ht ise RL TT ga ON A CERTAIN CLASS OF GROUPS OF TRANSFORMATIONS IN SPACE OF THREE DIMENSIONS, INAUGURAL DISSERTATION FOR {pieOw DEGREE DOCTOR OF PHILOSOPHY OUR IS UNIVERSITY OF J,EIPZIG. SUBMITTED BY HANS FREDERIK BLICHFELDT LEIPZIG, JUNE 13, 1808. ¥ f : ; mate ee + sp ; é Lf “ Qn; F) ‘Sy i * . Bs . * } : . ay eo: : : 4 e b * Press of NEEDHAM BROTHERS Printers Stationers BERKELEY, CALIFORNIA. * + . i= ‘ ‘ h } ‘ : = . rs + i= i F Ps ; wall pe) ae a3 . y CONTENTS. Introduction, definitions Ate BS 48%. . The real groups similar to the eight-membered ye) By General properties and classification of the groups The eight-membered groups sroups . . - groups - - - . The seven-membered groups . The six-membered groups . The real-primitive groups in three variables . Some geometrical properties of the eight-membered PAGE. a Hy } 5 he A hoe eh, Sei i } Pea ty got ta “age cam The fundamental axioms characterizing the Euclidean and the Non-Euclidean Geonietries have been thoroughly investigated by Professor Lie in two articles, entitled ‘‘Ueber die Grundlagen der Geometrie,” pp. 284-321 and pp. 355-418 of ‘‘ Leipziger Berichte” for 1890. He shows that a certain class of Continuous Groups of Point-trans- formations is intimately connected with this question, and determines such groups tor space of three dimensions ; the groups being defined by the following properties: two points have one, and only one, invariant ; s > 2 points have no invariants independent of such two-point invariants. Following the plan of these articles of Prof. Lie, the writer proposes to determine and investigate some of the finite groups tn space of three dimensions, for which not less than three points possess invariants, and for which s points, s > 3, have no invariants independent of such three-point invariants. We shall consider only those groups whose transformations are analytical functions of the co-ordinates x, y, z, as well as of the occurring parameters. * T) ‘‘Berichte ueber die Verhandlungen der Koeniglich Saechsische Gesellschaft der Wissenschaften zu Leipzig. Mathematisch- Physische Classe.’’ Vol. 42. 2) To this class of groups belongs the group of Euclidean Motions and Similar Transformations : DMM tain We Sh yees we. OF mane sae Meeiigiy OX ner OR OY J Ox Cae o7 OF ee Bey eee Two points have no invariants for this group, but three points have two, namely, two apgles of the plane triangle which has the three points as vertices. Sah The notation is that of Prof. Lie. A group formed by p independent infinitesimal transformations Xf = 8 (a, 9,2) Ltn oy, at +6 (2 Kaye we shall call a o membered group. ‘The term ‘‘ Bahncurve’ we shall translate by ‘‘ pathcurve.” By the ‘‘ combining” of two infinitesimal transformations X;f and X,/ we shall mean the forming of the ‘‘ bracket-expression”’ (Klammeraus- druck’’): (XS; Yaa) =e OU ae X Gye A. GENERAL PROPERTIES AND . CLASSIFICA~ TION OF THE GROUPS. The number of independent three-point invariants for any one of the groups considered can not be greater than three, as is easily seen from the following consideration. Let the ¢ infinitesimal transformations ; oF Xf ee Gi (ney, =a) ve ae th rn z) a see % (x, 4, 2) 5 Se as) sO) ee form a group with the required properties. We have then p (Ae hn) 2 Cis Aa Jie alee oe 1 oe Cx. being a constant. Theinvariants of the three points 2,,/y,, 2,4, Vj, 2) 0a g, are the solutions common to the partial differential equa- tions We fa XO PE XO f+ XO f= ol i =, 2, ee where Age i (ea, Vny By ee. 1); (a Dny a) of OVn Beer (ay Vay 20) Ue Hae Ory, Zn Now, were there more than three independent solutions, the number of independent equations would be less than six. In this case then, there would certainly be less than six independent equations of the system ee XO fA X, OF So ees a: written in the six variables 7,, y,,.2,; %,, y,, z,. Now, these equations form a complete system of partial differential egua- tions, 1. e., for every value of the subscripts 7 and &, igs oleae aus fi i (Wis WS) They would accordingly have at least one common solution, which would be an invariant for the two points 1, Vy) 25 Uy, Vy 2%) 5 —Coutrary to the hypothesis that not less than three points should possess invariants. The assump- tion that any one of the groups considered has more than three invariants for three points is therefore false. Hence: The groups under consideration fall into three classes, according as the number of the three-point invariants ts one, two or three. There being no invariant relations for less than three points, the groups must be two foid transitive. There will therefore be three conditions imposed upon the indepen- dent parameters of such a group to fix one erbitrary point in space, and stx conditions to fix two arbitrary points. SEs aad Consider now the first class of groups, where three points are connected by one invariant relation, say Zi (7, Jiy 2 ) Vos Vay zy ; Vs, V3) 2.) —— constant 5 Hs, Vis 25 Vas Vos %n3 Te, Vos 2% DEIN the co-ordmatessocmmrae three points. If by the general transformation of such a group, the co- ordinates +7,,.,,°2, 3°", etc: take the values’ 6.’ (oy, Gee etc., we have the relation vi (",, Vy 43 %ay etc. ) = fats vs Ce — etea), or, as we shall for brevity write it, L553 223)3) ee) ee BMD Fi ik | Let us now fix two arbitrary points, as %,, y,, 2,3 2, Vo).