UC-NRLF 
 
 Ubc mnivetsttp ot Cbtcago 
 
 A NEW BASIS FOR THE 
 METRIC THEORY OF CONGRUENCES 
 
 A DISSERTATION 
 
 submitted to the faculty of the ogden graduate school of 
 
 science in candidacy for the degree of doctor of 
 
 philosophy 
 
 Department of Mathematics 
 
 BY 
 
 LEVI STEPHEN SHIVELY 
 
 Private Edition, Distributed By 
 
 THE UNIVERSITY OP CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 
 1921 
 
EXCHANGE 
 
Ube TDlntrersit^ ot Cbtcago 
 
 m 
 
 A NEW BASIS FOR THE 
 METRIC THEORY OF CONGRUENCES 
 
 A DISSEKTATION 
 
 submitted to the faculty of the ogden graduate school of 
 
 science in candidacy for the degree of doctor of 
 
 philosophy 
 
 Department of Mathematics 
 
 BY 
 
 LEVI STEPHEN SHIVELY 
 
 Private Edition, Distributed By 
 THE UNIVERSITY OF CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 1921 
 
St \ 
 
A NEW BASIS FOR THE METRIC THEORY OF 
 CONGRUENCES. 
 
 Introduction. 
 
 In the classical theory of rectilinear congruences as developed by 
 Kummer/ the analysis is based upon two fundamental quadratic 
 forms, formed from the expressions for the coordinates of the 
 surface of reference and from the spherical representation of the 
 congruence. Thus, when studied by this method, the congruence 
 is in reality defined by its relation to two surfaces. Important 
 results of the *Kummer theory are the existence of the two focal 
 surfaces and of two one-parameter families of developables. These 
 results have been used by Wilczynski as the basis of his general 
 projective theory of congruences.^ In the present paper also, the 
 focal surfaces will be regarded as known by means of a parametric 
 representation, such that the ruled surfaces u = const, and v = 
 const, of the congruence are its developables. 
 
 The first section is devoted to the introduction of certain local 
 coordinate systems, all closely related to the general line of the 
 congruence. These coordinate systems are used to study the prop- 
 erties of the congruence in the neighborhood of one of its lines, as 
 well as the properties of its Laplacian transforms. Homogeneous 
 cartesian coordinates are introduced in the second section and the 
 equations (DY of the Wilczynski theory are derived. The coef- 
 ficients of these equations being expressed in terms of the funda- 
 mental quantities of the focal surfaces, and their derivatives, there 
 is here afforded a convenient means for investigating the relations 
 
 ^ Kummer, AUgemeine Theorie der geradlinigen Strahlensysteme, Crelle, 
 Jour, fiir Mathematics, 57, 1860, 189-230. 
 
 2 Wilczynski, Sur la theorie generale des congruences. Memoires publics par 
 la Classe des Sciences de FAcademie Royale de Belgique. Bruxelles, 1911, pp. 86. 
 This memoir will, in what follows, be referred to as the Brussels Paper. 
 
 3 Brussels Paper, p. 16. 
 
 1 
 
 46^853 
 
2 New Basis for Metric Theory of Congruences. 
 
 between the focal surfaces and the congruence. These two sections 
 form the basis for the present investigation. 
 
 In the third section we obtain new proofs of some of the most 
 important general theorems on congruences. These results are 
 typical of those obtained by means of the Kummer theory. The 
 fourth section takes up the axis and ray congruences and the cor- 
 responding curves on the focal surfaces, all of which were recently 
 discovered by Wilczynski.^ A number of new theorems concerning 
 these configurations, mainly metric in character, are proven in this 
 section. In the final section we discuss the equations of the 
 Darboux conies of a point on one of the focal surfaces and consider 
 some special cases. 
 
 It was at the suggestion of Professor Wilczynski that this research 
 was undertaken and I take this opportunity of expressing my 
 appreciation of the helpful encouragement he has given as the 
 work progressed. 
 
 1. The Congruence Referred to Local Coordinate Systems. 
 
 Let the focal surfaces, which are assumed to be non-degenerate, 
 be denoted by Sy and Sz, and let the ruled surfaces u = const, and 
 V = const, of the congruence be its developables. Then Sy will be 
 given by equations of the form 
 
 Vk = fk{u, v) {k = 1, 2, 3), 
 
 where the parametric curves u = const, on the surface Sy are the 
 edges of regression of one family of developables of the congruence 
 and where the curves v = const, are conjugate to u = const. 
 Consequently, making use of the customary notation of the theory 
 of surfaces, we have D' = and the equations of Gauss^ for Sy will 
 assume the form 
 
 yuu = Aiyu-\- A2yv + DX, 
 
 (1) Vuv = Biyu + Biyv, 
 
 yvv = Ciyu + C2yv + D"X, 
 
 1 Wilczynski, The General Theory of Congruences, Transactions of the Ameri- 
 can Mathematical Society, 16, 1915, 311-327. This paper will hereafter be cited 
 as Gen. Th. Cong. 
 
 2Bianchi, Vorlesungen iiber Differential geometrie, p. 89. Eisenhart, A 
 Treatise on the Differential Geometry of Curves and Surfaces, p. 154. 
 
New Basis for Metric Theory of Congruences. 
 where 
 
 Ai = 
 
 11 
 1 
 
 i"l 
 
 (2) ft-ifl-ft-lz' 
 
 o-^in ^-i?!' 
 
 are the Christoffel symbols formed for the surface Sy. The coor- 
 dinates Zk, {k = 1, 2, 3), of P 2, the point on the second focal surface 
 Sz, which corresponds to P^ must satisfy^ the condition 
 
 2 =2/ -^2/ 
 
 'V) 
 
 that is 
 
 (3) yv = Bi(y - z), 
 
 and we shall have Pi ^ 0, since otherwise the surface Sy would 
 degenerate into a curve. Let A[, A'2, - - - , C2 be the Christoffel 
 symbols for the surface Sz, and let us denote the fundamental 
 quantities of the first and second orders for Sy and Sghy E''^\ • • •, 
 Z)(^)'' and E^'\ • • •, D^'^'' respectively. 
 
