<.'i^_\,^'i'T;>j 
 
 
 ^ii^M 
 
Ube XHniversitp ot Cbicaao 
 
 SOME SPECIAL CASES OF THE FLECNODE 
 
 TRANSFORMATION OF RULED 
 
 SURFACES 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY 
 
 OF THE ODGEN GRADUATE SCHOOL OF SCIENCE 
 
 IN CANDIDACY FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 DEPARTMENT OF MATHEMATICS 
 
 BY 
 
 JOHN WAYNE LASLEY, JR. 
 
 Private Edition, Distributed By 
 
 THE UNIVERSITY OF CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 
 1922 
 
XTbe TaniverBiti? of CblcaQo 
 
 SOME SPECIAL CASES OF THE FLECNODE 
 
 TRANSFORMATION OF RULED 
 
 SURFACES 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY 
 
 OF THE ODGEN GRADUATE SCHOOL OF SCIENCE 
 
 IN CANDIDACY FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 DEPARTMENT OF MATHEMATICS 
 
 BY 
 
 JOHN WAYNE LASLEY, JR. 
 
 Tl 
 
 .O . 
 
 Private Edition, Distributed By 
 
 THE UNIVERSITY OF CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 
 1922 
 
■•• t- 
 
 Li 
 
INTRODUCTION 
 
 The general theory of non-developable ruled surfaces may be 
 made to depend^ upon a system 
 
 (i) 
 
 y"-\-piiy'-\rpi2z'+qiiy+qi2Z = o, 
 2" -\-p2iy'-\-p22z'-\-q2iy+q22Z = o 
 
 of two second order linear homogeneous differential equations, 
 where the coefficients pik, qik are functions of x, and where the 
 strokes indicate differentiation as to x. According to the funda- 
 mental theory of differential equations these equations have a 
 general solution, a pair of functions y and z of x, which can be 
 expressed as linear combinations' with constant coefficients of the 
 four particular solutions {ji, z^, {{ = 1,2,2,, a) for which the determi- 
 nant 
 
 y'l y'2 
 
 z[ z', 
 
 yi y^ 
 
 Z\ Z2 
 
 (2) 
 
 D = 
 
 y\ 
 
 z\ 
 
 ys 
 
 Zt 
 
 
 does not equal to zero. 
 
 If we interpret ji and z,- (i = i, 2, 3, 4) as the homogeneous 
 co-ordinates of two points in space, we obtain two curves 
 
 yk=yk{x), Zk=zk(x) (^=1,2,3,4). 
 
 The line joining those points which correspond to the same values 
 of X generates a ruled surface. This ruled surface will not be a 
 developable as a result of the hypothesis that D is not equal to zero. 
 Let us transform the system (i) by putting 
 
 (3) ri = ay-\-0z, ^ = yy-\-8z, ^ = ^{x), 
 
 where a, /3, 7, 8, ^ are arbitrary functions of x for which 
 
 ' E. J. Wilczynski, Projective Diferential Geometry of Curves and Ruled Surfaces, 
 Leipzig, 1906, pp. 126 ff., hereafter referred to as W. 
 
 492860 
 
FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 This transforms (i) into a new system of the same kind. The inte- 
 grating ruled surfaces of the two systems are the same, but they 
 are referred to a different pair of directrix curves and a different 
 independent variable. 
 
 The fundamental invariants of (i) are^ 
 
 (4) 
 
 where 
 
 (5) /■ = 7^„+M3 
 
 and 
 
 
 
 , 
 
 
 Mi 
 
 
 Mi 
 
 (6) 
 
 M, 
 
 
 M, 
 
 
 , 
 
 
 Vi 
 
 
 Vi 
 
 (7) 
 
 V2 
 
 
 v^ 
 
 
 f 
 
 
 w 
 
 
 w 
 
 (8) 
 
 W: 
 
 
 W- 
 
 6, =P-4/, ^4-i = 8&4^;'-9^;'+8/^. 
 