% 3 [foots ie ee am Be a yl ee so that @, == 7, == 9,32, = £6 eee The number of independent parameters of the group is thereby reduced by stx, as we have seen above. The relation (1 ) now becomes LEGER 8) ToT ee 3) ES a ee eine ee ea showing that any, other point as +,, ¥,, 2, in space is re- stricted to move on a surface. Two conditions more im- posed upon the parameters of the group will therefore fix this point. The equation ( 2 ) then becomes the identity ER Mak aw a oe ae The motion of any other point as 2,, y,, 2, is now re- stricted by the relations Jil Dead) oa oN aR Rp 2, 4’), alia, 3) 4) = hoo. a 4’), Pay3,4) = 2 (203, 49s ee Oe If it is impossible to eliminate x,, y,, z, from these equa- tions, then this point x,, y,, 2, cannot move in a continuous manner, and we shall regard it as being fully fixed. In this case therefore, the group is reduced to the identical PeaMaioeiialOn et 2), =), 2, ==2,,' by Imposing upon its parameters 6 + 2 = 8conditions. Zhe group ts accordingly eight-membered. | On the other hand, if it is possible to eliminate x,, y,, z,, from the three equations (3), but not from any two of them, as. Wee yom, (1,2 4") 3-2 (1, 3, 4) = 7 A 33 4), then this point x,, y, 2, can still move on a curve. One condition more will therefore suffice to fix it, and it is evi- dent that any fifth point x,, y,, z, being bound by the independent relations Pe 25) 4115 24/5), /:(1, 3) 5)=2Z (13,5), Were, 5) = / (1, 4,.5') would be fixed also. This group is therefore 6+24-1=09- membered. 5) In the case where only one of the equations (3), as eo) =) ie, 4), iS. independent, itpis, clear, that dfter fixing any number of points as %,, y,, 2; etc.; ey. Vers Z,,, the relations restricting the motion of any other point Ky Ves By Mee a Ca, UNK), A) 0 Sarl ee eed would be equivalent to only one, Uae) =n oeienl so that this point +, x, 2 would still have two degrees of freedom. But, as we are only considering fzzte groups, 7.e. groups with a finite number of arbitrary parameters, we easily see that this case must be excluded. —-[o-— A group belonging to the first class ts thus produced by either nine or eight infinitestmal transformations. Consider now the second class of groups. Here a group has two invariants for three points, say vi (V5 ays Xo; Zo} V3, V35 24) VAR Oar 21,4 Coy y 2) Bos X3, Voy 2.) which we shall for brevity denote by Z (1,2, 3) are 3) Four, points%,, 9), 2,3 ->.'.3 4, 2, have theming 7 (1, 2,3), 7(1, 2,4), 21, 3,4), 7 (2, 3,4), ) J (1, 2, 3), J (t 2,4), J, 3, 4), J (2, 3,4), of which at least the following £'(1,52,-3)) Ps 2y-4) ef (L230 Ly ee) oe rr would be independent. (4) Now, could the remaining invariants of the system (4) be obtained from these, all the invariants of s points would be obtainable from. 7 (1, 2, 7), \/ (is 2.2). 4—=3, 4, rr That is, we should have but 2s — 4 independent invariants of s points; in other words, there would be only 2s—4 independent solutions of the system of partial differential equations We f= ROP pK Of rete ee 1 ST SR an eS where a eee (tate ee erry aay 9 a 2 ) 1) n 1 ny ns Bn dX, o = ‘¢ ea vny Ynys Zn): 26 21 X, f, (== 1; 2) + 14) Pp being ‘the: anfinitesimal): transis tion of the group considered. ‘Taking s sufficiently large, the equations of this system would be independent of each other, and their number po would accordingly be 3s — (2s—4) s+ 4. The group having a defincte number of parameters, we conclude that the system of invariants (5) is not sufficient to determine all the invariants of the four points. We must add to the system at least one invariant more, say /(1, 3,4). Then the following invariants of Ss points : nears) ye /-( Ly 2 753.0 2 (ary Ofna me MTD ve it Ine auege be i as A in all 35—- 7, are easily seen to be independent of each other, and therefore Pi Sn8)S (3 San 7)) et 7 Moreover, since three points have two invariants, Pee Sie 357 276 Ol Pee. The groups of the second class have thus seven tnfinttest- mal transformations that are independent. We come now to the groups of the third class. Let us denote the three independent invariants in the three points PO VE Ao kah i ceee ey a eae by 2) (tore. 3), Pi Aie ee ewe my 2 eis s points have plainly the following independent invariants: TAD? NOT iit) thoy Th) Seto Us geil) sn nadie oh A gos, 6, 95; the number of which is 3s—6. Hence p = 3s — (3s—6), pe tO: Moreover, three points have three independent invariants, and therefore » = 3.33, or p = 6. The groups of the third class are accordingly six-membered, _ We saw that the groups of the first class are either nine- or eight-membered. In the first case, let Va BE yo Sj (ae; J) 2) ee = eth (A Vs z) 3, have no in- variants independent of such three-point invariants ; —no three mnjinttesimal transformations can have the same pathcurves.” Suppose there were three such tranformations, X,/, X,/, X,/, so that X, f=, (1y2,) HS ASHP; (4,2) GS The partial differential equation r ¥ of of (Laat of = —— Cc. 4 Sass ) — ——- J — Oe es E(2I,2) i (x,y, aes r Sila, 9,2) 0 has two independent solutions, wz, v, say. Let us change the independent variables in the group to x,w,v. Then Xf, X,fand X, f will take the forms Me ST ae ale cd er it, v) ox’ L, 0x Ys Ox Meniuvatialis Oimime three: pOomts 144. °24,, Vy jo vy, dace X,, U,, V, are determined as the solutions of a number of differential equations, among which we should find the following : 1) See page 371 of ‘‘ Leipziger Berichte ’’ for 18go. 2) It will be noticed that, in the general case where a group inz variables has no invariauts for less than m points, the invariants of s >m points being all dependent on the m-poliit luvariants, 20 m in- finitesimal transformations can have the same pathcurves. The proof would _be similar to that given above. — {1 6h— be ip, a + 90h Brae we=0 > adie ee eee Oar sage " ot Mor (3) af —— | p,. Bert a ERA, ean J The determinant 2 + #,'?. 7%, .