LIBRARY 
 
 OF THE 
 
 University of California. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
THE 
 
 PRIMITIVE DOUBLE MINIMAL SURFACE OF 
 
 THE SEVENTH CLASS 
 
 AND 
 
 ITS CONJUGATE. 
 
 Submitted in partial fulfillment of the requirements for the degree of Doctor of 
 Philosophy, in the Faculty of Pure Science, Columbia University, 
 
 BY 
 
 GRACE ANDREWS, 
 
 
 ^■:^, 
 
 /; 
 
 NEW YORK. 
 
 1901 
 
THE 
 
 PRIMITIVE DOUBLE MINIMAL SURFACE OF 
 
 THE SEVENTH CLASS 
 
 AND 
 
 ITS CONJUGATE. 
 
 Submitted in partial fulfillment of the requirements for the degree of Doctor of 
 Philosophy, in the Faculty of Pure Science, Columbia University, 
 
 BY 
 
 GRACH ANDREWS 
 
 ^ OF THE 
 
 UWfVERSlT\' 
 
 or 
 
 NEW YORK. 
 
 The Evening Post Job Printing House, 156 Fulton Street. 
 
 (evening post building.) 
 
 I 9 o I . 
 
A- 
 
 ^^ 
 
 ^vlv. 
 
THE PRIMITIVE DOUBLE MINIMAL SURFACE OF 
 THE SEVENTH CLASS AND ITS CONJUGATE. 
 
 In a paper entitled "On Certain Algebraic Double Minimal 
 Surfaces," the author, Dr. James Maclay, discusses the minimal 
 surfaces determined by the following conditions : (i) The gen- 
 erating minimal curves, conveniently designated by (a) and 
 (/^) shall coincide, that is, the curve (a) shall be its own con- 
 jugate ; (2) to one point of the infinitely distant circle shall 
 correspond one tangent to the curve ; (3) the curve shall have 
 only two infinitely distant points. 
 
 These points may, without loss of generality, be taken to 
 correspond to 7i = and u = co. When this is done and 
 the functions F{!() and /(?/), [/'"{?() =- F{u)l that define the 
 surface in connection with the equations of Weierstrass are 
 determined so as to satisfy the prescribed conditions, they are 
 found to take the forms 
 
 (I) A (^0 
 
 A- 
 
 --2- 
 1 
 
 
 1 
 
 
 + 
 
 ^0 
 
 + 
 
 ^1 
 
 IC 
 
 
 
 cj 
 
 u' 
 
 
 
 ^M 
 
 -+ (- 
 
 -i)^ A' 
 
 
 2 
 
 ^M 
 
 = 
 
 — 
 
 -/< 
 
 (/' 
 
 + 
 
 0(/' 
 
 + 
 
 2) 
 
 (2) F,{u)=^:£6^ 
 1 
 
 On assuming the a to be included in the A 's and putting 
 
 equation (2) becomes 
 
 (4) F, {u) = 1 3^. 
 
 1 
 
 The elements ^^{u), (/< = i, 2 .... k) composing Fi,{ti) 
 themselves define double surfaces. To a given value of ju 
 correspond an infinity of surfaces arising from the variation 
 of the arbitrary constant A^. These, however, are all similar 
 since s^ can be thrown into the form 
 
 (5) ^, = c, ' + ^''l^^f' " ' . (^. real). 
 
 On the other hand, except when k ^ 1 and F^ (u) reduces to 
 3t the surfaces defined by F^ (u) are not all similar. Since 
 the class of F^. and s^. is the same, being expressed through 
 the equation C = 2;/ + 3, F^ may appropriately be called the 
 general ; 5^, the primitive surface of the class {2k + 3). 
 
 210501 
 
The present paper is a study of the primitive double sur- 
 face and its conjugate corresponding to // = 2.* 
 
 I. — The Primitive Double Surface, /x = 2. 
 
 I. T/te Equations of the Surface. — The function defining 
 the surface is 
 
 (6) ^•^ = -^' (^2=1) 
 
 or, 
 
 (7) &2 = ''-—^A&2" =^2). 
 