 e,,={p-^){K-r^)-\-{ir-2jy, 
 
 till M22 W12 W31 
 
 Vij —V22 fij V21 
 
 Wii—U'22 W'l2 U21 
 
 ^ = Wli + M2J, J = UiiU22, — Ui2U2l, K = Vj{U22 — Vi2V21, 
 
 = 2pu — 4qii-\-p\i-\-pi2p2i , 
 
 = 2/>i2 — 49i2-f /'l2(/'li + /'22) , 
 
 ^2Pji—4q2i-\-p2i{pii-\-p22) , 
 
 = 2p22 — 4q22-{-Pl2-i-pi2P2l , 
 
 ■■ 2Mii+^i2«2i — />2,M,2, 
 2Ui2-\-{pll-p22)Ul2-pl2iUll — U22), 
 
 ■■ 2Mji— (/>„ — /) J «« + />„ (Mii — M J , 
 
 • 2Mj2 pI2'^2ll p2l'^^I2 > 
 = 2V'ji-\-Pl2V2t — P2lVl2, 
 
 = 2f 1',+ (pii - />«)»« - piiiVii - f 2j) , 
 
 = 2V2i—{pii—p22)V2l-\-p2l(Vll — V22) , 
 = 2V'22 — pl2'V2i + p2lVl2- 
 
 A tangent plane intersects a surface in a plane curve which has a 
 double point at the point of contact. If one of the double-point, 
 tangents is an inflectional tangent to this plane curve, the point 
 of contact is called a flecnode, a name due to Cayley.' From the 
 point of view of Salmon,^ followed here, the inflectional tangent at 
 this point intersects four consecutive generators of our surface. 
 On each generator there will be in general two flecnode points, 
 
 ' W., pp. 102, 104, 112. 
 
 * A. Cayley, Collected Mathematical Papers, II, 29. 
 
 J Cambridge and Dublin Mathematical Journal, IV (1849), 252-60. 
 
INTRODUCTION 3 
 
 since four consecutive generators of a ruled surface have two 
 straight-line intersectors. As x varies these two points trace a 
 flecnode curve of two branches. The tangents to the asymptotic 
 curves through the flecnode points generate a flecnode surface of 
 two sheets. For any system of form (i) the flecnodes are obtained 
 by factoring the quadratic covariant 
 
 C = Ui2Z^ —U2iy^-\- {uij,—U22)yz. 
 
 When a ruled surface 5 is referred to its flecnode curves, W12 = U21 = o. 
 We may without loss of generality take ^„ = P22 = o. Under these 
 conditions the system of differential equations for one sheet F'-^'' of 
 the flecnode surface may be written'^ 
 
 (9) 
 
 Pi2 P12 
 
 p"-\-[2(q,^-^q22)-pi2p2i]y'-2^ p'+ 2q',,-p,2q2i-4T^qii \y 
 
 Pl2 I Pl2 J 
 
 — gjaP = 
 
 where 
 
 p = 2y'+pi2Z, (x = 2z'+p2iy. 
 
 In precisely the same way the system for the second sheet F<~'^ 
 may be written, if in (9) we transpose the subscripts and replace 
 y and p by z and a, respectively. This system we shall denote 
 by (10). 
 
 We shall call /^^'^ the first flecnode transform of S. We shall call 
 F'-~^^ the minus first flecnode transform of S for the reason that^ 
 the first flecnode transform of F'-~''^ is S. The minus first transform 
 of F^'^ is S. The first transform of F^'^ we shall call the second trans- 
 form of 5 and denote it by F'-^K Continuing in this way we obtain 
 a suite of surfaces which we shall call the flecnode suite. Questions 
 naturally arise as to the cases in which this suite either terminates 
 or returns again into itself. This paper concerns itself with some 
 of these questions. 
 
 The author wishes to express to Professor Wilczynski his deep 
 appreciation for his genuine interest and helpful criticism. 
 
 ' W., p. 153. = IMd., p. 178. 
 