¢,° could not vanish unless the relation a + 8.09 + 7G? = 0 were satisfied, the co-efficients a, f, y, being functions of Gh es as ] es: 1) Hn Uy) Uy} Agy Uy Vqy Oly. ~ Phe quantities 200 ers aaa are, however, functions of +,,z,, v,, only, and accceainee independent of x,, 7,3; 23, %¢s,V, Hence a, 6, y should, be constants; but this would lead to the relation aX, f A: A Agha contrary to the implicit assumption that X,/, X,/, X,/f are independent infinitesimal transformations. Since therefore the determinant 2 + o,% 7,° ,® is not - zero, the equations (7) are equivalent to the system ) | ) somo. =o, = =o. The itivariants in the three points %,, 4, 0, ; 4.7 2X4, U,, V, would accordingly be free from the variables 1,, x,, %,. Now, the invariants of s points, s > 3, are all, according to hypothesis, made up of the three-point invariants, and would thus be free from x,, %,,...., 2. Among the differ- ential equations defining these invariants we should then necessarily find the following : 3 cs ap aa eae os Dei ecen coe eat tea s independent equations in all. But, the group considered being finite, containing p infinitesimal transformations, say, it could not give rise to more than p independent differential equations in any nuiimber of points. By choosing s suffi- ciently great, we see the impossibility of the system (8), and our assumption that there could be more than two in- finitesimal transformations with the same pathcurves is contradicted. We now go back to the nine-membered groups of the form fas A (x, 9) E+ Nea MRERCLE oipeke Ane No three infinitesimal transformations may have the same pathcurves, and all the nine differential equations ee hee fs Xf Gy fe, must not be independent of each other. Referring to the work by Prof. Lie” on groups in three variables, we find without difficulty that no nine-membered groups of the required form comply with the conditions given above. There are no nine-membered primitive groups in three variables.”) Thus we see, that no nine-membered groups satisfy our requirements, and the three classes of groups are therefore respectively exght-membered,; one invariant for three points, seven-membered, two invariants for three potnts; six-membered; three tnvariants for three points. 1) ‘‘Theorie der Transformationsgruppen,’’ Vol. III, Chap. 8. 2) See Vol. III, Chap. 7 of ‘‘Theorie der Transformations- gruppen.’’ Prof. Lie had already given the primitive groups in “Archiv for Mathematik,’’ Vol. 10, 1885. —T3— B. THE EIGHT-MEMBERED GROUPS. There are no primitive, eight-membered groups in space of three dimensions.” If a group belonging to this class is of the form KSEE (Lt mle nL Gay, De Uma Yo 2, SRA Mah the shortened group Paes aees must be three-membered, otherwise we should have one or more invariants in the three points 7,,,,2,3 Xo. V2.3 Xs) Var 2 involving x,, x,, x, only; namely the solutions of the differ- ential equations Wife E(x) + &, ( 4) + Ela) ee =o, 2—=T1,.. The invariants of s points would therefore contain only the co-ordinates. 4,,:2%,5..: +). 4% But) then the dittemesacs equations determining these invariants would be composed at least of the following : 2 ae a ki Of Ot Ueno Bye By are he, OF Aote e in all 2s, This is impossible, since the total number of these equations can not be more than 8. 1) See Vol. III, Chap. 7 of ‘‘Theorie der Transformations- gruppen.’’ . All groups of the form wo ; of ne ASE &i (1,9) S-+ mx) talane, is to ee aie and the shortened group XH ale + n(x, n¥ being primitive, are, as before menuoHea: already deter- mined, as are also the groups having a single imvariant sys- tem of surfaces, += constant, but no invariant system of curves of the form + —constant, y= constant. Only the following two types are eight-membered and have no three infinitesimal transformations with the same pathcurves : Bik eh ition eee 22? P27, xp ya, Ce ety eee Aah AD 3% feel yt ae B P) G7, 9; ADIGA YP, ed =, ep eg x IP 7 A/T. (Here we have for shortness written /, g, 7, in place of f of of O07, Oz, respectively, which we shall largely do in the following analysis. ) It remains now for us to determine the groups having an invariant system of surfaces x=constant, and having be- sides an invariant system of curves of the form +—constant, y=constant. Such groups will be of the form. Xf= &(a)p + mx,v)9 ae A Vie, eT 2, = aS ——70)—-— We have seen that the shortened group must be at least three-membered (p. 18). Moreover, the shortened group X f= El(a)p + n(xy)¢ must be at least six-membered; otherwise we would have three infinitesimal transformations of the form (a y,2)7, which would have the same pathcurves. Now, all groups in two variables x, y, have been deter- mined by Professor Lie.”» For the required shortened groups iy (a= Ei(x)p —|- nlx, V)9, eS Lye eis oem 6, eS we find the following types: The group Xf eight-membered. q 9, x9, xg, x'g, p, xp 2V9, xP > 4xy¢. Gi XQ; BQ)" KO, 99; Pr Zep 3g, ee The group Xf — seven-membered. G3 XG, 29, XG, Pp; 2XP 7 399, Gp 1 310g. G49) BINT, PP VG ee en ae The group X:f six-membered. G5 KG = iG SP; AP AYO, (ape we yg G XQ, IQ Pp, 2XP + 9, XP + XV. INGE I a Pata, aot 1) ‘‘Mathematische Annalen,” Vol. 16. —- 2 ik —_— The eight-membered group Xf. The shortened group X, / has here either of the forms: TAG ed, OG Pip 2y9, 2p -- 4x9; Bee edt, V7 Py 2rP + avg, 1p + 3cy¢. We have now to determine functions QP, eter on x, y, 2, so that the two sets of infinitesimal transformations: Xi f= 9+ ar, f= x9 Pr, Kf xg + yr, AG =aAg + 07, X, f= p &, Xef = 2xp + avg + ir, XSEH XP + 40g Ur, Xe f= xg + Ir; MSH qt ar, Kf x9 fr, XY f= xg + yr, X f= x09 + Or, Xf=p + tr, Xi f= 2xp + 3ya + 1, Xi f= xp + 3xvg Mr, Xf yg Ar; form groups; 7. é. the relations 8 XS, aay) = 2s Paisthel pies BeSRR Shee of 8, 1 where ¢,,, is a constant, must be satisfied. - Both groups have infinitesimal transformations of the form gtar, x¢q+ Br, eg+t yr, x¢+ or, p+ e&, 2xp + kyg +, p+ kaya + ur, k = 4, or 3. For the variable 2 we may substitute any tunction of x, y, 2, provided this is not a function of x and y merely. It is then plain that by the substitution of 2’ for z, where 2’ = P (4, y, 2) is a solution of the differential equation we change p+ & into f, without at the same time altering the form of the other infinitesimal transformations. Having done this we write z for 