 The first Laplacian transformed congruence consists of the 
 tangents to the curves v = const, on Sy. Let its second focal 
 surface be Sp. Similarly, let S^. be the second focal surface of the 
 minus first Laplacian transformed congruence, which consists of 
 the tangents to the curves u = const, on Sz. By the same method 
 used for obtaining (3) we find 
 
 (4) zu = B'^iz - y), 
 
 (5) ' yu = B2(y - p), 
 
 (6) z, = B[{z -a). 
 From (3) and (1) we find that 
 
 /yN _ {Bl)u — P1P2 
 
 \' ) ^u d2 yv) 
 
 ^ Eisenhart, loc. cit., p. 405, eq. 43. 
 
4 New Basis for Metric Theory of Congruences. 
 
 where Bi 4= 0, since Sy is supposed to be a non-degenerate surface. 
 Eliminating Zu and yv from (3), (4) and (7) gives 
 
 (8) B', = B,-^-^\ 
 
 From (5) and (I2) we find in the same way 
 
 .o\ (^2)1; — B1B2 
 
 (9) Pv = ^2 Vw 
 
 By means of (3), (5) and (6) we find the following distances 
 between corresponding points on the focal surfaces: 
 
 (10) 
 
 We have also 
 
 yP.= 
 
 Bi 
 
 yPp = 
 
 ■ylE(y^ 
 B, 
 
 p - 
 
 Vg(^) 
 
 B[ 
 
 fiy) jriz) 
 
 COS CO = , , . , T , cos CO 
 
 (11) . ^^^^ . , H^'^ 
 
 sm CO = , , , — , sm co 
 
 ■VE( !/)(;/( J/) ' ^jE(^z)Qiz)' 
 
 ^ff{y) = ^E{v)Q{y) _ fiv)^^ JJiz) = ■^E^z)g(z) — 7^(2)2^^ 
 
 where co and co' are the angles PzPyPp and PyPzP^ respectively, 
 each of these angles being so defined as to be less than 180°. 
 
 We shall now introduce the first of the several local systems of 
 rectangular coordinates which we shall use subsequently. This 
 system, which will be designated by (I), is defined as follows. The 
 origin is the point Py. The {-axis coincides with the line of the 
 congruence through Py, so directed that Pz lies on its positive end. 
 The ?7-axis is in the tangent plane to Sy at Py, and is so chosen 
 that the angle between its positive end and PyPp is acute. The 
 ^-axis coincides with the surface normal at Py, its positive direction 
 
New Basis for Metric Theory of Congruences. 5 
 
 being chosen in such a manner that the relative orientation of (I) 
 is the same as that of the original axes. 
 
 Referred to the original axes, the direction cosines of the axes 
 of (I) are, 
 
 iyk)v 
 
 J-axis : 
 
 ^iG^y) ' 
 
 (12) ,;-axis : ^^J^^^ [.F^^Kyk). - G^^Kyk)u^ 
 
 f-axis : — Xk 
 
 H(y) 
 
 {yk+i)u (yk+2)u 
 (yk+i)v (yk+2)v 
 
 (k = 1,2, 3; 2/4 = yi, y^ = 2/2). 
 
 With reference to (I) the lines PyPp and PzPp have the direction 
 cosines 
 
 F(y) TJiv) 
 
 P P • 
 
 ^ f" ' '^E^y)Q(.v) ' ^Eiy)Q(y) ' ' 
 
 (13) BiF^y^ - B^G^y^ BiH^y^ 
 
 ^'^^ * hi ' hi ' ^' 
 
 - (^1= ^G^y\B\E^y^ - 2BiB2F^^ + BlG^y^)), 
 and the coordinates of P2 and Pp are 
 
 (14) 
 
 P.- (^^,0,0). 
 
 
 p 
 
 The focal planes of the given congruence, passing through the line 
 PyPz, are the tangent plane to Sy at Py and the osculating plane 
 of the curve u = const, at the point Py. Their equations are 
 
 (15) r = 0, and D^y^' -slG^^^r) - CiH^y^^ = 
 respectively; the direction cosines of the normal to the latter are 
 
 ,,,, ^ D^yy'ylG^ CiH^y^ 
 
 (16) 0, 
 
 We also find that the focal planes of the first Laplacian transform. 
 
6 New Basis for Metric Theory of Congruences. 
 
 through PyPp, are the first of (15) and the plane through Py normal 
 to the direction 
 
 (17) h ' h~' hi ' 
 
 The axis of the point Pj, is the line of intersection of the planes 
 which osculate the curves u = const, and v = const, passing 
 through Py. By means of (16) and (17) we compute the direction 
 cosines of the axis of Py, which are the following: 
 
 A3 ' h ' ' h^ 
 
 where 
 
 ;^3 = JAlG^y^'D^yy' - 2A2CiF^y^G^yW^yW^yy' 
 
 + ciE^y^G^yW^y^^-{- G'^yW^y^^D^y^'^ , 
 
 Hence the angle between PyPz and the axis of Py is 
 
 _, CiF^yW^y^ - AiG^yW^yy' 
 
 cos- ^^ ,^ 
 
 while the angle between PyPp and the axis of Py is found to be 
 
 , -ylG^y^CiE^yW^y^ - A2F^yW^yy') 
 
 cos ^ p== . 
 
 h-iE^y'> 
 
 Since a congruence is normal if and only if the focal planes are 
 perpendicular we have, by reference to (16) and (17) the following 
 criteria : 
 
 A necessary and sufficient condition that the given congruence 
 be normal is that Ci = 0; and a necessarj^ and sufficient condition 
 that the first Laplacian transformed congruence be normal is that 
 ^2 = 0. 
 And similarly, 
 
 A necessary and sufficient condition that the minus first Laplacian 
 transformed congruence be normal is that CI = 0.^ 
 
 1 These criteria obviously agree with the theorem (see Eisenhart, loc. cit., 
 p. 401): A necessary and sufficient condition that the tangents to a family of 
 curves on a surface form a normal congruence is that the curves be geodesies. 
 