I. CASES OF TERMINATION 
 
 A flecnode tangent is not ordinarily tangent to the flecnode 
 curve. If it were, since it is at the same time tangent to the 
 asymptotic curve which passes through the flecnode point, the 
 flecnode curve would be a straight line. The corresponding sheet 
 of the flecnode surface then degenerates into a straight Une. In 
 this event any ruled surface made up of the Unes intersecting 
 this line may be called its flecnode surface. If we try to continue 
 the flecnode suite, we are powerless to determine which of these 
 surfaces to select. We shall say in this case that the flecnode 
 suite terminates with its first transform, since it cannot be con- 
 tinued in unambiguous fashion. A necessary and sufficient condition 
 that the flecnode suite terminate with its first transform is dio = o, 
 i.e., the given ruled surface has a straight-line directrix. 
 
 Since the ruled surface has a straight-Hne directrix let us take 
 this line as one of the reference curves Cy. Let C^ be an arbitrary 
 curve. We may write 
 
 ^^^^ \Zi=/i(^), Zj=/j(x), Zj=fj{x), Z4=I. 
 
 Let us compute the system (i) for which these are the funda- 
 mental solutions. We obtain^ the following system, 
 
 z''+[xf','-f--x^-^+f^)y'-^^z'^ 
 whence by (6) 
 
 ««=2j/3,:«:K/»-^/o+2(:»^'+2)(/r--^')-3;p-Wx''-/n 
 
 •h2(xfr-fn, 
 
 ' W., p. 128. 
 
 4 
 
CASES OF TERMINATION 
 
 where \f3,x\ indicates the Schwarzian derivative of /j with respect 
 to X. The flecnodes are given by 
 
 (12) V = yy f=-«2i>'-«222- 
 
 (13) 
 
 The second flecnode point J" is given by 
 
 Si^ W21 ^22/1 ) 
 
 Sa^ U21X W22/2 > 
 
 S 3 ~ W22/3 , 
 
 f4= ^22 • 
 
 We obtain the following result: The equations of the most general 
 curve which can serve as the second branch of the flecnode curve of a 
 ruled surface with a straight-line directrix can he obtained without any 
 integration. 
 
 If now we make upon y and z the transformation indicated by 
 (12) we refer our surface to its flecnode curves. The resulting 
 equivalent system of form (i) is 
 
 (14) 
 
 77" =0, 
 
 f"+( W2l-2^^^^'+/>22W2I-/'2lW22 ]n 
 U21U22 
 
 + <H 
 
 -2U2,- 2W^^— +^22«2I-/'22— ]V 
 
 Uo<2 W22 *^22 Uo 
 
 1 I W22 , W22 U2i . . 
 
 + ( P22— — 2-J- ]i=0. 
 
 Carpenter has shown^ that when one branch of the flecnode 
 curve is rectilinear the second branch of the flecnode curve may 
 be plane. In this case he has shown that p2i can be determined 
 except for two constants of integration. He has found, moreover, 
 that the second branch of the flecnode curve may be a conic, 
 in which special case the equations of form (i) take the form 
 
 (15) 
 
 y =0, 
 z"-\-icy-iz=o. 
 
 We have found that d^o = o is a condition for the termination of 
 the flecnode suite with its first transform. Let us suppose 0io5^o 
 
 ' A. F. Carpenter, "Ruled Surfaces Whose Flecnode Curves Have Plane Branches," 
 Transactions of the American Mathematical Society, XVI (1915), 529, hereafter referred 
 to as C. 
 
FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 and that the suite terminates with the second transform. We 
 shall then have ^io = o. We desire to express this condition in 
 terms of the invariants of the original form (i). 
 
 The differential equations for F'-^^ are given by (9), but in a form 
 which is not convenient for the computation of its invariants. 
 We proceed to transform (9) into an equivalent system for which 
 Pil = p2l = Uii = U2l = o. To do this we first refer the surface to 
 its flecnode curves. Furthermore, we shall specialize the inde- 
 pendent variable so as to make w = i. To refer the surface to its 
 flecnode curves we put 
 
 (16) 
 
 The resulting system is 
 
 y=y, Y=-^y+p. 
 