 24 
 
 When f)^ is substituted for f {u) in the second form of the 
 equations of Weierstrass, namely 
 
 x = ^{i-u'')f'{tc) + uf{:ii)-f{tc) + -^ (i-^) A"i^') 
 
 + vf,' {V) -/i (V) 
 
 + ivf,' {v)-if^{v) 
 z = lif" iu) -/' {ii) + vf{ {v) -// (7/) 
 
 the following equations for the surface are obtained : 
 
 _ I r u^ ^ u-^ u^ + u.-^-\ I r_ 7/* + 7/ -^ 
 ^ ~TL 4 "*" 2 J2L 4 
 
 -f 
 
 4 
 
 V -\- V - 
 2 
 
 ] = ^ + ^1 
 
 tl 4 2 — 2-1 '1-4 
 
 ^^— +^ — J-tL— T^ 
 
 1? — u ^ , 7/'* V ^ 
 
 + =N+N,. 
 
 i 3 
 
 The paper already cited determines the plane lines of 
 curvature and the straight lines of the general surface F^ (//), 
 the properties of the minimal curve and the infinitely distant 
 elements of the primitive surface ; the nature of the geodesic 
 in the .i-j-plane by which the primitive surfaces /' even are 
 characterized and the remaining curves of intersection in the 
 same plane. 
 
 2. Plane Lines of Curvature and Straight Lines of ike 
 Surface. — When a surface is represented upon the unit sphere 
 
 * The double surface corresponding to /n = i has been treated by Schilling in the Inaugural 
 Dissertation, " Die Minimalflachen fiinfter Klasse." 
 
through parallel normals, plane lines of curvature pass into 
 small circles with planes parallel to the lines of curvature and 
 right line asymptotes into great circles with planes perpen- 
 dicular to the right lines. It is therefore possible by forming 
 the equations of the stereographic projections of these circles 
 on the plane Z = o and combining them with the differential 
 equations of the plane lines of curvature and straight lines to 
 determine the latter. 
 
 When this is done, it is found that no non-meridian circles 
 represent curves of the desired type, except the pole iiv = ^, 
 which as it involves u — o corresponds to a line at infinity, 
 and the equatorial circle of the sphere iiv — \ = o. The 
 relation uv — i = ^ determines upon the surfaces }x even, a 
 line of curvature in the -i'_j'-plane and upon the surfaces /i odd, 
 the ^-axis. 
 
 It is found that meridian circles cannot represent curves 
 of the desired types unless the arbitrary constants y^^ involved 
 in Fk {u) fulfill certain conditions. When every A^x. but one 
 is zero, /\. reduces to 3^. In this case, the surface possesses 
 (/i 4- i) lines of curvature lying in planes that pass through 
 the .c-axis, and (/< + i) straight lines lying in the A'j/-plane 
 and passing through the origin. When /< is even, the planes, 
 among which is found the .r;:r-plane, pass through the lines, 
 which include the .v-axis. 
 
 As all the plane lines of curvature on the surface are 
 represented on the sphere in great circles, they are geodesies, 
 and by a theorem due to Schwarz determine planes of sym- 
 metry. By a second theorem, the straight lines are axes of 
 symmetry. 
 
 It follows from these results that the surface /^ = 2 has 
 four planes of symmetry, including the xy and .r.c-planes and 
 three axes of symmetry, including the x-axis. 
 
 3. The Minimal Curve. — The order, class and rank of the 
 minimal curve of the primitive surfaces are found to be 
 2}x + 4, 2/i -f 2, 2yM -\- 4, respectively. 
 
 The stationary points of the curve are furnished by the 
 values of 11 satisfying. 
 
 (10) '^^{7i)=o;ox,u^^'^-^ = {—x)^^\ 
 
 When /< is even, a pair of conjugate points corresponding to u 
 and — II are found to be symmetrically disposed with respect 
 to the xy-plane. The middle point of the line joining a pair 
 lies on the surface. 
 
 The plane of osculation at the infinitely distant points of 
 
6 
 
 the curve falls into the plane at infinity and the tangent line 
 becomes the tangent to the imaginary circle. 
 
 When /^ = 2, the order, class and rank are 8, 6 and 8 
 respectively, and equation (lo) takes the form 
 
 (11) U^ = —\. 
 