2’. Then from i gt eee we see that Oa ae Ox ana? since we can have no- infinitesimal transformations of the form or. , Thus g+ av=q+/A(%2)r. We can now find a substitution 2’ = @, (y, 2) (where @, is not a function of y merely), which will change g + /, 7 into g and leave f un- altered. We then drop the prime, and we have thus deter- mined the infinitesimal transformations Pp: q. Combining these in turn with +g + fr we get a 33 (p, XG, = OC) 7 Oras erie 6 eatin hy Hence OG Bet Ore HeeOy. Sakae so that +g + fr 1s of the form Bg ae he Oe By a change in z alone we can cause f(z) to become a constant, a, say, which is either o or 1. (In the following analysis we shall denote constants by the letters a, 4, c, etc.) Again, from the bracket expressions Py 24D IG OT) = fp ae, 02 Gy 24p A RG 4 7) = kg sy me (49 + ar, 2xp + kyg + 1) = (k— 2)ag + & : : | as yr, we find that 1: = /,(z), when @ = 0; and 1 = (£—2) z+ ¢, when a = 1. Let a=o. In this case we have obtained the infinitesi- mal transformations Pr % %9, 2xp + kyg + flzyr. Combining g with 2° + kayg + ur we get : (Qa Pp i RAVG + ur) == kag + care Hence, ; GU Te ceo O. Then from 0% 7 (19, 2) + kayg + ur) = (k—-1) 29 1 ry ae we obtain the infinitesimal transformation +g. Now we have three of these with the same pathcurves, Peet Ligs which is not permissible. [he case a = o is thus excluded. Sieg 2 Next, let a— 1. In this case we have reduced to simplest form the infinitesimal transformations BG 9 11, 2xb + kyg + R+ 2) e+ 0/7. Since £ is not equal to 2, we can change 2 so that ¢c disappears. Let us combine /, g, and 2x%p + kyg + (k — 2)zr in turn with 274 + kayg + ur. ‘The results are 2xp + kyg + oy, sae +- ve 2x°p +- 2kayg tp Dats b/s 6 t Gas 4 ys +k 2G oe Each of these infinitesimal transformations must be of the form l(aqg-+ 7) + m (2xp + kyg + (k — 2) 2r) +n (xp +t keyg + 47). Hence nu — (k ote Rye Then from the bracket expressions ag 7, XP + kayg - ((k — 2) x2 + ky) 7) = (k — 1) (#9 + 247), (x'q + 2nr, x'p + kayg + ((k — 2) xz-+ ky) 1) = (k— 2) 9 327), (x"q + 3¥'7, XP kayg + (Rk — 2) ay + hy) 7) = (k — 3) (9g + 42°7), we get the remaining infinitesimal transformations of the group where & = 4, which is thus given by: py g, xgtr, 2g + ear, 89+ 327, x'¢g + 427, | ap -- 2yq + ar, P+ 4nyg + (2e2- 4y)r. | In the case of the group where &£ = 3 we have yet to. determine A in yg + Ar. From or (PEUR An) eT; oA @, 99 + ANv=ot+ ie we find that A = f(z). | Then from 3 (ag +7, v9 + fr) = x9 + ey (xp + ay9 + er, 9 + fr) = CE — for. we finally see that /,(z) = z. The group has therefore the form: Pd, xa +47, ¢ + 2xr, x2¢ + 327, vq + 27, 2xp + 3y9 + ar, xP + 3xyq (eZ + 39) 1. The seven-membered group Nh The types of the shortened PrOUp TAI fe U el, Bo hairs) are here the following: Q XG XG, ©, P, 2XP + 3IG XP + 3479; 9, ~9, 9, V9, p, 2XP + 2g, KP + 2x99. Besides adding to each of these infinitesimal transforma- eae ae tious of the shortened group a term @; (4%, y, 2) 7, we must add an infinitesimal transformation Av in each case to com- plete the groups. We can evidently put A = 1, which is equivalent to changing z. The groups are thus of the form KIN, F Apne? Sepa + Now, from (7, Xf + Pr) ae we see that 9; (4, y, 2) =az + VY (x, »). ‘Then from (Xft faz+ Wir, Xft faz + Win == (Ko, Xf) + Mey, we easily see that the infinitesimal transformations of the ‘‘derived groups’”’ do not contain z in the coefficients of 7. The derived groups of the above-written shortened groups Rife Tao hag op a heve an common the infinitesimal transformations BP 7; %9, #9, 2xp + kyq, 6 + kxyg, k = 2, or 3. The complete groups have therefore in common infinitesi- mal transformations of the form Porar, g+ fr, xq+ yr, e¢g+ 6r, 2xp + kyg + &, Xp A kxyg@ iy rs hk 2, OF == 3 the functions a, /, etc., not containing 2. We can at once put a=o. Combining then f# with q + Br we get ; 07 (p. q+ 67) = Er, which shows that asa ihe pcre so that p Se + f(y). By a change in 2 of. the form z= 2’-+ F(y) we can cause f(y) to disappear without at the same time altering the form of the infinitesimal transformations other than g + fr. , We now have the‘infinitesimal transformations putting 2 for 2’) By G+ axr. The groups have now the forms Yr, p, g-axr, xq+yr, q+ 67, 2xp+ 3r¢g + &, xD oa 3XVG +e 17’, KG + 8 CORO a ieee Veg Or, .2np apg fF, wD 209g + 1, YE UT; where y, 0, € and z do not contain 2. Then by forming bracket expressions, etc., we find with- out much difficulty that the following group only has no three infinitesimal transformations with the same path- curves : | | E LEP GOATS AE OIG A ei sep a VG, xp + 2kyq yr hee The six-membered group X,f. The different types of the group Xf, 7 = 1, 2, ...-. Os are as follows: FAQ 49, Py 22 pc IG Pr Oey: 9X9, IQ P, XPT VG *P 1 IG; I VG: I Uo: Pr AP EL: Each of the complete groups we are seeking will contain two infinitesimal transformations of the form a(1%,y,z)r, f(x,y,2)r. | We can evidently put 6 = 1 at once. If (7, ar) does not vanish, then, according to Professor Lie, we can so choose the variable z that these two infinitesimal transfor- mations take the forms 7, 27. Combining them with any other infinitesimal transfor- mation &(x)p ae MXV)9 a o(4,9,2)7 of the group we get e (1, \G(2)) + eo SC 2 a er, Che E(x)p ae MXI)G ne C(4,y,2)7) ce (eS a o)7, which gives us the equations oO Taira d€ =a + bz, 2— —C =a + vz. OZ Hence, € = 4 -+ #2, which result, for our purpose, is | _ equivalent to € = 0, since we already have the two infini- tesimal transformations 7, 27. We therefore simply add 7 and zr to each of the shortened groups X; PI oat TD haa sacs , 6; but then it is easily seen that in each case there would be three infinitesimal transforma- tions with the same pathcurves. We need therefore only consider the cases where Tar keaa Oo: so.that a= f, Gr, 9). Let Xi f+ —, (*, v, 2)r be any infinitesimal