 ■y-_.^y_r-(g-i|+,..),+|-r=o, 
 
 (17) 
 
 + 
 
 2!^-#+2(9„+9,,)-^,/>.,]y 
 
 -( 
 
 L P 
 
 'K. 
 
 .Pl2 
 
 Ip'l 
 
 p 
 Pl^p 
 
 iP'A pi. 
 
 f. ^*t:-S('"+'">- 
 
 Pl2P'21 
 
 4 
 
 ■f 2gxib' 
 
 +9.JF = o. 
 
 By the choice of suitable multiphers we may transform (17) into a 
 system which preserves the condition Wj" = w"j = o and satisfies 
 the further condition pfi=pf2 = o. We accomplish this by putting 
 
 (18) 
 
 y=p.2v, Y=Y, 
 
 where />i2 7«^o on account of the assumption ^wt^o, a transformation 
 which reduces (17) to 
 
 P 
 
 ,+^K=o, 
 
 (19) { 
 
 Y"+ \2p'A-^+2p,Mr,+q,,) -p\2p2^ T,' 
 
 Ap'^2-^+h§+pUq^^+q22)-pr2p2.p'u-hP\2pn+2q[,p,Ari 
 
 Pl2 Pl2 J 
 
 11 
 2PI2 
 
 -\-q22)Y = o. 
 
CASES OF TERMINATION 7 
 
 We are now in a position to compute conveniently the invariants of 
 F^^\ In particular we have^ 
 
 where we use the upper index i systematically for the quantities 
 referring to the surface /^"'. We find 
 
 (20) e=-2^^-§-2(9„+9,,)+^./»... 
 
 Pl2 P12 
 
 Since d^o is an invariant of the system (17) which is geometrically 
 determined by (i), it must also be an invariant of (i). We have^ 
 
 (21) 
 
 P12 I 
 
 Pl2 lOPIo 
 
 Pl2 4''I0 
 
 Substituting these values in (20) we obtain 
 
 (22) C' = ^(-i60x'o-8^;^xo+4^9C-^,+ i2ei2+04-i)- 
 
 I of 10 
 
 In order to put ^"o into a form in which its invariant character 
 will be apparent we seek to replace the derivatives occurring in 
 (22) by invariants and to put into evidence the isobaric property 
 which has been masked by the assumption ^4=1. We find that 
 
 C = -^(3^S-2M,o-20|^A+20|Mx4-^4^9+O4.i)- 
 
 is an invariant which reduces to (22) under the assumption ^4 = 1. 
 Consequently, 
 
 ^3O = 3^S-2Mao-2e|eA+20fMi4-^4^9+O4-i = O 
 
 is a necessary and sufficient condition for theflecnode suite to terminate 
 with its second transform. 
 
 One naturally inquires about the case d[^''^ = o. It is obtained 
 from the foregoing case ^"0 = by merely changing the sign of d\. 
 
 'W.,p. 119. ' Ibid., p. 120. 
 
8 FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 If ^30 = 0, we may use the methods of the first part of this 
 section to compute the second branch of the flecnode curve on F*". 
 This curve is at the same time a branch of the flecnode curve on 5. 
 We can obtain both branches of the flecnode curve on S, by apply- 
 ing the minus first flecnode transformation to F"'. Thus we see 
 that the eqtiations of both branches of the flecnode curve on S may 
 be determined without any integration if the flecnode suite terminates 
 with its second transform. 
 