New Basis for Metric Theory of Congruences. 7 
 
 The axes of our second local coordinate system (II) are the lines 
 of curvature tangents and the surface normal at Py. In order to 
 define this system completely, we note that the differential equa- 
 tions of the lines of curvature on ^S^ are 
 
 dv — \idu = 0, 
 
 dv — \2du = 0, 
 where 
 
 ^ ~ r / TvG)^mv)myy' i 
 
 (19) ' ^ ^ ^-^ 
 
 X2 = 
 
 14-^/l-J — — - 
 
 ^^\-L-r(^(j/)2)(i/)"_ G<<yW^y^yj' 
 
 We shall refer to the curves through Py, defined by (18), as h and 4 
 respectively. Taking as positive that direction of the tangent to h 
 whose direction cosines are 
 
 (y.)u + My.). (it =1,2, 3), 
 
 and denoting by 6 the angle between the {-axis and the positive 
 tangent to h, taken in the direction which rotates the {-axis through 
 90° into the ?7-axis, we have 
 
 cos d = , — = a, 
 
 ^G^y)(E^y) + 2\iF^y^ + X'G^(^>) 
 (20) ' 1 ^ 
 
 smS = = B. 
 
 ■ylG^y)(E^y) + 2Xii^(^> + X^G^y^) 
 
 To justify the choice of the negative sign in the last equation we 
 observe that 
 
 H(y) 
 cos (p = — 
 
 ^G^y\E^y^ + 2Xii'^(^> + \\G^y^) 
 
 where <p is the angle from the 77-axis to the positive tangent to h 
 measured in the same direction as 6, and that sin B = cos (p. 
 
8 New Basis for Metric Theory of Congruences. 
 
 Let the positive direction of the tangent to I2 be that one which 
 agrees with the positive 77-axis when system (I) is rotated about 
 the f-axis through the angle 6. Then the positive fi, 771, fi-axes 
 of system (II) are defined to be the positive tangents to h and hy 
 and the positive f-axis respectively. The transformation from (I) 
 to (II) is 
 
 fi = a? + iSry, 
 
 (21) 771 = - /5^ + ary, 
 
 Ti = r. 
 
 Our third system of coordinates, designated by (III), is obtained 
 by taking as origin the point Pz and as axes the lines through P ^ 
 which are parallel to the corresponding axes of (I). If the coor- 
 dinates of the point (J, 77, ^) referred to (I), be (J', -q' , ^') when 
 referred to (III), we have the following equations of transformation: 
 
 (22) ,' = „ ' 
 
 . V = r. 
 
 We now pass to the system (IV) which is obtained by rotating 
 (III) about the J'-axis, in the direction from the ry'-axis toward the 
 f'-axis, through that angle d\, which brings the ly'-axis into the 
 plane PyPzP^ in such a manner that PzP„ projects upon the positive 
 end of the ry'-axis. The new positions of the axes of (III) are 
 defined to be the J'', f]", r''-axes of (IV). 
 
 . The plane PyPzP^ is both the tangent plane to ^S 2 at P 2 and the 
 osculating plane ofu= const, on *Sy at the point Py. The direc- 
 tion of its normal is given by (16). Hence 
 
 sin ^1 = € 
 
 (23) ' ' 
 
 COS 01= e , , , 
 
 ■\Q{y)J)iv)"^ j^ C^H^y)^ 
 
 where e has one of the two values d= 1. The equations of trans- 
 
New Basis for Metric Theory of Congruences. 
 formation from (III) to (IV) are 
 
 (24) " 
 
 The equations which enable us to pass from (I) to (IV) are 
 
 (25) V' = 
 
 e(r)CiH^y^ - iD^yy'<G^) 
 
 '\Q(.y)J[)(.yr'^ -|_ C2jj(y)2 
 The inverse of (25) is 
 
 ?= r + 
 
 (26) 
 
 r 
 
 B, ' 
 
 e(- -n^D^yy^JG^^ + ^CiH^y^) 
 
 Since the line P^P^. makes the angle 180° — co' with the positive 
 J''-axis, we find from (10) and (11) the coordinates, referred to (IV), 
 of P, to be (- P(^)/P; -ylE^, m^^B', <W\ 0). Referred to (I), 
 Pa = {fi, ^2, ^3) where 
 
 (27) r2 
 
 _ B[ VE(^)G^(^) - PiP(^) 
 CiH^y^H^'^ 
 
 B[^E^'\G^yW^yy'^ -{- ClH^y^')' 
 
10 New Basis for Metric Theory of Congruences. 
 
 J)(y)"JJ{z) ^Q(y) 
 
 n = 
 
 B[^E^'\G^yW^yy''' -{- C\H^ 
 
 2/)2 
 
 We shall now determine the ambiguity in the sign of e. Since 
 the lines PzP„, PzPy and the surface normal of Sz at Pz whose 
 direction cosines referred to the original axes are 
 
 (28) - 
 
 H(^) 
 
 {^k+l)v i^k+2)v 
 
 (k = 1, 2, 3; 2:4 = zi, Z5 = Z2), 
 
 have the same relative orientation as the original axes, the same is 
 true of the J''-axis, the T^'^-axis, and this direction of the surface 
 normal. Consequently the sign of e must be so chosen as to make 
 the f"-axis coincide with this direction of the surface normal. 
 