transforma- tion of either of the groups we are considering, the corre- sponding infinitesimal transformation of the shortened group being X,f Combining this with + and fA (4, y)r successively we get OD | OD, ener og Y) (ise eR X;fi)r. Hence, i , , dD ert ee ea — Xf = A’ a mee SE giving Eee Rh = a+ i+ of Two of the shortened groups Bye a 1 a eas , 6, contain g, xg. Accordingly we should have in those cases Yea, + bf + Geka wet a bhatt from which equations it would follow at once that (. would be a function of x alone. The same two shortened groups contain ~, 2%p + yg, Moa LOT 2. Therefore, a a 4 fp eft sgh — at BAL Cif By means of these equations we find without difficulty that : ‘ ee Now change z into whereby 7 is changed into (m«-—+ x)r. By this change we would get the infinitesimal transformations (mx -+ z)r, (hx + l)r, instead of hx el * i mx + 2 m These we can replace by the two, Pea since they must be independent. The shortened group not yet considered is g, yg, y’¢, p, xp, x*p. From the equations OF, 2 Off 2 E : By TST Cal 9 D5 ore ae aa we get at once He ans l f= Ke, rps ere Again, from Fama + A+ eff, 52 = A+ BLA CR we get fe eee we Sine 7 pea fk It is plainly sufficient to consider only one of these cases, since we get the other by interchanging x and y. We shall therefore, for the two infinitesimal transforma- trons, av, fr, use the two 7, xr. Two of the groups we wish to determine have in common infinitesimal transformations of the form Lely PA V7, Gi} Ov, oag 4A ery, BAD TT RVG aS OY, 2p + kaxyg + ur; k= 2, 0r = 1; while the remaining group is of the form r RIDES ME, PO, WG a ap yD TM AMES ame By combining any one of these infinitesimal transforma- tions with 7 and «7 we find without difficulty that the functions y, 0, €,1, and A are of the form az + @ (x, y); and that x= (6+ 4)2+ 4% (x, vy). Then by building bracket-expressions, and by choosing suitable functions z+ f (x, y) to replace 2, we finally obtain one group of the form required, namely: | vr, xr, p, 9, x9, v9 + (ay + bz)r, a (6 — 1) = 0, 2xp + yq+2r, xb + xyqg + x2zr; bis not equal too. | The eight-membered groups are now completely deter- mined. We have the following types: Pp: gq, +9 = Y, Xp Sh oe ae Es yp — eases xp = VQ; vp + xyqg+ (y — xa)r, ayp ty’¢ + 2 ly — ¥a)r. Te | Pp» J, +79; xp paras 6 Di,

3, are not independent of such three-point invariants. The first requirement is fulfilled by any one of the groups Myf GAO Yep a Mikes WC oi ieee Dian Cae a seat if the eight linear partial differential equations Xf Ay f DG Sag jae 0, 2= E325. ieee yo where XP = Er, 16) + nena) + Gn Set a are all independent. We readily find that they are inde- pendent in each case. We shall now consider the second requirement. Let us for brevity denote the invariant of the three points ,, 1,, 2,; Ay Sok Ent Hes Voy Ze BY L (a, b,'c}. If the determinant d/(1,2,4) d/(1,3,4) 0/(2,3,4) dle OX, : OX, OX, o/(1,2,4) d/(1,3,4) d/(2,3,4) OV OV OVs 0/(1,254) 0Z(13,4) 8/(2,3)4) 024 ; O24 . 02, i were different from zero, then it would be impossible to eliminate the variables +,, 1%, 2, from the invariants Tt. 2A) ee onan), /. (1,3; 4); eenerally speaking, it would be impossible to eliminate »,, y,, 2, from / (1, 2, a), 7 (1, 3,2), 7 (2, 3, @); and therefore the invariants of the system Pp sie) Rsk WAG aire Caaria een (Owe TXD F Urea hy MUL eaed PLCS Jy ke TET S55) ee LO) JE OE ACRE BN alee 58 Uae He tN would be independent of each other. This system contains 3s — 8 invariants. ‘The groups potie -Smetiberedys POInts aa. We eyo 1a, oh a Pe would have just 3s — 8 independent invariants, for which we can consequently take those of the system (9), provided the determinant J is not equal too. A sufficient condition for any one of the groups that it may possess the second property is thus that the determinant must not vanish. The invariants /(1,2,3) for the different groups are as follows: — Group A. TCA YP { Ly, ees the a AGS ie 454 . Ly, ueAy or 2,(x, a x5) | . L”, Mae dais 2,(4, at x;,)] Lys nite Bikes 2, (x, «ie x;) | [4 aot ae 2,(4, nec A))| \ ( [yn TE Deere, Z(4) he x;)| Group B. 7(T,253) 1 (253 4) Cay 0,4) =e 013 2,3) in the case of group 1’; and V G20} + V e308} + Vara) = Niet in the case of group Z when 6 = —1. As four points must have four independent three-point invariants'for the groups required, it is plain that the groups above must be omitted. Thus only the following satisfy the requirements: P,Q *9 XP — IG, VP, APT IGT, XP + IY TAT, IP IG /P BQ *g+7, 2¢o + axr, x9 + 32°77, vq + 27, xp + Ig, UP + 32y9 + (ee + 3y)r. PV UF Ty XQ, BR QAGAT, I eee ae x*h +- 2xXxyg ee P: QQ, 7, XG; AT, 2xp og -}- art ap — XV + Ete : ya + (ay + bz)r; a (6 —1) = 0, bis not equal to 0, ov —1. All the groups 1-4 are therefore such that three points have one invariant ‘within each of the groups, while s points, s>3, have no invariants independent of such three-point invariants. Moreover, any group in space of three dimensions that possesses these properties must be similar to one or other of ais Wace the groups 1-4. We must, however, not forget that we have hitherto regarded x, y, 2, as complex as well as real variables; if we restrict x, y, z, to be real variables, the group-types 1-4 are not sufficient. We have thus yet to determine all real groups similar to the groups I of the variables. 4, by means of imaginary transformations GC THE REAL AZGROU PS “SIMILAR TO.VEEE HIGHT-MEMBERED GROUPS. The principles involved in the solution of this problem are clearly exhibited in the memorable article by Professor Lie, ‘‘Ueber die Grundlagen der Geometrie,” pp. 355-418 of ‘‘ Leipziger Berichte’’ for 1890, before mentioned.” It is of importance for the following to note the systems of surfaces or curves that are invariant within the groups 1-4. We verify without difficulty that the groups 1-3 have one invariant system of curves, viz.: pI “== constant, ? y —— «¢ and that the group 4 has the two systems: ° ‘6 x == constant, } x == constant, | y eee ce \ ) oy when a = 0; if ais not equal to v0, then it has but the one system: + = constant, y — constant. In addition, the groups 2-4 have one invariant system of surfaces, namely: “ ==.constant. 