II. CASES IN WHICH THE GENERATORS OF THE 
 
 SECOND AND MINUS SECOND 
 
 TRANSFORMS INTERSECT 
 
 Let us inquire whether the generators of the second and minus 
 second flecnode transforms can intersect. Consider the quantities 
 
 (23) 
 
 pki) = 2y' — 2^ y — p , 
 
 II =2p'-{-[2(q,^-hq22)-pi2p2i]y-2^p, 
 
 formed from (9) just as p and cr were formed from (i). Since the 
 original system has been taken in the form for which pji = P22 = W12 = 
 U2x = o, we have 
 
 2qiI=Pl2 , 2/>i2/>2I— 4(911+922) =«Ii + «22 , 
 
 pi2Z = p—2y', 2p'+^i2<r = Wii3'- 
 
 If we substitute these in (23) we find 
 
 (24) 
 
 ,(!) = 
 
 P12 
 
 y-px2z, 
 P'. 
 
 ^uy——-p—pi2(r 
 
 Every point on the line p"V has the property^ that the straight 
 Une joining it to the corresponding point on the generator of F"' 
 is an asymptotic tangent of i^"\ This correspondence is expressed* 
 by the fact that the corresponding variables undergo cogredient 
 transformation. In particular, the line a^^^Y will be the tangent 
 to the asymptotic curve through the second flecnode point, and 
 consequently a generator of F^^\ if, and only if, o-"' is that point 
 on the line p"V corresponding to F as a point on the Une yp. But 
 we have 
 
 F = <}'-w«P, 
 
 'W., p. 146. 
 
 ' Loc. cit. 
 
lo FLECNODE TRANSFORMATION OF RULED SURFACES 
 and therefore 
 
 Using (24) we find 
 
 (25) (rW = (-2 ^ <-|«mW )y-p^,uSz-\-2 ^ «(')p+/;,.MW(r . 
 
 \ Pl^ I Pl2 
 
 Any point on the hne (r*"F is given by the expression 
 
 (26) [(-2 ^^ <-^«w^^Aa+</3l>;-/>,,<az+ (2 ^ w«-7^W^V 
 
 Similarly any point on the line a^~'^^Z is given by 
 
 (27) 
 
 — /).-i/(-« 
 
 /'2i«^r"73'+ 
 
 l«^(-i)_2?£i«(-»)y+ 
 
 «ir"5]i 
 
 +/>,,«( -I) 7P+ /a ^ m(-^)7-m(-')5 W , 
 
 where the upper index — i refers to the system (10). The point 
 Pz is the second flecnode point on F^~'\ and o-^~*^ is the point on 
 the line p^~^V corresponding to it. The quantities p^~'^ and v are 
 obtained from (10) just as p and <t are from (i). 
 
 In order that a point on the hne given by (26) may also be on 
 the Hne given by (27), we must have 
 
 (28) 
 
 P12 
 /'i2«(''a+*-2 ^ m'-'>co7+w(-"cj5 = o , 
 
 p2l 
 
 where co is a proportionaUty factor. This requires the vanishing 
 of the determinant of the system, which after a combining of 
 columns can be written in the form 
 
CASES IN WHICH GENERATORS INTERSECT 
 
 II 
 
 (29) 
 
 or 
 
 (30) 
 
 2 
 O 
 
 ^21 P21U21 
 
 PI2U 
 
 (I) 
 
 o 
 
 2 
 
 o w^-') 
 
 4pi2p2i(ulT'^u^'^-u^^^u'-'^y-{uu^-'^u^^^y = o 
 
 We proceed to express (30) in terms of the invariants of S. 
 We have u = u^~^'' = u'-^K Our condition (30) can be written 
 
 (31) 
 
 Now we have^ 
 
 4pi2p2i{ui^ " + M^iO' — W4 = ' 
 
 U' = i 
 
 4q=-u, 
 
 P12 P'2: 
 
 BaOq 
 
 \Pl2 p2l/ 4^10 ' 
 
 M.=|, »;r»+«Si'=«(g-|;). 
 
 Substituting these values in (31) we find 
 (32) t?i8=^9-^Ao=o. 
 
 In the foregoing we have assumed d^r^o, 6 ^0^0, cases dis- 
 cussed elsewhere in this paper. 
 