 4. TJie Infinitely Distant Elements. — These, as is known, lie 
 on straight lines obtained by joining the infinitely distant 
 points of one minimal curve to the infinitely distant points of 
 the other. They arise therefore from the four combinations 
 
 {\) II =0,V = \ (2) ?/ = CO ; V = ^\ {^) U = 0, V = CO ; (4) 
 
 ?^ = 00 , V = 0. When these values are introduced into the 
 equations of the primitive surfaces 
 
 X + iy = ^- ^ , j-— - + ^^ '- 
 
 , , . — « - '^ - 2 , (— l)^ «AX (_ l)'^ + ' Zy'^ + '' 7/ - '^ 
 
 (12) x — ty^ —J— — H j H 
 
 /< + 2 /f /' + 2. ^l 
 
 ^ ^ /i + I. /' + I. 
 
 it is found that combinations (i) and (2) give rise to the line 
 
 (13) Z = 0. T = 0. 
 
 and (3) and (4) to the lines 
 
 (14) X -\- iy = 0. z = 0. 
 
 (15) X — iy ^ 0. T =^ 0. 
 
 respectively, r = ^ is the equation of the plane at infinity. 
 The lines are not simple lines of the surface. This may be 
 shown in the case of the first line as follows : assume ?/ = 00 , 
 ?; = 00 , and regard only the terms of the highest degree in 
 equations (12). We have then 
 
 / i)^ + 1 ^M- + 2 
 
 (16) 
 
 .r + 
 
 z> 
 
 
 + 1 ^^,x + 2 
 + 2. ■ 
 
 from which 
 
 we 
 
 derive 
 
 
 (17) 
 
 
 
 X 
 X 
 
 + zy _ / 
 
 — iy ^ 
 
 X — z y 
 
 H + 2. 
 
 2 
 
 To every point of the straight line correspond (yu + 2) 
 values of the ratio — , when ?^ = oo , t-- = oo . 
 
 V 
 
 But now assuming u = o, v = o, and regarding only the 
 terms of lowest degree, we have 
 
 ^-»x-2 _ Z^-^-2 
 
 (18) x + iy = — ,, ■ ^ , x — ty = 
 
 /< + 2 /< + 2 
 
and therefore 
 
 (19) -iL±i>="''^"^=.(^y"' 
 
 ^ ^' x — iyu-*^-^ \v ) 
 
 To every point of the straight line correspond (/< + 2) 
 values of the ratio — , when 11 = 0, v = 0. Apparently the 
 
 V 
 
 line is of multiplicity 2(/' + 2). But for a given point, to 
 every pair of values Ji, v satisfying (16) and (17) correspond a 
 pair — T' ~', — u ~^ satisfying (18) and (19). It is character- 
 istic of double surfaces that every point corresponds to two 
 pairs of values of the parameters, namely, u, v and — v" , 
 — jr^ therefore the multiplicity of the line under considera- 
 tion is (/< -(- 2). 
 
 The two imaginary lines (14) and (15) are also of multi- 
 plicity {i-i + 2). 
 
 5. The Geodesic in the xy-Plane. — By employing the 
 equations of Schwarz defining a minimal surface in terms of 
 a curve through which the surface passes, the following 
 relation is obtained between p the radius of curvature of the 
 geodesic and the parameter u 
 
 (20) p = li- S(ii {u). 
 
 It ^b 
 
 Through the substitutions ii = e'^ and qi = -^, (20) 
 
 may be changed into 
 
 (21) P = 4 sin Tp. 
 
 In this form p is recognized as the radius of curvature of a 
 hypocycloid generated by means of a fixed and a rolling circle 
 with radii equal respectively to 
 
 1.1 {/ii + 2) /' + 2 
 
 ip is the angle between the line of centers of the two circles 
 and the .f-axis. 
 
 From the relation between R and r, it is seen that the 
 curve has (/< + i) cusps, one of which lies on the ,t'-axis. 
 Since a cusp occurs wherever p = o, a. comparison of (20) and 
 (10) shows that the points in the ,t-j/-plane furnished by pairs 
 of conjugate stationary points of the minimal curve are the 
 cusps of the hypocycloid. 
 
 6. The Curves of Intersection in the xy-Flane. — It is of ad- 
 vantage to determine these for the surface /' = 2 by a special 
 discussion and to reserve until later a statement of the more 
 general results reached for the surfaces /< even. 
 