1) See in particular pp. 404-416. Before we proceed to determine all real groups similar to the groups 1-4 by means of imaginary changes of the variables, we shall determine the real groups similar to the group 4, as the result will be used later in this investigation. Let the variables of such a group be denoted by 4, 14, 2%. By means of substitutions of the form is GF, Cag odie 21) sts (ay, Va Bid Vee 2ECe, 2 == ete sae — Pp, Py, etc., being vea/ functions of x,, 1, 2, and z denot- ing as usual |“-1, = the group 4 Pg x9 + 7, tp — V9 — 227, Wp — 27, XP + VG, xp + wyg-+ (y— xz)r, ayp + y'¢ + Ay — x2)r, must be transformed into a real group. Let us fix a real; arbitrary point 2; —="a, 9) ==, eee within the sought group. The number of its independent, infinitesimal transformations will evidently be reduced to five, which will all be real, forming a five-membered group. To this group will correspond the subgroup of the group 4 obtained by fixing the point corresponding to +7, = a, yi = 6,, 2, = ¢, and.determined by the equations a D(a, b,, C1) = ZP,(Q,, On oy ee a, V 7 etc. = b, Fagen el eae a, b, c being, in general, complex numbers. Now, substitutions in the group 4 ot the form we a = a Oe ae, leave that group unaltered, after we drop the primes. For the -poi1t corresponding to7 4, = a), 9. == Bj 12, == may therefore take the following DOH) 0 ius Gh a Ce ees cans Ss AY acces and by fixing this point within the group 4 we obtain the subgroup: De a Ore ee Ti eee Ep apg + (y— 22)r, yp ty gel — x2). This group is thus, by means of the substitutions (10), transformed into a read, five-membered subgroup of the sought group, which subgroup we shall call G’. It is easily verified that the group (11) has one and only one invariant system of surfaces, oe constant. 1 To this system must therefore correspond a real system belonging to G’ (an imaginary system could not belong to a real group unless the conjugate-imaginary system also belonged to that group). Let this system be given by Ji —— = constant, 1 so that we have a relation a ee x raves \Geae jf; being a real or a complete function. Having established this relation, we go back to consider the group 4. It has two invariant systems of curves, pee constant, l 2 == constant, \ Sa ; ’ ph 66 2 66 \ ae re Ane aes iy, Aise= The required group has therefore also two such systems, one of them being given by —— 40— J F A a = constant, 2(%1,.91,21) + 7p. In2) if = constant 25% ares If this system is not a veal system, 7. ¢., if @, is not identical with #(@,), the conjugate-imaginary system ) Albee constant, Y, — z7p,=—= constant, “2 | A would also be invariant, and should therefere correspond to the system 2 == constant, y — x2 = constant. That is, we should necessarily have ah = Ale, y — x2). But this relation, in connection with a Rp gee Bh would lead to the absurdity a = f(2, 7 na XZ), co Consequently, the system (12) can not be an imaginary system, and so we may designate it by © Os == constant, 2+, — constant. ; Hence we infer that the variables x, y are changed into 4, y,. Accordingly, the shortened group f= Ese DID AOA oye Eh 2 ee of the group 4 is changed into a shortened group Tye Gi (41, WP _ am + bh cptd ~ax,+d,’ That is, 2-469 or y oF Gat, be atiSniee S82 oe. Now, a substitution of the form ig Maa Os eye Bs a Teed os dyn en eee bgt ge does not alter the form of the group 4, after we drop the primes. Hence it is obvious that we can put , fees ape ips ME Group 4 has also at least one invariant system of curves, a = constant, y = constant. If this is transformed into a real system, say +, = constant, v, = constant, the shortened group X,f—= &(x)p+ nlx,y)¢, 7 = 1, 2, ae , 6, of group 4 is transformed into a shortened group es Ei (X1)p~r + 1 OX, Vi) 915 (Bree Viggo GO : 6. of the required group. As in the case of the groups previously considered, the group 4 is then a type of all real groups similar to it. On the other hand, the system + = constant, y = con- stant may be transformed into an imaginary system, say a4 == constant,.y, 22) == constant: The system x, == constant, 4; =-"72;"== constant would then also be invariant, and would accordingly correspond to the system : «x = constant, 2 ==constant, ne ar, = Ca Bx,)qy + C6 ae Dx)n, A, B, etc., being constants. This gives (x) = A’+ B's, and the relations (14) are therefore the following: FS Ay. Yrs yi aay), wit a teen ia; I 2 ; , or Cain Nea 8 a Rane seet cit 2); I ys, f ; 4, => 2 The remaining infinitesimal transformations of the group 4 are now without difficulty changed into the following: 201 Pr + NG + AN, XP Ig: Ae, {ie bho te, OF { y,2( —i+6)+ 2401+ b) hin, when we take account of those already determined: 9,, 7, A915 AY, Pi- If now 6 = m + 7m, the infinitesimal transformation fant + 4) + ta(r — 4)} a + {yd (— 1+) +a +d}n breaks up into the two (i + m) (ng ani) + 2 in — nd, 2Cng + 217) Bie 20) eds Vi) 3 which must not be independent, in order that we should not have more than eight infinitesimal transformations. Accordingly, Tis 9707 ab i9e" and, since 4 is not equal to 0, or — I, we may not have // ee Ee Camels The only real group that we have thus obtained other than the groups 1-4, page 36, is the following (omitting the subscripts, and putting c = ay. 1+ m’** | VS Sh Ds Gy KG, 2k PA VG, a erg yO + ar + clzg— yr), xp + xyQ -- xer. The real, eight-membered groups are now completely determined, and we can therefore give the following result: All the real, eight-membered groups satisfying the require- ments given on page 