 If ??i8 = o, we may recover again the determinant of the system 
 (28) whose vanishing is a necessary and sufficient condition that the 
 system have a non-trivial solution. The quantities a, (3, C07, co5 are, 
 in fact, proportional to the cofactors of the elements of any row in 
 the determinant of the coefficients. Since the cofactors of the 
 corresponding elements in the rows are proportional we are led to a 
 unique set of ratios unless all of these cofactors vanish, i.e., we 
 have a definite point of intersection. Consequently, ??i8 = o is a 
 necessary and sufficient condition for the generators of the second and 
 minus second transforms to intersect. 
 
 ' W., p. 119. 
 
12 FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 If we choose the independent variable so that ^4 = 1, the condi- 
 tion (30) becomes 
 
 (33) 4(Pl2p21-p21p'uy-pl2p2l = 0. 
 
 We may solve this relation in symmetric fashion by putting 
 
 P21 
 Then {;^^) gives upon integration 
 
 (34) Q=cJ^. 
 
 If we assume pn = p22 = Ui2 = U2j. = u—i=o (32) becomes 
 
 (35) ^-^10=0. 
 
 Under the same assumptions Carpenter^ has shown that Cy is a 
 conic if, and only if, 
 
 (36) e,+2e',o=o. 
 
 We shall assume d^y^o, for if ^9 = then by (35) 0io = o, a case dis- 
 cussed elsewhere in this paper. Differentiating (35) we have 
 
 (37) ' 26,6^-6^0=0. 
 
 If now we eliminate d[o from (36) and (37) we find, since 6^9'^ o, 
 
 (38) 4^;+i = o. 
 
 Integrating (38) we get 
 
 69=-i(x-\-c). 
 From (35) we find 
 
 6^o=Mx■^cy. 
 
 Using the condition^ that Cy be a plane curve we have 
 
 Moreover, we have assumed ^4=1. Thus we have obtained 
 expressions for the four fundamental invariants in terms of the 
 
 'C, p. 515. ' Ibid., p. S16. 
 
CASES IN WHICH GENERATORS INTERSECT 13 
 
 independent variable and one arbitrary constant. We may now 
 compute^ explicitly the coefficients of (i). They lead us to the 
 system 
 
 (y"-hdz'-iy = o, 
 
 (39) I 2-+ j^(:,+,)y+_I_(^_,)3,=o , 
 
 where c and d are two arbitrary constants. However, one of these 
 constants is not essential since 
 
 y = ay, z=bz, ^ = x-\-l, 
 
 the most general transformation which leaves our conditions undis- 
 turbed, serves when 
 
 ^=d^ ^='' ^=r 
 
 to remove d. The resulting system is 
 
 (y"+kz'-\y=o, 
 
 I 16 16 
 
 For every value of k, (40) defines a class of mutually projective 
 ruled surfaces. We see then that there exists a single infinity of 
 classes of mutually projective ruled surfaces the generators of whose 
 second and minus second flecnode transforms intersect, which have 
 the additional property that the flecnode curve consists of two distinct 
 branches, one of which is a conic. It can readily be shown by a 
 direct test that the system (40) has all of the properties attributed 
 to it. 
 
 Let us inquire whether in addition to the foregoing the second 
 branch of the flecnode curve may be plane. The necessary condi- 
 tion is^ 
 
 (41) 2Mio-3^io+^o=0. 
 
 But the values of d^o determined above clearly do not satisfy (41). 
 So we conclude that there are no ruled surfaces whose second and 
 minus second flecnode generators intersect, which have the property 
 
 ' W., p. 120. 2 c, p. 516. ■ 
 
14 FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 that their flecnode curve consists of two distinct plane branches, one 
 of which is a conic. 
 
 We proceed to compute the invariants of F^'^ and F^~^^ in terms 
 of the invariants of S for the case t?i8 = o. For this purpose we shall 
 use the form (19) in which the surface is referred to its flecnode 
 curve and multipliers for the dependent variables have been chosen 
 so as to remove certain derivatives. The equations for F'-'^^ may 
 be obtained from (19) by transposing the subscripts provided the 
 dependent variables f and Z are put in place of 77 and Y. We 
 find the following values for the invariants of system (19) 
 
 L F« /*" • J 
 
 (42) 
 
 P12 Pl2 
 
 e^:i=i6(-2^+s§+P^2P»). 
 