 The directed line whose direction cosines, referred to (I), are 
 given by (16), is that one which, referred to the original axes, has 
 the direction cosines 
 
 (29) 
 
 {yk+i)v 
 iyk+i)v^ 
 
 {yk+2)v 
 {yk+2)vv 
 
 (k = 1, 2, 3; 2/4 = yi, 2/5 = ^2), 
 
 where h is a, properly chosen positive number. Now the lines 
 whose directions are given by (28) and (29) are parallel. Our 
 problem therefore reduces itself essentially to that of determining 
 whether the signs of corresponding quantities from (28) and (29) 
 are the same or opposite. 
 From (3) and (4) we obtain 
 
 (30) 
 
 Zu ^ — TT 
 
 Bi^"' 
 
 and by differentiating this and (4) with respect to v, eliminating Zuv 
 and reducing by (3) we find 
 
 1 
 
 (31) 
 where 
 
 ■^yv n yvv 
 
 K= 1 + 
 
 
 B2 \BiJv 
 
 Upon substituting in the first of the quantities (28) for (2:2) „, (2^3) «, 
 
New Basis for Metric Theory of Congruences. U 
 
 (z2)v, and (z3)v, their values- as obtained from (30) and (31), it 
 reduces to 
 
 {y2)v {yz)v 
 {y2)vv {yz)vv 
 
 This shows that the directions given by (28) and (29) are the same 
 or opposite, according as the sign of B'2 is negative or positive. 
 For a non-degenerate surface Sz, B'2 cannot vanish. Now ac- 
 cording as e = + 1 or — 1, the positive f -axis coincides with the 
 directed line, whose direction is given by (29), or the opposite 
 direction. And since in (IV) the t''-axis coincides with the surface 
 normal of directions (28) it follows that, according as B'2 is negative 
 or positive, e takes the value + 1 or — 1. 
 
 Our last system of local coordinates is defined by its relations to 
 the lines of curvature on Sz. The equations of these lines are 
 
 dv — \'idu = 0, 
 (32) 
 
 dv — \2du = 0, 
 
 where X'l and X2 are formed from the fundamental quantities for Sz 
 in the same way that Xi and X2 are formed from those for Sy (see 
 (19)). In what follows we designate those curves through Pz 
 defined by (32), by /! and /^ respectively. Furthermore, the 
 positive tangent to ![ is defined similarly to that for h. Then, 
 if d^ denote the angle from the positive tangent to u = const, on 
 Sz at the point Pz, to the positive tangent to ![, measured in the 
 direction from u = const, towards v = const., we have, as in (20), 
 
 cos 6' = , ^- = a 
 
 ylG<^)(E^^) + 2X;i^(^> + x; (?<^)) 
 
 sin d' = 
 
 VG?(^)(^(^)-f- 2x;i^(^) + \?G^'^) 
 
 Also, if (p' be the acute angle from the ?7''-axis to the tangent to 
 u = const., then 
 
 cos (p 
 
 sm (p 
 
 
12 New Basis for Metric Theory of Congruences. 
 
 A rotation of the axes of (IV) about the f ''-axis through the angle 
 ip' -f Q' brings the f and ry^'-axes into coincidence with the lines 
 of curvature tangents, the positive ?7"-axis coinciding with the 
 positive tangent to l'\ . The respective positions of the axes of (IV) 
 after this rotation are defined to be the axes of (V). Let 
 
 a" = cos ((p' + (90 = p==-- , 
 
 r = sin {<p' + 6') = , ^ -. 
 
 Then if we denote by {^i, rji, fl') the coordinates, referred to 
 (V), of the point whose coordinates referred to (IV) are (^'\ rj'', ^'') 
 we have 
 
 fr = a''r + r>?", 
 
 (33) vi = - P"^" + a'W\ 
 
 2. The Differential Equations (D) of the Congruence. 
 
 Wilczynski has shown^ that the projective theory of congruences 
 may be based upon a completely integrable system of partial 
 differential equations of the form 
 
 yv = mz, Zu = ny, 
 (D) yuu =ay-\-hz-\- cyu + dzy, 
 
 Zvv = a'y + h'z + c'yu + d'zv. 
 
 He has shown that such a system has four pairs of linearly inde- 
 pendent solutions {yk, Zk), (k = 1, 2, 3, 4), and that if (yi, • • •, 2/4) 
 and {zi, • • • , 24) be regarded as the homogeneous coordinates of 
 two points Py and Pg, the locus of these points, as u and v vary, 
 will be the focal surfaces Sy and ^Sz of the congruence. It is the 
 object of this section to obtain the differential equations (D) of our 
 congruence, with coefficients expressed in terms of the fundamental 
 quantities of the focal surfaces and their derivatives. We shall 
 ^ Brussels Paper, pp. 9-19. 
 
New Basis for Metric Theory of Congruences. 13 
 
 exclude from present consideration those congruences whose lines 
 are the tangents to asymptotic curves on either of the focal surfaces. 
 In our notation, this is expressed by the conditions, D^^)" y>^ 
 and Z)(^> 9^ 0. 
 
 We can then eliminate X and y^v from the first and third of 
 equations (1) and the equation obtained by differentiating (3) with 
 respect to t. This gives 
 
 (34) yuu= ^^{y — z)^ ciyu + diz^y 
 where 
 
 7)(y) 
 
 ai = A,B, + ^(^. [(5i - C^)B, + (5i) J, 
 
 Proceeding in a similar manner with the corresponding equations 
 for the surface »Sz we obtain 
 
 (35) Zvv = cl[{z — y) -\- c[yu + d[zv, 
 where 
 
 a[ = C[B', + -^^ [(5; - A[)B', + (5;) J, 
 
 ^1 - ~ J)iz) ^2, 
 
 di — 62 T\( y\ A2. 
 
 
 Let now {yi, 2/2, y^, 1) and (21, Z2, zz, 1) be homogeneous cartesian 
 coordinates of the points Py and Pz respectively. The system of 
 equations consisting of (3), (4), (34) and (35) is satisfied by these 
 coordinates since it is obviously satisfied by 2/ = 1, s = 1. To 
 obtain from these equations the system (D) we set 
 
 V = fBidv, 
 q = fB'zdu, 
 
14 New Basis foe Metric Theory of Congruences. 
 
 and make the transformation 
 
 y = e^y, 
 z = e^z. 
 Then (3), (4), (34) and (35) become 
 
 yv = mz, Zu = ny, 
 
 (36) yuu= ay-\-bz-\- cyu + dz^y 
 
 Zvv = a'y + h'z + c'yu + d'z^, 
 where 
 
 n= - e^^B2, 
 
 a = A2B1 + ^^[(5i - C2)B, + (Bi), - CpJ 
 
 + AiPu — pi — Puuy 
 
 (37) ^=-'''d^'^^' 
 
 a' = - e^^ ^B',C[ + ^ { (5; - A[)B', 
 
 + {B',)u+B',pu]'^, 
 
 ^' = ^^c; + ^[(5; - A\)B'2 + (5;). - ^;gj 
 
 d = C2 jyjy A2 2qv. 
 