5, are similar to the groups 1-5 by means of a veal transformation of the variables. The two groups 4 and 5 are similar through an imaginary transformation. D. SOME GEOMETRICAL PROPERTIES OF THE EIGHT-MEMBERED GROUPS. Since two points have no invariants, the motion of a point in general position would not in any way be restricted if we fix any given point 7,, 7,, 2,. All the groups being imprimitive, we should, however, expect that certain sys- tems of points or some isolated points would be restrained to less than three degrees of freedom, when a given point is fixed... Thus the points on the line <== 35), 3 =e fixed point being x,, y, 2)), have only one degree of freedom, as they must remain on that line. ‘This follows at once from the fact that all the groups 1-5 have the invariant system of curves (lines) a = constant, > 7 ==: constant; 7, 2it ese curves are interchanged by the transformations of the 2ToUups. Since three points are connected by one invariant rela- tion, a point in general position is free to move only on some one surface, when two points, also in general position, are fixed. This surface we shall with Professor Lie call a Pseudosphere, and the two fixed points we shall call its focz. To every pair of foci corresponds a one-fold infinity of Pseudospheres, given by the equation 1A Mi it Vey 235 XV, &) == constant, I(%, try 213 ete.) being the invariant in the three points Bi ie Ae ey, ye. +( We stppose the two fixed POLBES EO De) °9/5° S15 25,4; So) As the invariant / for the different groups are all given (pp. 34, 35) except in the case of group 5, we can thus with- out difficulty write down the equations of the Pseudospheres. 2g 3 3a Let us now consider group 1. If we use e >” fora new a we change 7 into %<27, and the invariant 7 becomes | Deed a Hs) ON I) Se | | Eee The Pseudosphere Se ie ea Oa Veet Ths (68 is accordingly a plane. When the foci are any two points lying in the plane ax + by —1= a, the Pseudosphere becomes ax + by — 1 = cz, Thus it appears that all Pseudospheres with the same pair of foci form an axial pencil of planes, and any given Pseudosphere has for foci any point-pair in a given plane. The last remark is also evident from the fact that the group is ‘‘systatic,’’” 7. ¢.,-when any point x,, 7), 2) 15 fixed, then is every other point on the line Pe RG AMET TY also fixed, which is seen directly from the infinitesimal transformations of the five-membered group obtained by fix- ing the point 41, "1, 21: (w— 41)9, (4 — 1) — (9¥— Ni), (9 — IP, (4 21)"P + (4 — ma) (9 — NG +(e er, (7 — 11) (¥ — Wn) PY = Id TAY See Therefore, by fixing the two points 11, 1, 21; 4), Vo; 2o5 all the points on both lines 2 =, 9 = 3152 =e y= jy, Yemain invariant; and accordingly also every straight line joining these two lines, since the group is projectivé, as is easily seen. © It is therefore clear that every point in the plane of the EWO? LIES (A ey AY ig : Y= yl, must be fixer when the. Wo, Poms B15. Vax 2 tga ay ae eee eye The Pseudosphere of either of the groups 2 or 3 is a ruled surface of degree 3. “The generators are parallel to the plane + = 0. ‘The foci lie on the Pseudosphere. In the case of group 4 the Pseudosphere is transcendental or of degree 24. We shall accordingly consider the group only for special values of @ and 8. Let 6— 1, anda—o. ‘Then the invariant 7 becomes C21 2) Gi = 4%) — (i — 4%) a = %) Oe 25 2) ar, On EI (xy ny X,) wer On Sg) Gy 7 x,) 1) ‘‘Systatisch.’? See p. 386 of ‘‘ Leipziger Berichte’’ for 1890. Seay eee The Pseudosphere is a plane passing through the two fixed points. Thus it appears that for any given Pseudo- sphere we may take any pair of points on a given line as foci. We should therefore expect that when two points are fixed, any point on the straight line joining them would be restricted to move on that line. ‘This is also evident from the fact that the group is projective. E. THE SEVEN-MEMBERED GROUPS. The groups of the class we shall now consider have two invariants for three points. These groups we can determine in a manner similar to that which we employed in the case of the eight-membered groups. We find, however, a great maty more groups, principally owing to the fact that if a group here has an invariant system of surfaces + = con- stant, so that it is of the form: ee ey, 2g GAY e)Y, 2 SRT, Sy oy Psy the shortened group, et = e(0)p, need not be three-membered, as in the former class of groups. It must, however, be at least two-membered, in order that the two independent three-point invariants may not be functions of x,, x,, x, only. For, if that were the case, the invariants of s points, being, according to defini- tion, made up of the three-point invariants, would all be rreeqiront the VATA Dlesi vip, Vs y06 i c/n 5, Vek Sry Sano Maw es Sen LEMS would lead to a contradiction, as we saw on page 18, the groups being finite. Se a We would find that every group of this class satisfying the definition given on page 5 would be similar to one or other of the types given below. ‘The classification is, as in the case of the eight-membered groups, based upon the geometrical figure left invariant by the transformations of the respective groups. A. Primitive Groups. Here we have the previously mentioned Group of Euclid- ean Motions and Similar Transformations: Ll. 1B, 9, % 2Q— It, 2D: — KV, VP — 29, 2. - Vg ee B. ILmprimitive Groups. a. ' One invariant system of curves: Y Bit IFAT, XO + UXT, tf — IG; IP + VI, xp + vq + 22r. Bx Dy Gy Vy XQ) 4D 99, YP Bi ee Ps XO +7, HQ 2x7, WG 34°, 2xP + 3y9 7 BP, xp 4- 3ayg + (ve + 3y)r. Bo, xg tr, eq + 2xr, yg + 27, xp — 27, xp + 2ayg + 2970. EO; p iS ze 13, 14. -, gq, VY, XQ, 9 a AY, xp “Fig, xp 2xyg + (ax + y)r. Pe Ty AE LIT, IG 227) Bap Vg, xp + «yg + Yey'r. Dy Gielen LO OL BY te VI, ap + kyg + [2(k — 1)2 + ax’*]r, &— 2) a=—o0. LE GI LG AT NT NSE ay)r, xp