 \ Pz2 Pa 
 
 The corresponding invariants for F^~'^ may be obtained from these 
 by transposing the subscripts. If t?i8 = o, we have 
 
 [ p^2p^^=\{2e',o+e^) , p2xpa=li2d',o-e,) , 
 
 (43) 
 
 911+922=^(16^10—^4.1) , 
 
 P^2 16^, 
 
 (8^10^10+2^9010+^10 — 4^10) > 
 
 l^ = -7V(32Oxo'-480ioOi'o-I209^io+l609Mio+24^i'o+^9U, 
 Pu 04tfJo 
 
 giving the following values for 0"', dg\ etc., in terms of the invari- 
 ants of S, upon the hypothesis that the independent variable is 
 chosen so as to make ^4=1, 
 
 ^9" = ^(-32^o^x'o'+48Mx'oC+I209^io-l6Mxo0i';-240x? 
 
 (44) 
 
 - ^9^10+ 20x0^4-1+ ^9^0^4-1+ 2^o^;x) , 
 
 ^"0 =-^(-l6Mx';+I20.'?-0xo+O4.x) , 
 10^0 
 
 ^i'J=^(-i6Mii+8e,0,'o+.exo+2O0iS+i6^o) 
 
CASES IN WHICH GENERATORS INTERSECT 15 
 
 For the computation of the invariants of F'-~^'> the equations (43) 
 remain vaUd if we transpose the subscripts in the left members and 
 replace 6^ by — dg in the right members. Consequently, equations 
 (44) with the latter change give us the corresponding invariants of 
 
 If we abandon the hypothesis 64 = 1, we find the following 
 
 /.--) J —^4 ^9^10 ~ ^0^15^4-1+ ^4^9^10^4-1 ~^o^is) > 
 
 I 
 
 where^ 
 
 -10 z-/)2"(~ 2^10^20+3^5 ""^^10+^0^4.1) , 
 
 lOtflo 
 
 C = i-(-2M.o-4^!^As+^5^io+5^s) , 
 
 ^IS — 5^io^4~ 204^10 , ^15 = 5^4.1^4—2^4^4.1 , 
 
 To verify (45) it is sufficient to note that these formulae reduce to 
 
 (44) when ^4 = 1, and that the right members of (45) are invariants 
 of 5. 
 
 The invariants of F'-~'^^ are given by a system obtained from 
 
 (45) by replacing dg by -6^. 
 
 Since 0io = 0i7" we conclude that in case the generators of the 
 second and minus second flecnode transforms intersect, the minus 
 first flecnode surface belongs to a special linear complex if, and only 
 if, the first flecnode surface belongs to a special linear complex. In 
 this event the second and minus second flecnode surfaces are straight 
 lines having a point in common. 
 
 Let us inquire whether the corresponding absolute invariants 
 of F'-^^ and F^~^^ may not have the same values. Equations (44) 
 and the corresponding ones for F''~^'> show that it will then be 
 necessary that 
 
 ' W., p. 112. 
 
 [ 1209^10— 16^9^10^10 — ^9^io~l~^9^o^4-i = • 
 
i6 FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 We shall again assume that 6g9^o. Since t?i8 = o we must then 
 have also Bio9^o. The equations (46) reduce to 
 
 (47) ^10=0, 04.1=1. 
 
 Equations (44) show that in this case 
 
 so we conclude that the absolute invariants of the first and minus 
 first flecnode transforms are equM if, and only if, these surfaces belong 
 to special linear complexes. 
 