 Equations (36) constitute the system (D). They are satisfied by 
 corresponding pairs of the homogeneous coordinates {e~^y\, e~^y2, 
 
New Basis for Metric Theory of Congruences. 15 
 
 e~^yz, e~P) and (e~^Zi, e~%, e~^Z3, e~^) of the points Py and Pz 
 respectively. 
 
 3. Some of the Principal Results of the Classical 
 Theory of Congruences. 
 
 This section will be devoted to the proof of a few of the general 
 theorems on congruences which are ordinarily proved by means 
 of the Kummer theory. Use will here be made of the coordinate 
 systems which were established in section 1. 
 
 Let Py and Pz be points in the neighborhoods of Py and Pg 
 obtained by giving to u and v the increments 8m and 8v respectively. 
 Then by Taylor's theorem, (3), (4), (5), and (6), 
 
 Y = y-\- yubu + y^bv + • • • 
 = y-\- B2(y - p)bu + Bi{y - z)8v + • • • 
 
 Z = z + Zu8u + Zv8v + • • • 
 
 = z + B^iz -y) + B[{z - a)8v + • • -, 
 
 where the terms not written are of higher order than the first in 
 du and 8v. Denoting by ({, tj, and (^i, rji, fi) the coordinates of 
 Py and Pz when referred to (I), these equations, together with (14) 
 and (27) give, up to terms of the second order, 
 
 J = a8u + P8v, Ji = ao + oii8u + ^i8v, 
 
 7j = y8u, rji = yi8v, 
 
 r = 0, ri = 8i8v, 
 
 where 
 
 
 V^' ' B, 
 
 
 ^G'^y^' V^(z)((^(?/)2)(2/)"^-|- C^H^y^^) 
 
 «o = —B — , 8i= e 
 
 ^1 ' V^(^)((;f(2/)2)(y)"^+ (72^(^)2) * 
 
16 New Basis for Metric Theory of Congruences. 
 
 The line which is the common perpendicular to PyPz and PyPi 
 has direction cosines proportional to 
 
 (38) 0, 8it, y - yit; 
 
 (-S) 
 
 and the abscissa of the foot of this common perpendicular upon the 
 latter line is 
 
 «o7(7 — TiO 
 
 (39) ? = 
 
 {y-7ity+dlt' 
 
 In order to find the values of t for which this abscissa is a maximum 
 or a minimum, we differentiate with respect to t and set the result 
 equal to zero. This gives 
 
 2 
 
 (40) t'-2^i + Z^^=0' 
 
 7i 7i + Oi 
 
 which is the differential equation of the curves in which the principal 
 surfaces of the congruence intersect the focal surface Sy. Solving 
 
 (40) we obtain 
 
 (41) ^ J 
 
 ^^ = ^(i- vwtl)' 
 
 If we use these values of t in (39) we obtain the abscissas of the 
 limit points of the line PyPz. They are 
 
 Q!o7i^ 
 
 (42) 
 
 The midpoint of the segment joining these limit points has the 
 abscissa |q;o = ^('sG^/Bi). This proves the well-known the- 
 orem :^ 
 
 The midpoints of the two segments bounded respectively by the limit 
 points and by the focal points coincide. 
 
 ^ This theorem and the others of this section, together with Hamilton's equa- 
 tion, are given in sections 4 and 5 of the previously cited memoir by Kummer. 
 
New Basis for Metric Theory of Congruences. 17 
 
 Again, if the values of h and ^2 from (41) be substituted suc- 
 cessively in (38) the directions of the common perpendiculars at 
 the limit points are obtained. It is easy to verify that these direc- 
 tions are perpendicular. Since these perpendiculars and the line 
 of the congruence to which they are normal determine the principal 
 planes, we have the theorem : 
 
 The two principal planes through each line of the congruence are 
 perpendicular. 
 
 Let us denote by df and di the distances between the focal points 
 and the limit points respectively. Then df = VCr^^V^i- Also 
 from (42) 
 
 
 ^G(y)D^yy'' + C'.H^yy' 
 
 Therefore 
 
 Q(y)J){yy 
 
 df j)^yy''<G^y'> 
 
 di ^Q{v)D{v)"^ _|_ clH^y^^ 
 
 sm r 
 
 by (15) and (16), where r is the angle between the focal planes. 
 This gives the theorem: 
 
 The ratio of the distance between the focal points to the distance 
 between the limit points is equal to the sine of the angle between the 
 focal planes. 
 
 In particular, when the congruence is normal, Ci = and this 
 ratio is unity. Thus, the limit points coincide with the focal 
 points in a normal congruence and conversely. 
 
 In order to deduce Hamilton's equation, let co be the angle between 
 the common perpendiculars whose directions are given by (38) for 
 t = t and t = ti. The calculation shows that 
 
 
 
 f2 
 
 = COS^ 
 
 CO, 
 
 
 which is equivalent to 
 
 
 
 
 
 
 ? = 
 
 ■■ f 1 COS^ CO + ^2 
 
 sin^ 
 
 CO, 
 
 This is Hamilton's equation. 
 
 
 
 
 
18 New Basis for Metric Theory of Congruences. 
 
 4. The Axis and the Ray Congruences. 
 
 The developable surfaces of the axis congruence intersect ^^ in 
 the axis curves. Since Sy is not degenerate, m 9^ 0, and thus 
 these curves are determined by the differential equation '} 
 
 (43) -5^2 _ j5^5^ _ c'ddv^ = 0. 
 
 m 
 
 In terms of the fundamental quantities of the focal surfaces, the 
 coefficients of (43) are found by (37) and the definitions of c^i and / 
 to be 
 
 
 '~]^B,B, + 2{B,)u- (^1), 
 
 + 
 
 l^^C^'^lJ^- (^2).-^^l0g;^(^,J, 
 
 (44) I = A^Bi + ^, (Bl - B,C, + (5i), - B^C^) 
 
 + A,B, - {B2)uy 
 c'd = B,B, + {B,)u - (^1). + [^(yy'C^)^' 
 
 A necessary and sufficient condition that the axis curves be conju- 
 gate is 
 
 which reduces to 
 
 m 
 
 Q2 , D(i/) 
 
 (45) (5i)"-(52).-^-^-^log5^,= 0. 
 