byr. BW 1% HY X’7, xr* a’g + (%2x*° + ay)r, xp + 399 + 42r. Ps Uy %, x9 + yr, xr, yg + 227, xp — ar. Ty) 27, Pg XT, XY xT, HG (hat + ay), xp + avg + 62r. Two invariant systems of curves: BG, 9+ 1, e+ 227, x°¢ + 32°7, x'¢ + 42°F, ap + kyg + [(# — 1)2 + alr; (k— 1)a = 0, DG xg +7, ag + 2xr, 9 + 34°, 2g + a'r, xp + (sy + )9 + Ge + 5r'yr. oa d ! git Ye tatoo oy ee 7 | , fe eS es se Pp; q; ag+r, xg + 2x7, Ja A ee ‘i 7 Ad ie ae xp — er. a | PD *9 X97, XG + 347, VOT 27, Hp ae PO 29,4 7a 7; xg + 347, xg + 6x°r, a 7 xp a (57 -+ x°q) + (32 at 10x*)r. ogy 18 Bs 9 49, PG +7, 74307, 09 + 6er . xp + keyg + [hk — 2)2 + alr ere Pi 5% xq, xq + ar, Og +327, ae 19. | ae 4 xp + kyg + (A — 1)2 + ax]r; (k — 2)a = 0 DP q; 1%, XQ, xg + i 6 x9 oe By, 20. | ren tp Gy + 49 4 eee 21. DU 1 49, ¥"¢ + xr, yg + er, xp — are : : 22 Dp; g YT, XQ, AY, X " + aZr, xp a Sy oT bs a is not equal to o. hk Re x4, ar, yg azr, xp ed ber; ‘oe 23. a a is not cue) to o. ta ' a 2 | * wh ¥ rope i oe, WO ee eer ie % * y 4 ; . 4 . ae & 4 ee aad jas : i 4 Me s3 hy ; 24. | P: J, 7, XY, AT, VO ae Aa, Xp + (22 ne ai lee | c. A o' invariant systems of curves: Oe 2, g, %, xg, xr, 2xp + yg + 27, b+ xyq + x2r. d. ‘Two invariant systems of surfaces: 20; Pip, py ae PA ad 5 VT, VE I. | 27. | VY, XT, Pp xp, 7, V9, vg TIT | 28. | Ket VF 4 Bi xp aE azZr, J, 2V oe ar, vq + V2r. | We shall not investigate these groups in detail nor shall we attempt to determine the canonical forms of all the real groups similar to them. It might be of interest, however, to find whether there would be any veal primitive groups besides group I (7. é., groups that do not possess any real invariant systems of curves or surfaces), belonging to this class. This problem we shall proceed to solve—in fact, we shall consider the more general problem ‘‘to find all the real-primitive groups in three variables that are not similar to the primitive groups.’’ F, THE REAL-PRIMITIVE. GROUPS IN THREE VARIABLES. A real-primitive group in three variables might leave an imaginary system of surfaces invariant. But it is evident that it must at the same time leave the conjugate-imaginary system invariant. Let then the two systems be represented by x1 + ty, = constant; x, — zy, = constant. Therefore the system of curves Ky + 2y, == 1eonstant, 41)=— 277) == constant; is invariant, which system, being equivalent to x; = constant, 7, = constant, is a real system. ‘The group could therefore not be real- primitive, and we are thus led to the conclusion that a veal-_ primitive group in three variables can have no invariant system of surfaces. The next question to serene is whether such a group can have an invariant system of imaginary curves, say a-—-7z@ = constant, y -+ 26 = constant, so that it would also have the system a-— if = constant, vy — 70 = constant. These two systems could not be identical unless they were real (cf. page 44). Hence, a real-primitive group in three variables must have at least two invariant systems of curves, if any. i ae All imprimitive groups in three variables having no invariant system of surfaces have been determined by Professor Lie.” By examining these we find that one group only has two invariant systems of curves, namely the group P; q) a ay r; Xp fey Gi ees Boe 2"7, Xp + Id; xp + xyg + (y — x2)r, xp + 99 + 2(y — x2)r. We have already examined this group (pp. 38-43) and. found no real-primitive group similar to it. Accordingly: There are no real-primitive groups in space of three dimensions except those that are fully primitive, 7. e€:, have no invariant systems of curves or surfaces, real or complex. We now go back to the seven-membered groups. Ac- cording to the preceding analysis, there are no _ real- primitive groups belonging to this class except group 1 (page 54) and the groups similar to this group. | Group 1: +S GY AOR Ai De zp, ip — x9, xp +99 + er leaves invariant the differential equation of the first order ax’ + dy’ +- dz’ = o, and this one only. In other words, there is a cone of line- elements of the second degree connected with every point x, ¥, 2; which cone in this case is imaginary. But it may be transformed into a real cone and can as such be repre- sented by the equation ax* + dy’ — dz’ =o. 1) ‘‘ Theorie der Transformationsgruppen,’’ Vol. III, chap. 8. The corresponding group: 4 Ba ror +20, 27+ 2p, 29 — Id, Re, xp +99 er ¢ neleneiae to this ee io G. THE SIX- MEMBERED GROUPS. The groups belonging to this Glass are very numerous, being all the groups in three variables for which two points i have no invariants, and for which Ea points: have thre« (cf. page 11). RS ea - There are no primitive groups contained i in ‘this class, and therefore no real-primitive (cf. page 59) Nd 6 1 —— Wi Yeu I, the undersigned, was born at Illerup, Denmark, Janu- ary oth, 1873. I attended the public and private schools of Copenhagen until 1888, in the spring of which year I passed the ‘‘ Almindelige Forberedelses Examen” at the University there. In the fall of the same year I went to the United States of America, where I have resided since. I became a citizen of that country by virtue of my father becoming a citizen while I was yet a minor. After having followed the profession of civil engineering for some time, I entered the Leland Stanford Junior Univer- sity, California, in the fall of 1894. There I devoted myself principally to the study of Mathematics under the guidance of Professors Allardice, Little and Green. The degree of Bachelor of Arts was conferred upon me in the spring of 1896, and that of Master of Arts in the spring of 1897. In the fall of 1897 I entered the. University of Leipzig. Here I have remained since, and have attended lectures given by Professors Lie, Mayer, and Drude; semi- nars by Professors Lie, Mayer, and Engel, and laboratory work in Physics under Professor Wiedemann. ne ba DICHEELDT: