to find all the six parametric groups in space, which are defined by real analytical equations and for all real points satisfy the conditions Brands: The six equations: 5 8 ers al fh (a Yu Z,) ea Winn e SiG ete b. C5 Yo» 25) mas as Xx, 02, (k = Ge 2. e . Gk can have one and only one solution. ‘Therefore = DS Se OR va or) eet a while the 5 rowed sub-determinants do not all vanish. We can put this criterion in another form. We mul- tiply the W,- by such quantities ¥, ... <2 apaegmee the co-efficients of ig sf, Aa in the expression y¥, W, -+ .+yW, W,f, shall be equal to zero. We obtain three equations in the y’s with co-efficients depending on X1, Yi Z,, which we can solve for-three of the Ws.Guuie other three are indeterminate. We choose three of the many expressions Vif = 2, WF so that no relation 0, (%,Yp%) Vip +o, Vif +o, V,F =O exists, and set 8x, if dy ioe 2 ane as 4. 7») SF aa oF ly The determinant A, —2=+ {7',@, vanishes with A, and the two rowed sub-determinants vanish with the five rowed sub-determinants of A.. The geometric sig- nificance of this is, that a pair of points can not be trans- formed into every other pair but only into o’ pairs, and that is exactly the significance of the condition that two points have one and only one invariant. We have then the following criterion: If three independent infinitesi- mal transformations, Y,¢=ap+Bq+y¥r, of a six- parametric group X,¢ .. . X,F, leave a point invari- ant, then two points have one and only one invariant, when the determinant of the sub-group vanishes identi- cally, while the two rowed sub-determinants do not all vanish. It is evident, that after holding one point x,, y,, z, sta- tionary, all the other points move on o’' invariant sur- faces. In the case of the Euclidian motion, these sur- faces are the oo’ spheres with the centre x,, y,, z, which we will call pseudospheres. In the general case, we will therefore call the o' invariant surfaces pseudo- spheres, with the centre x, y, z, All these satisfy the three partial differential equations, Yer, Gk) eae) only two of which are independent. It is evident that a, B, y are not independent of x,, y,, z,, else there would be only oo’ pseudospheres, which would violate the axiom of unrestricted motion. We see then that there are at least «” pseudospheres in space. It is also evident, that there is not even one partial differential equation NS ge Af =0, whose coefficients are independent of x, y, Z,, which is satisfied by all the pseudospheres in space. It follows easily then that among the infinitesimal trans- formations, no two occur, which satisfy the relation $, Sif + $, X, f= 0. If we hold a point stationary under our six-parametric group, then the line-elements, through the point, will be permuted among themselves by means of a projective group with at the most three parameters. We know also that a projective group in a plane, with not more than three parameters, has either an invariant pcint or an invariant conic. If we regard the co-ordinates of the line-elements dx, dy, dz as homogeneous co-ordinates in a plane, we come to the conclusion that our six-para- metric group leaves either a line-element or a cone in- variant. In the former case our group is imprimitive and leaves a set of co” curves invariant. In the latter case our group is primitive and has the invariant differ- ential equation: a,, dx’ + a,, dy’ + a,,dz’+ 2a,,dx dy pee GyGZ rig o, eda All the six-parametric primitive groups satisfying our conditions have been determined in an earlier publica- tion by Lie and are similar either to the Euclidian or nou-Euclidian groups by means of a real transformation. If the six-parametric group is imprimitive then it can occur that o° surfaces remain invariant and it is in that case easy to prove that the surfaces are interchanged among themselves by means of a three-parametric group. For if we