 If we confine our attention to those congruences whose focal surfaces 
 are distinct so that Z>^^^ as well as D^^^' is different from zero, 
 the Codazzi equations^ 
 
 Z)?^ - 5i7)(^> + A^B^yy' =0, 
 (46) 
 
 Z)ir>" + CiD^y^ - B^B^yy = 0, 
 
 1 Gen. Th. Cong., p. 316. See eqs. (7), p. 315 for the definitions of di and I. 
 
 2 Eisenhart, loc. cit., page 155. 
 
New Basis for Metric Theory of Congruences. 19 
 
 enable us to express (45) by the equation 
 
 ( , ^^\ _( n P^\ 
 K^^D^y^ )u~\ 'B^yy).' 
 
 This condition is obviously satisfied when the congruence and its 
 first Laplacian transform are both normal. But this is equivalent 
 to the condition that ^Sj, be a surface of Voss. Hence we have the 
 theorem : 
 
 The axis curves with respect to a congruence consisting of the tangents 
 to one of the two families of conjugate geodesies on a non-developable 
 surface of Voss, are conjugate. 
 
 We shall also prove that 
 
 If a congruence consists of the tangents to the lines of curvature of 
 one system on a surface Sy, this surface being either a quadric, or a 
 surface of revolution of constant total curvature, then the axis curves 
 are conjugate. To prove this theorem we observe that, in case 
 F<iy) = 0, (45) becomes 
 
 Jliy) Jmy) 
 
 — — — — = U'V 
 
 B^yy^G^y^ ' 
 
 where C7 is a function of u alone and F is a function of v alone. 
 This condition is clearly satisfied by an isothermic surface whose 
 lines of curvature are isothermal-conjugate. The fundamental 
 quantities of the quadrics^ show that they have these properties. 
 Furthermore since all surfaces of revolution are isothermic, and 
 since the lines of curvature of all surfaces of constant total curvature 
 are isothermal-conjugate, the part of the theorem relating to 
 surfaces of revolution of constant total curvature also follows. 
 The foci of the axis of Pj, are given by the factors of^ 
 
 c'dxi - mlyr^y^ - dmT^y^\ 
 
 the point P^iv) being the intersection of the axis with the line PzP„. 
 These foci are on the same or on opposite sides of Pj, according as 
 — (dm/c'di) is positive or negative. The fifth of the integrability 
 
 1 Ibid., pp. 239-241. 
 
 2 Gen. Th. Cong., eq. (10), p. 316. 
 
20 New Basis for Metric Theory of Congruences. 
 conditions^ of system (D) gives the equation 
 
 IB1B2 - (5i) J (^ 1 - p(y)vp(,) j 
 
 If we consider the case in which the axis curves are conjugate and 
 use (48) the value of — (dm/c'di) reduces to 
 
 2)(y) 
 
 ^B^B,+ {BOu-{A,).-\-(^^.C,^J 
 
 This is positive or negative according as D^^^ and D^^)" have 
 opposite or the same signs. Hence we have proved the theorem: 
 
 // the axis curves on a surface Sy are conjugate, the foci of the axis 
 lie on the same or on opposite sides of Py according as the total curva- 
 ture of Sy at Py is negative or positive. 
 
 We next proceed to a consideration of the ray curves on Sy. 
 Their differential equation is^ 
 
 (49) dndu' - Uubv -'!^di^=0. 
 
 m 
 
 In this equation I is given by (44), and 
 
 ^^ = 5ur'[^i^2 - (5i)«], 
 (50) 
 
 — = B1B2 — {B^v- 
 
 ifv 
 
 Thus we see that if D^^W^^^" ?^ 0, a necessary and sufficient con- 
 dition that the ray curves be conjugate is 
 
 (51) {B,)u == (52).. 
 
 Now 51^2 — iBi)u and B1B2 — {B2)v are the Laplace-Darboux 
 
 1 Brussels Paper, p. 17. See also Gen. Th. Cong., p. 313. 
 
 2 Gen. Th. Cong., p. 318. 
 
New Basis for Metric Theory of Congruences. 21 
 
 invariants^ of the equation (I2). This gives the result found by 
 Wilczynski •? 
 
 " A conjugate system on a non-developable surface has equal 
 Laplace-Darboux invariants if and only if its ray curves also form a 
 conjugate system." 
 
 Let us interpret the condition for conjugate ray curves in the 
 case in which the original parametric net on 8y consists of the 
 lines of curvature. Then (51) becomes 
 
 where TJ and V are functions of u alone and v alone respectively. 
 But this condition is satisfied when and only when the surface Sy 
 is isothermic. Hence 
 
 A necessary and sufficient condition that the ray curves with respect 
 to the lines of curvature on a non-developahle surface be conjugate, is 
 that the surface be isothermic. 
 
 Wilczynski has shown that a necessary and sufficient condition 
 that the axis curves, for a non-degenerate surface Sy, coincide with 
 the original conjugate net on ^S^, is that Sz be developable. He 
 shows further that when this condition is satisfied it also follows 
 that Sp is developable. We shall consider the corresponding 
 question for the ray curves. Reference to (50) shows that when 
 j)iv)D{yy' ^ Q the ray curves are parametric when and only when 
 the invariants B1B2 — {Bi)u and B1B2 — {B2)v are both zero. 
 Since these are the conditions that Sz and Sp degenerate into curves, 
 we obtain the following result : 
 
 Necessary and sufficient conditions that the ray curves on a non- 
 developable surface Sy coincide with the parametric curves are that the 
 surfaces Sz and Sp be degenerate. 
 
 To determine whether the foci of the ray of Pj, lie on the same 
 or on opposite sides of P 3 we proceed similarly to the method used 
 for the corresponding problem of the foci on the axis. Here we 
 find that they are on the same side or on opposite sides of Pa ac- 
 
 J Darboux, Lemons sur la Theorie Generale des Surfaces, Tome 2, pp. 23-25. 
 2 Gen. Th. Cong., p. 319. 
 
22 New Basis for Metric Theory of Congruences. 
 
 cording as — (mn/dnii) is positive or negative.^ When (51) is 
 satisfied 
 
 2)(y)" 
 
 ^2(p-3) ^ 
 
 drrii B\ 
 
 Therefore 
 
 If the ray curves on Syhe conjugate, the foci of the ray of Py lie on 
 the same side or on opposite sides of Pz according as the total curvature 
 of Sy at Pyis negative or positive. 
 
 Reference to (45) and (51) shows the truth of the following the- 
 orem: 
 
 Any two of the three conditions: (a) the axis curves on Sy are 
 conjugate, (b) the ray curves on Sy are conjugate, (c) the parametric 
 curves onSy are isothermal-conjugate; for a non-developable surface Sy, 
 imply the third. 
 
 For further investigation of the case in which the conjugate net, 
 in which Sy is met by the developables of the congruence, consists 
 of the lines of curvature, we shall have need for the Gauss and 
 Codazzi equations. In this case they are^^ 
 
 :^)„-^(VG-)„ = o. 
 
 Let us consider a congruence which has the property that it and 
 its first Laplacian transform are both normal. If, in addition, the 
 lines of curvature are parametric, we then have E^p = 6r?^ = 0. 
 With these conditions the first of (52) gives D^^W^y^" = 0; hence 
 Sy is developable. We may state this result as follows: 
 
 // a congruence and its first Laplacian transform are both normal 
 
 1 See eq. (22), Gen. Th. Cong. 
 
 2 Bianchi, loc. cit., p. 94. Eisenhart, loc. cit., p. 157. 
 
New Basis for Metric Theory of Congruences. 23 
 
 and they meet their common focal surface in its lines of curvature then 
 that surface is developable} 
 
 The converse of this theorem is not true.^ However the truth of 
 the following theorem, which resembles somewhat the converse of 
 the preceding, will be established. 
 
 If a surface is developable, then one of the two congruences which 
 consist of the tangents to the two systems of curves of a conjugate net 
 on the surface, is a normal congruence. 
 
 Taking the given conjugate curves as parametric we have one of 
 the two cases: Z)(^> = 0, D^y^ 9^ 0; D^y^ 9^ 0, D^^^" = 0. In the 
 first case (46i) gives A2 = 0, showing that tangents to the curves 
 V = const, form a normal congruence. And in the second case 
 (462) gives Ci = with the result that the congruence consisting 
 of the tangents to the curves u = const, is normal. 
 
 5. The Darboux Conics. 
 
 The surfaces upon which the conjugate system determined by the 
 developables of the congruence have equal Laplace-Darboux invari- 
 ants have been studied by both Darboux and Wilczynski. We 
 have already seen the interesting geometrical interpretation of this 
 property by the latter, namely, the conjugacy of the ray curves. 
 Although Darboux's interpretation is entirely different from that of 
 .Wilczynski, the two are essentially equivalent. Darboux's theorem 
 is that if these invariants are equal, there exists in the tangent plane 
 to Sy Sit Py Si conic having second order contact at Pz with the 
 curve V = const, and having also second order contact at Pp with 
 the curve u = const., and conversely. Whether these invariants 
 are equal or not, there are two conics (distinct if the invariants are 
 unequal) which Wilczynski has called the Darboux conics of Py, 
 the first of which has second order contact with v = const, at Pz 
 
 ^ The excluding from consideration of surfaces for which D^^^" = is not 
 vital to this argument. 
 
 2 That there are surfaces for which the hypothesis of the converse is satis- 
 fied is shown by the example : 
 
 E(y) = a, F^y^ = 0, G^y^ = v?f{v), D^y^ = 0, D^y^ = 0, D^y^' = u, 
 
 a being a constant, not zero, and f{v) an arbitrary function of v alon e, not iden- 
 tically equal to zero. 
 
24 New Basis foe Metric Theory of Congruences. 
 
 and first order contact with u = const, at Pp; and the second has 
 second order contact at P^ with u = const, and simple contact at 
 Pz with V = const. We shall hereafter refer to these as the first 
 and second Darboux conies of the point Py with respect to the given 
 conjugate net on the surface Sy, 
 
 Let us proceed to find the equation of the first Darboux conic 
 of Py. Denoting by Pz the point near Pz, obtained by giving to u 
 the increment 8u, we have by (4) and (5) 
 
 (53) 
 
 = z+B',(z-y)8u-h hL{iB2)u+B2]{z-y) 
 
 Let (J, 77, f) be the coordinates, referred to (I), of Pz- Then from 
 
 (53) and (14), 
 
 (54) ^= vM-\- ••., 
 
 where 
 
 
 1 ^G^y^ f _,, . ^,2 . B^B'^F 
 
 T = o 
 
 2 Bi 
 1 BoH^y^ 
 
 [mu+B'U^-^\, 
 
 V = 
 
 2 V^(y) * 
 If we impose upon the conic 
 
 (55) aif + 2A1J77 + 61772 _^ 2g,^ + 2/177 + Ci = 
 
 the condition that it be satisfied, up to and including terms of the 
 second order in 8u, by (53) we find, upon expressing ^1, /i and Ci 
 in terms of ai, hi and 61 that it reduces to 
 
 (56) aif + 2^1^77 + 61772 - 2aiQ:{ " ( ^' + 2^iq: ^ 77 + aio" = 0. 
 