III. CASES OF PERIODICITY 
 
 It has been shown^ that each sheet of the flecnode surface of S 
 has S itself as one of the sheets of its flecnode surface. Thus 
 the minus first transform of 7^'" is S. The first transform will 
 ordinarily be a new surface F^'\ Let us inquire whether it too 
 can coincide with S. If so, the flecnode suite will be periodic, of 
 period two. In this event the two sheets of the flecnode surface 
 of F^" must coincide. Then the two sheets of the flecnode surface 
 of 5 cannot be distinct, else' they would be distinct on /^'". Thus 
 we see that ^4 = is a necessary condition. It is evident geo- 
 metrically that this condition is also sufficient. We conclude then 
 that the flecnode suite is periodic, of period two, when, and only 
 when, the flecnode curve meets every generator in two coincident points, 
 or is indeterminate. 
 
 This theorem may also be established analytically. The 
 ruled surfaces for which the flecnode curves are indeterminate are 
 quadrics. In this case we have not only ^4 = 0, but Wi2 = W2i = Wii — 
 U22 — 0. The flecnode surface in this case coincides with the original 
 quadric generated by the generators of the second kind. The 
 second flecnode surface is the original surface generated again by 
 its first set of rulings. 
 
 We may now assume 6 ^9^0 and ^lor^o since the cases excluded 
 by this hypothesis have been considered already. The generators 
 of the first and minus first transforms are^ generators of the second 
 kind on the quadric which osculates our original ruled surface along 
 a generator. They are distinct since we have assumed dj^9^o. 
 Consequently, they cannot intersect and, therefore, the flecnode 
 suite cannot be periodic, of period three. 
 
 This theorem, too, may be established analytically. We pro- 
 ceed to determine whether the flecnode suite can be of period 
 four. We shall assume again d^j^o and ^io?^o. We require that 
 the line joining the point Y to the point Z be a generator of F^'\ 
 
 ' W., p. 178. 'Ibid., p. 147. 
 
 17 
 
1 8 FLECNODE TRANSFORMATION OF RULED SURFACES 
 
 In particular, it is necessary that the point (r"\ given by (25), shall 
 be a point on this Une. Now, any point on the line YZ is given 
 by the expression 
 
 Consequently, we must have 
 
 (48) 
 
 M2'a+ 2 
 
 -<+^MW^*Mw = 0, 
 
 P12 
 
 Since the three quantities a, ^, 00 cannot all be zero, it is necessary 
 that the second order determinants 
 
 (49) 
 
 
 P12U21 
 
 
 < 
 
 Pl2 
 
 } 
 
 «(^> 
 
 Pl2 
 
 shall both vanish. Now pi2^o, else the hypothesis ^10 5^0 is con- 
 tradicted. Moreover, w"Vo, since the flecnode curves are dis- 
 tinct on S, and are therefore distinct also on F^". It is necessary 
 then that 
 
 (50) 
 
 M^r'V^)-w'VM^-^>=M=o, 
 
 However, w = o implies ^4 = 0, contrary to hypothesis. So we 
 must conclude that the flecnode suite cannot he periodic, of period 
 four. 
 
VITA 
 
 John Wayne Lasley, Jr., was born in Burlington, North CaroHna, Sep- 
 tember 2 2, 1 89 1. After attending the public schools of his native town he 
 entered the University of North Carolina in 1906, graduating with the degree 
 of Bachelor of Arts in 1910. He received the degree of Master of Arts from 
 this institution in 191 1. In 191 5-16 he attended the Johns Hopkins Uni- 
 versity, studying with Professors Bateman, Coble, Cohen, and Morley. Dur- 
 ing the summers of 1917, 191 8, 191 9, and the year 1919-20, he was in residence 
 at the University of Chicago, studying with Professors Birkhoff, BUss, Dickson, 
 Moore, and Wilczynski. Since receiving his Bachelor's degree during the 
 years not specified above he has taught in the University of North CaroUna in 
 the capacity of instructor, assistant professor, and, associate professor. In 
 connection with this work he studied with Professors Cain and Henderson. 
 To all of his teachers he is indebted for inspiration and instruction. To 
 Professor Wilczynski he is particularly gratefxil for sympathetic interest and 
 helpful advice during the preparation of this dissertation. 
 
 19 
 
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