introduce the invariant ¢(x,y,z) as a new variable x in our group, then the coefficients of all the p’s will be simply functions of x and the shortened in- finitesimal transformations €, (x) p form a group in one variable, which transforms the surfaces, at the most, bya three-parametric group. It is easy to see that the short- ened group is neither one or two-parametric, for in the first case there would exist only o' pseudospheres in space and in the second case two of the infinitesimal transformations would have the same path curves, both of which are excluded. We have the result then, that the set of surfaces are transformed by a group with three parameters. We now go back to the case where our group leaves a line-element through a general point invariant. There exists then an invariant simultaneous system, dx dy az dz Bixty zy mn Cxuy, Zz) Ry oneyaae That is: the set of curves determined by the solutions, $ (x,y,z) =¢, and ¥ (x,y,z) =¢, are invariant under our group. If we introduce these in- variants, as new x and y in our group, we obtain, Reh eee Pt HX, ¥) de, Ca) t. The shortened transformations, Vif = & (X,Y) p + 0, (x,y) q form then a group which shows how the o? surfaces are transformed. Ifthe group had less than five par- ameters, there would exist two infinitesimal transforma-_ tions with the same path curves, which is excluded. The set of straight lines is then transformed either five or six-parametrically. Is the group Y,¢ primitive and s1x-parametric, then it has the form Pp; q, xq, >. OF y q, y p. 1 If it is imprimitive, then it has at least an invariant set of curves ¢(x, y)=c. ‘This, considered as an equation in space, represents a set of surfaces which is invariant for the group X,¢ and is transformed three-parametric- ally. ‘The curves ¢ (x, y) =c are therefore transformed in the same way and we have only to find the groups Y, F, which transform the invariant set of curves ¢ (x, y) three- | parametrically. The six parametric groups which con- form to this condition are the following : Dy: Dre ae i aC Le q, Reqhorx Ge pox p -+y ay Sposa ae Lil. yee Geo, a iy Gh. xX Diao ee Tye If the group Y, ¢ is primitive and five-parametric, it has then the following form: P, 4, xq, Xp—yq, YP. V. If imprimitive : PG XG, 2xptyq, x ptxy4q. VI. The groups sought for have the form: ee Vine eee where the Y,¢ have one of the six forms above. By bracket expressions we determine the @’s in all the cases and then examine the groups to see that they fulfil our conditions. The groups, obtained from the shortened groups I and III, do not conform to our conditions and from the others we obtain the four following: emcee sv Cart kor, ie et 2aet wl ¥ dec. 2k v1, Pp 4, xq+r, x'q+2xr LT xptyqtAr, x*pt2xyqt2(Ax+a2y)r. Heavier x T, 2 oT; Bile Peete YG YP: Pp, q, Tf, Ue aan Re rH. Og mane WF Live xp+xyqtayy't. As yet the problem has been solved only under the condition, that the quantities x y z are complex. We wish however to find all the real finite groups which satisfy our conditions. Ifa real group satisfies the con- ditions in regard to the invariants of two or more points, the group obtained by considering the variables x yz as complex quantities will also satisfy the conditions and is therefore six-parametric, andis by a real or complex transformation similar to one of the groups already formed. It is necessary therefore to find all real groups which are similar to the above mentioned groups by imaginary transformation. The groups found similar to II, HI and IV are similar by means of a real substi- tution. Further, all groups similar to I by means of — 14 — real or imaginary transformation are similar by means. | of a real substitution, except the following: Pad ox Deed yp X de NF (x’—y’)pt+2xyqt(2k,x—2my)r 2x y Pp Peer ie ak Viet 2 ee which is similar by means of the substitution X,=xtiy y,=x-ly 4,= fe ; : * k +im in