New Basis for Metric Theory of Congruences. 25 
 
 We shall now determine the ratios ai : h : 61 such that (56) has 
 first order contact with the cuspidal edge at Pp. Denote by Pr 
 the point near Pp obtained by giving to v the increment 8v. Then 
 by (3), (5) and (9), 
 
 R= p-[- Pv8v + ~^dv^+ ", 
 
 ^^^) = p + ilf (p - y)bv + ^ { (if. + M^){p - y) 
 
 where 
 
 M = Bi- 
 
 B, 
 
 Let (Ji, ryi, ^1) be the coordinates, referred to (I), of Pr. Then it 
 follows from (57) and (14) that 
 
 fi = «i + iSi5i) + yi^v^ + • • •, 
 (58) 'r]i = \i-{- ^ihv-\- vih^-\- •• •, 
 
 fi= 0+..., 
 
 where 
 
 JP(y) 
 
 ""'^b^Tg^' 
 
 1 
 
 7i = 2l 
 
 Xi = 
 
 (M, + M2) ■;^^-y=^+ M^lG^y^ I , 
 
 b^4g^^ 
 
 Ml 
 
 b^4g^^' 
 
 _MPM_ 
 
 b.4W^' 
 
 H(y) 
 
 Pi = UM. + M') 
 
 B2<G^y^ 
 
 Now let ai -.hi : 61 be such that (56) is satisfied up to and including 
 first order terms in bv, by (58). Thus the equation of the first 
 
26 New Basis for Metric Theory of Congruences. 
 
 Darboux conic of Py is (55) where the coefficients are given by 
 
 h = my^[G^y\Bi)u - BlF^y^'], 
 
 b, = BlF^y^^ - 2{Bi)uF^yW^y^ + BlG^y^\ 
 
 (59) • , 
 
 gi= - BiH^y^'^lG^, 
 
 /i = myWG^\BiF^y^ - B^G^y^), 
 ci = G^y^H^y^\ 
 
 In a similar manner we find the equation of the second Darboux 
 conic of Py. It is found to differ from that of the first only in the 
 values of hi and 6i. The values of these coefficients for the second 
 conic are obtained from (59) by changing {Bi)u to {B2)v This 
 agrees with the fact that the two conies coincide when and only 
 when the Laplace-Darboux invariants are equal. To the results 
 which are deduced in the following paragraphs for the first Darboux 
 conic, there evidently correspond similar results for the second, 
 which can be stated analytically by changing (J5i)„ to iB2)v in the 
 expressions for those of the first conic. 
 From (59) we find 
 
 aj)i - hi = G^y^'H^y^'lBlBl - (5i)3. 
 
 Hence the species of the conic depends upon the invariant B1B2 
 — {Bi)u and upon B1B2 + (5i)u. If the invariant B1B2 — {Bi)u 
 is not zero, that is if iSz is not degenerate, the conic is a parabola 
 when and only when B1B2 + {Bi)u = 0. The conditions that it 
 be a circle reduce to 
 
 El Gl 
 
 ^ = 7rV0, 
 
 (60) 
 
 The determinant 
 
 E 
 
 {Bi)u 
 Bl 
 
 G 
 
 F 
 G' 
 
 A = 
 
 h 
 bl 
 
 h 
 
 - G^y^'H^y^'lBiB2 - (Bi)uJ 
 
 vanishes when and only when B1B2 — {Bi] 
 theorem, 
 
 vanishes. Hence the 
 
New Basis for Metric Theory of Congruences. 27 
 
 A necessary and sufficient condition that the first Darboux conic of 
 the point Py degenerate into a pair of straight lines is that the focal 
 surface be a curw. 
 
 When the condition of this theorem is satisfied the equation of the 
 conic becomes 
 
 [5iF(^> J - {BiF^y^ - B2G^y^)v - H^'^ V^J = 0. 
 
 This is the equation of the line PzPp, the ray of Py. 
 
 In case F^^^ = 0, the equation of the first Darboux conic reduces 
 to 
 
 (61) BlE^y^e + 2 <¥^^G^\Bi)uiri + BlG^y^-q^ 
 
 - 2BiE^y^ ^W^i - 2B2 ^E^G^y^T] + E^y^G^y^ = 0. 
 
 . / B2^lG^^ Bi^Wy^ \ 
 
 Its center is the pomt [b^b2+(B,).^ B^B.+ iBOuJ' ^^ 
 
 {Bi)u = 0, that is, since i^^^^ = 0, if E^^^ is the product of a function 
 of u alone by a function of v alone, the axes of the conic are paralled 
 to the lines of the given congruence and of its first Laplacian trans- 
 form which intersect at Py. Then aA - h\ = E^^^G^^^BlBl 
 and the conic is an ellipse. 
 
 By means of the Gauss and Codazzi equations it may be shown 
 that there is no surface, aside from the trivial case of the plane, 
 for which the first and second Darboux conies of every point with 
 respect to the lines of curvature, are coincident circles. For, 
 suppose they are both circles and that they coincide. The first 
 of (60) implies the existence of a function (p{u, v) such that 
 
 £'"> = L>Pu + fiu)J, G = [± ^. + g{v)J 
 
 where f{u) and g{v) are arbitrary functions of u and v respectively. 
 Now, by the second of (60) and the coincidence of the conies, 
 
 [(^^1 _[{G^-\ _. 
 
 But these last conditions, together with (52i) show that D^^W^y^" 
 = 0. Suppose D^y^ = 0. Then by (522), since (V^), 9^ 0, it 
 follows that D^y^" = 0. Similarly it may be shown by (523) that 
 D^yy = implies D^y^ = 0. 
 
VITA. 
 
 Levi S. Shively was born near Cerro Gordo, Illinois on October 
 10, 1884. His early education was received in the public schools 
 of Cerro Gordo. After spending several years in Mt. Morris 
 College he entered the University of Michigan in 1906, from which 
 he graduated two years later with the degree Bachelor of Arts. 
 During the years 1908 to 1915 he was Professor of Mathematics 
 in Mt. Morris College. He entered The University of Chicago 
 for graduate study in the summer of 1915 and during nine succeeding 
 quarters was a student of Professors Moore, Dickson, Bliss, Wilczyn- 
 ski, Lunn, Moulton, MacMillan and Birkhoff. He received the 
 degree Master of Arts in the summer of 1916, the thesis for same 
 having been written under the direction of Professor Moore. The 
 present thesis, in candidacy for the degree Doctor of Philosophy 
 was prepared under the direction and inspiration of Professor 
 Wilczynski. 
 
 28 
 
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