which, both k, and m can not vanish, and in which [ioe TE 0) kc AEE, el We have then this group V to add to our list. Kowalewski by a similar discussion found besides. the Euclidian and non-Euclidian groups in space of four dimensions the following three, which satisfy the condi- Hons. A. Bate Py — ¥%_P3, Pe ar X, Ps, %,P2 %, Py — ¥, Po. %_Pi; X, De tee eRe py oy UO Daa KE ee Vie Pi» Po Ps, %, Po — ¥3 Pi» %2P3 — Xz Pos X3P; — X, Ps, WER: Eh. 2S D7 BU, 2x,U Se Pi, Pr Ps, Xi P2— ¥: Py, X2P3 + X3 Py, Xs Py + X, Psy; U, 2x, bio (xy + so X, ) Pir 2 x,U =e Xai ea Poy 2,00 CS xe ee pe VIII. where A Dy che Daisies se Pata a xy 7K, KS GA Pires ble DISCUSSION OF THE GROUPS IN SPACE OF THREE DIMENSIONS. f Definition.—A general point under a transitive r- parametric group in n-dimensions is one for which all the n-rowed determinants of the matrix do not identi- cally vanish. A general point, under an instransitive group, is one for which the largest non-vanishing deter- minants of the matrix do not all vanish. Definition.—A group is systatic, when by holding a general point fixed, all the points of a certain manifold- ness remain at rest. All of the groups under discussion are systatic, because all the transformations are per- mutable with r. We can see this directly, for by hold- Poem come point. x = y =z —0,..which is” a pen- etal point, all the: points xo y =o zZ==Z remain at rest. ‘There are therefore o' points which can not take Smeposttiouseaiter Yolding xy —az =o xed. On the other hand a general point can do this. GROUP I. Dy, oGsee Der, Vet ekis, x'p-2xr, yqite2kyr. The invariant of two points has the form: [=2,+ Z,— log mao Xo k log ‘Yar Ne) io The three-parametric sub-group obtained by holding the point x sy == 7 =='0 fixed iis : | yq—kxp, x*pt+2xr, yqt2kyr. i The group is of course instransitive and any point, for which both x and y are not equal to zero, is a general point. If we hold the point x,, y,, z,, (X)3= Ojameme fixed the resulting sub-group has the form, (3 Xo Vo X k Yo Bs) p+ (y’—y Yo) q+ (7 z Ao ane RY. =r. c pe 2 The differential equations for determining the path curves are these: dheie: x’ (x ,— x’) : _= | o dt a x, CS meaay enyates ape ae: (y Yoo where [x’],_,—x, etc. We get then from tigse aaa equations and that of the pseudosphere x/= x ek Yot | Nicos aes eYot 12 / ZZ + log * +k log =. x bs However large t may become it is impossible for a point to return again to its initial position. By setting the coefficients of p, q and r equal to zero, we see that the only points remaining at rest are: (0, 0, z) and (X,, Vy Z). If however we choose x, = 0, or y, = 0, which we obvi- ously have a right to do, as in both cases the point is general, we obtain much simpler forms for the path- curves and, instead of all the points of two curves re- maining at rest, all the points of a certain surface re- main stationary. For x, = 0, y # 0, the sub-group has the form xp +2 xT. Here it is obvious that all the points of the surface X =oOremain at rest. The equations of the path-curves are easily found to be Stim? Sf , '=z+2log~. ss Each of the surfaces of the set y = const. is invariant. For y, = 0, X, # 0, the sub-group is yqt2kyr. All the points of the surface y = o remain at rest and the equations of the path-curves are ns y z=—Z2+klog—. Me Fach surface of the set x = const. remains invariant. pone) 18 ke GPOUP II. Dee Cre csr io eae Spey Cee po 2 XK yed ach A The invariant of two points is: l=z+2,- = Aide: (x,— x)’ ae EE xy ae By holding the general point x = y = z = 0 fea get the sub-group xpt+(y—Ax)q, x’q+e2xr, x p+2xyq tayo x) r. Examination of the matrix shows that all points, with the exception of those for which x and y are both equal to zero, are general points. If we hold the general point (0, Yor Zp) Yo # O, Z # O fixed, we obtain the sub-group x’q+2xr. The finite equations are then: x= xy ay Se bee 7 ee Here all points of the surface x = o remain fixed and each surface in the set of surfaces x = const. is in- variant. If on the other hand we take our point outside of the surface x = 0, there exists no surtacesavnen whose points remain at rest. The sub-group, obtained by holding x,, y,, z, (x, # 9, y, * ©) fixed, is: x (x — x,) p+} (Ax—y) pone’ Vt Xo +1 2(y+Ax)— 220.1 Saale x te) and wé see that only the points (0, 0, z) and (x, Yy Z) remain at rest. Grovp III. Rises. 4 fe r, f, So re gee De a ee The invariant of two points is: LS heme 7 Hn wd SA themed 054: The invariant sub-group obtained by holding the gen- Beta OMCs nye 7 OL xediis:: <0 yan a Ve ava HT By examining the matrix we see that all points, except those for which x and y both are equal to zero, are gen- eral points. The one-parametric sub-group, obtained by holding the second point X,, Yo) Z) (X) # O, Y) # O) fixed, is: CAB) jnt (2 Ja Yo Yo Yo from which the finite equations are: x’ =x + *o ese t, Y oe Yo yi =yt(x—%y jt, Yo Z=Z'. Here, in contrast to the other groups that we have examined, all the points of the surface x y,— x,y =o remain at rest. meio) a If we take x, = 0 the sub-group takes the simple form aie with the finite equations : SO eer oy meet Kn ons and the path-curves are parallel to the y-axis. All points in the surface x = o remain at rest. If we take Y, = 0 we get the sub-group x p with path-curves par- allel to the x-axis. In this case we see that, wherever we may choose our general point, all the points of a cer- tain surface are invariant, while with the other two groups that was not the case. Group IV. Dyoid (oe Xs ie tye KD vi, Ro ey Oat ay OTs The invariant of two points is: (Cy, We Vo)e Rt Beers Bras “ 20x) The sub-group determined by holding the general point. x= y = 72 — 0 fixed.is thefollowine: SoC 2x Dis 3G, IY; x*p+xyqttiy’r. If we examine the two-rowed sub-determinants of the matrix we see they do not all vanish except for the values x = 0, y = 0, so that we can choose any point for which both x and y are not at the same time equal ome ORY See to zero. We shall see that in this case wherever we may choose our second point there will be no surface the points of which all remain at rest. The one-para- metric group formed by holding x,, y,, Z)) (X)# 0, y)# ©) fixed has the form : 2x x, 2 x (2 eee ee BL) ( = )p+ )at( pds hi Xp Morex, —O the sub-group is: P(X Vi Vg) eh ee Wyo) it and for y, = 0 the sub-group has the form: ae) x) PE (XY ex yd eye. In each of the three cases the only invariant points are the ones on the two lines which are parallel to the z-axis and pass through the stationary points. If we determine the finite equations of the group, we find that asin all the other cases the path-curves are not closed. In this case the expressions are very complicated, but we need only to determine x to see the truth of the statement. GRovpP V. In this group, as we have seen, both m and k, can not be set equal to zero. If one of them however be set equal to zero, the other can be made equal to one and for convenience of discussion we can write our group ‘in the form : | PN Fis al ney et U, 2xU-—Sp, 2yU—Sq, U=xpt+yaq+ir, S== fy. The invariant of two points is: | y= x) Gi— v2) fev ets) and the sub-group, obtained by holding the general Point = y ==7 no xedus: XQ 12x — Sp, .2y US sig For the second point we may choose any point for which both x and y are not equal to zero. However ingeniously we may choose our points it is impossible to bring the one-parametric sub-group to a very simple form. In four dimensions and in fact in n dimensions, where we find groups analagous to this we shall see that for any value of n > 3, we can always choose our points so that the one-parametric group shall take the form of a rotation. The one-parametric group is in the case aU eos FC pane: = sane | uf oe + vA Xp ty, p 297 XY x 2 2 2x,Y¥— 2y,x +}x- 0 — 8 (x? — ; +( 0 0 yr ie San fein. tg vi yaya x ae from which the finite equations are the following: (x, tiy,) (& +i1y) x +1iy = fx,— x +i(y,—y)}e* +x +iy x? + y” A May abies Oe (gh gt aural Sa A & x’? + y’ However we may choose x, and y,, the only real points 1 Theory of Transformation Groups. Vol. III, p. 459. remaining at rest are those of the straight lines through the points (0,0,0,) and x,, y,,Z). parallel to the z-axis. From the finite equations it is easy to see that the path- curves are closed. Of all the one-parametric sub-groups there is only one, namely, that of group III, which leaves all the points of a certain surface invariant wherever we may choose the same general point x,, y,, z,. On the other hand there are two, namely, those of groups I and II, which have this characteristic, in case we suitably choose our general point x, y,)Z, Finally, there are two which under no conditions leave all the points of a surface invariant. CHAPTER, IL THE GROUPS IN SPACE OF FOUR DIMENSIONS. The only groups in space of four and five dimensions of those determined by Kowalewski, which satisfy the axiom of unrestricted motion in either sense be- sides the Euclidian and non-Euclidian groups, are the 1m- primitive groups we have numbered VI, VII and VIII, and two groups analogous to VI and VII in space of five dimensions. Kowalewski has shown that groups analogous to VI and VII exist in space of n dimensions. Further, from analogy between groups VII and V, he draws the conclusion that the one parametric sub-group, obtained from VII by holding three general points sta- tionary, has closed path-curves. By suitably choosing our general points we shall prove the truth of this state- ment, without being compelled to.resort to the long inte- gration which would be necessary if we chose our points entirely general. | | Group VII. Pir Por Ps, %1 Pp ~ X, Py, *X_ Ps — Xs Py, X3 Pi — X, Ps: i 2x Wp 2x,U —Sp,, 2x, U —Sp,, U=x, PtxPtaPpt+tpyHSs =. + eee The invariant of two points is: J gh Sa y,) I= | (xX, se! Na te Cx: = Vo or (xy he Our six-parametric group obtained by holding the general point P, = (0, 0, 0, 0) fixed is the following : X,P. 7 X, Py, X, Ps — X3P., X3 Py — X, Ps» 2x,U—Sp,, 2x,U—Sp,, 2x,U—Sp,. The matrix is in this case: — x, <7 O oO — x, Se O 2a O — x, fe) x7? — x? — x,’ 2X,xX 2x,x 2x 1 2 3 1 4 1 3 1 2 x%*—x7—x? 2xx 2x xX, X 2 8 1 2 3 2 2 2X,X x, —x7?—-x,” 2x Xs x 3 “2 3 1 2 3 For no values of the variables except x, = 0, x, = 0, x, = 0, do all the three-rowed determinants vanish. If we choose for our second point P, = (0, 0, z,, Z,), where Z, # O, we obtain the sub-group: X, P, — X%. Py, 2 Ut Dae Oe Diam, Dy) s Zp GAB tel Se ig 0) ody Ap b> Sees hu 9 Hor, our third ‘point we may take Py=— (0; 0; a’ ;, 0), where z, # z’, # 0, for in this case the matrix is: O O 10) 1@) — z”,+ 2,2’, O O fe) O — Zz", +.2,2', O O me) 6 in which all the two-rowed determinants do not vanish. We get then our sub-group in the form: X, P, — X_ Py. In space of five dimensions the result is exactly the same and, if we examine the analogous group in n di- mensions, we see that it is always possible to choose our general points so that our one parametric group shall take the form of the rotation: ae he Diy ee oat From this fact alone, however, we can not draw the con- clusion that our sub-group always has closed path-curves, wherever we may choose our three general points, as we shall see in the case of group VIII. Group VIII. Py, Pos Pg, %1 Po —~ Xo Pir Xe Ps 55 X3 Po» xy Pi teks u; 22S pr 2 = pi. 2 Kr Oy. 2 = X) Peete Sy Pst ky Pe Pa S= x + x, ae The invariant of two points is: : : : site (x, ne Yu) I= |v + Gy Gy) be The six parametric sub-group obtained by holding the general, poimt Pi=— (0, 0,°0, 0) fixed is: | X,P.— ¥, Pi, * Ps a X3 Py, Xs Py AP X, Ps, 2x,U—Sp, 2x,U—Sp,. 2%,U toe The matrix is in this case: — x, X; O fe) xe ae O x, Oo eh O x, —x,4+ x,’ rd pg a BEKO Xs 7190 2x, x; xX, — x, +x,’ PENS gb 2X, oie PES ED.D X, +x, +x,’ SS. Ponno wales of, the variables except x, == x, = x, ==.0, do all the three rowed determinants vanish. If we choose for our second point P, = (z, 0, 0, z,), where z,#o we obtain the sub-group: me Pact Xs Ps Dee it) aes CXED eae, P,) Z, Cee Ueat et) ai CS, Dy cts bey Zy. If we choose for our third point P, = (z’,; 0, 0, 0), where z, # z,' # 0, we have the matrix: fe) fe) fe) 12 1 fe) ara AS pig eeY A fe) = 12 / fe) O Come. Dy which shows that our point is general. ‘ We obtain then for our one parametric sub-group: X, Ps + Xs Pa which plainly has not closed path-curves. If, on the other hand, we choose for our second point P, = (0, 0, Z,, Z,), where z, # 0, we get the sub-group: X,P, — X, Pi Ser eal Dy rie carseat) , 2x,U+Sp, + Z, (X2 ps + X; p,). pes J) Further, by holding the point P, = (0, 0, z’,, 0) where Z, # Z', + O, we obtain for the sub-group, the rotation : X, Py — X,_ Pi- We see then that in this case the path-curves are closed, - while in the other they are not. This has its explana- tion in the fact that in general a triplet of points can not be transformed into every other triplet by means of a transformation of the group. Each triplet has three independent invariants; and it is possible to transform one triplet into another only when the three invariants of one are equal to three invariants of the other. If then in one case the path-curves are closed, they will be so in the other and vice versa. | | Let us examine now our group VII and see if it is pos- sible to choose such a point-triplet that our path-curves would not be closed. The special point-triplet we used in determining our sub-group was: P,=(0 fe) O el) P,=(0 O Zi jt) P, = (0 fe) ae O ) The invariants obtained by setting the co-ordinates of these three points in the general form for the invariant of two points are: For all positive values z, z’, and z,, the invariants are positive and never equal to zero. They are further in- dependent with respect to z,, z’, and z,, as we see from the functional determinant. If we choose any real positive values whatever and set them equal to the expressions above, we can solve for z,z’,z, If we set itis 2 2 a 2 —— , 2 n2 2 Z,e=a°, z’,=b’, (z,—2,7;e=ec and solve, we obtain the real values: ee baie a atts b? jae heey A Rey Fath hares f I—c I—c From the form of the general invariant, we see that all triplets of general points have positive invariants which can never be equal to zero. If, therefore, we choose any triplet of points whatever, we can always’ determine Zs Z's, Z, 50 that the invariants of the one triplet will be equal to those of the other and we can therefore trans- form one into the other by means of a transformation of our group. In our special case the path-curves were closed and we therefore know that they are closed in general. The one-parametric sub-group of group VIII has closed path-curves, as we have seen, when we hold fixed the triplet of general points: P,=(0 fe) O O:) PR=(0 0 % 4) P, = (0 O A Ta ‘The invariants obtained from setting these values in the general form of the invariant are: Lin Bie aah esa: ase au gare ye 2) I= Z 3 WAT pt We ape 2 I,, = (Zs AS € The expressions are all negative, and never equal to ~ Further, are independent with respect to these quantities as we zero for the possible values of z,, z’,, z see from their functional determinant. By the same reasoning as before, we see that if we choose any triplet of general points whatever whose invariants are all neg- ative, we can always determine z,, z’,, and z,, so that the invariants of the one triplet will be equal to the invariants of the other and we can therefore transform one into the other by means of a transformation of the group. We have then the result that all groups, obtained by holding stationary a triplet of general points whose inva- riants are all negative, have closed path-curves. The question then arises, whether these are all the sub- groups that have this characteristic. The question can be answered in the affirmative, for we know that group VII is similar to group VIII by means of the transfor- mation x’, 1 x,, and x, p, — x, p, is the only transfor mation of the group which remains invariant under this substitution. The invariants of the three fixed points are transformed into IL,=—-Ze 4 1,=—z,;, L.=—(@ —Z) eae We see then that only those point triplets whose invari- ants are all negative give us groups with closed path- curves. We have now only the group VI to discuss. This is analogous to group III in space of three dimensions and has not closed path-curves. ‘This I have not been able to prove as in the case of the two groups above by an ingenious choice of the general points. We can how- ever so choose our points that the sub-group appears in a very simple form. Group VI. P; — ¥,P3, Pe 2 X,P3, P3, XX, Po, X, Py — X, Py, X%, Pi, Xs Ps =e Xs, pi; — X, pe Xs; Pe a x) Wy; X, U, eae xy Py i Xa De 1 Xs Dette De The sub-group obtained by holding the point (0, 0, 0, 0) fixed is: XP. %,P, ~ %,P2, Pir X3 Py — X, 1 ey Doren Uy a, U As second general point we can choose (0, 0, Z,, z,) where Zz, # O, and we obtain the sub-group X,P,., %, Pi — X_ Py X,y Pi- As third general point we can choose (z,, 0, z,, z,) where z, # 0, and we get the very simple group X, Pi in which the path-curves are plainly not closed and run parallel to the x,-axis. All the points in the manifold- ness X,= 0 remain at rest. Group VI has its analogous group in n dimensions and it is easy to see that by suit- ably choosing our points we can always obtain the one- parametric sub-group in this simple form. It is impos- sible in this case to show by the form of the invariants that the path-curves are in general closed. To do this it is necessary to determine the sub-group that leaves the poiut-triplet :(0, 0, ©,'0), \(Z,, Z.) Zs) 2,), (29) 2 ee invariant and then calculate the finite equations of the group. In that case we would find that analogous to group III all the points of a plain manifoldness would | remain at rest. It is interesting to note that with all the groups, except V and VIII, the invariant of two gen- eral points can vanish without the points coinciding. If we interpret axiom C as meaning that after holding a general point fixed it is possible to define a space about that point, such that every point contained there- in can take o’ positions in space of three dimensions and o* positions in that of four,:then, as Lie simeuie publication of 1890 points out, these eight groups must be excluded and we have only the Euclidian and non- Euclidian groups. If however we give axiom C the in- terpretation that after holding a point stationary a gen- eral point can take o’ positions or o° positions as the case may be, then we must include the groups above. In this case we see that if we call the monodrom axiom to our aid, the groups V and VII would still have to be included. We.=see then that in one case the monodrom axiom is superfluous and in the other insufficient to de- fine the Euclidian and non-Euclidian groups. 1 Leipziger Berichte, 1890. NicD AS I, John VanEtten Westfall, was born on the 24 of June, 1872, in Dresserville, New York. My schooling preparatory for the university I received in Ithaca, New York, at the close of which, I entered Cornell Universi- ty, from which I received the degree of Bachelor of Science in 1895. Directly afterwards I came to Ger- many where I have studied three semesters in Gottingen and three in Leipzig. I have heard lectures under Klein, Hilbert, Schoenflies and Sommerfeld in Gottin- gen; and Lie, Mayer and Volkelt in Leipzig. : ‘ a % 5 * . rar. i § ‘ ; ro : ™ it 4 a , a Be ‘| i 4) . ae) 4s J ®, 7 A dat nent AL a ee