When for u and v are substituted A f ''' and A f ' * equation 
 (9) takes the form. 
 
 ;r = — i (A^ + A-^) cos 40 + i (A" + ;\-") cos 20 
 
 (22) J = — T (A* + A--*) sin 40 — 1 (A" + A-^) sin 20 
 
 2- = I (A^ — A~') cos 30. 
 
 The substitution p = A — A -^ obviates the difficulties that 
 arise from the fact that each point of a double surface corre- 
 sponds to one value of the parameter 0, combined with two 
 values of the parameter A^ namely A and — A ~\ By this sub- 
 stitution equations (22) become 
 
 .a: := — i (/3* -f 4P + 2) cos 40 + i (p" + 2) cos 20 
 
 (23) y = — \ {P* + 4P^ + 2) sin 40 — ^{p-^ 2) sin 20 
 
 ■s- = t P (P" + 3) cos 30. 
 
 The curves of intersection in the Ar;'-plane are determined 
 by the conditions 
 
 {A)p = o, {B)p^+2 = o, (C) cos 30 = ^.- 
 Condition (A) leads to the equations 
 
 X = — (cos 1p -\- ^ cos 21^') 
 
 (24) 
 
 y =^ — (sin ip — i sin 2ip). 
 
 where ip =' n — 20. These define a hypocycloid generated 
 by a fixed and a rolling circle of radii 3/2 and 1/2 respectively. 
 The radius of curvature p satisfies the equation 
 
 (25) p' = 4 sin 3/2 ^' = 4 cos 30. 
 
 The curve therefore has three cusps, one of which is on 
 the or-axis at x = — 3/2. The corresponding vertex is at 
 X = 1/2. 
 
 Since A = i and 7(v — i = o, when p = o, the curve is the 
 geodesic in the ,i'jj/-plane already mentioned. 
 
 Its degree may be determined by putting v = 11 ~' in equa- 
 tions (9) and substituting the resulting values of :r and j in the 
 equations of an arbitrary line. This gives an equation of the 
 fourth degree in li^. The degree of the curve is four. The 
 multiplicity is apparently two, but is really one, since the two 
 pairs of values of the parameters to which each point of a 
 double surface corresponds, namely «, v and — v~^^ — /< ~^ be- 
 come for points on the curve in question u^ v and — ?/, — v. 
 
Condition {B) leads to the equations 
 
 X = 2" (cos l}) -\- ^ cos 2^/') 
 
 (26) 
 
 y =\ (sin ip — i sin 2^'). 
 
 These evidently define a hypocycloid derived from the 
 preceding by means of the proportional factor — 1/2. The 
 curve has a cusp at x = 3/4, y = o and a vertex at ,i' = — 1/4, 
 J/ = o. It is double since each point corresponds to two vahies 
 of p and is isolated since these values are imaginary; for 
 an infinitesimal change in p makes the co-ordinates of the 
 surface imaginary. 
 
 Condition [C) implies the relations 
 
 sin 6(p ^ sin 40 cos 20 + cos 40 sin 20 =: 
 
 (27) sin 4(p cos 41?? 
 
 sin 2(p cos 2cp 
 
 The equations of the section therefore become 
 
 x _ y 
 
 (28) cos 20 sin itp 
 
 i (p^ + 6 fy + 6] 
 
 ^ o. 
 
 These are the equations of three straight lines of multi- 
 plicity 4, namely : 
 
 y = 0, 7 = ± V 3 .r. 
 Regarded as an equation in p", (28) is satisfied by 
 
 (-9) p' - - 3 ± ^ 3 + 
 
 4.r 
 
 cos 20 
 
 Two of the four values of p are real if C" ^h Other- 
 cos 2Cp > "^^ 
 
 wise all are imaginary. Comparison with equations (24) and 
 (25) shows that the projections of the cusps of the geodesic 
 hypocycloid upon the Ar-axis are given by 
 
 -^ =^ 3/2 cos 2 0, when cos ^(p ^-. 0. 
 
 The straight lines therefore proceed outward from the cusps 
 of the hypocycloid to infinity in connection with real parts of 
 the surface. Two real nappes pass through each. 
 
 7. Order of tJic Surface. — On adding together the degrees 
 of the vari'ous curves of intersection in the .vj-plane, regard 
 being paid to the multiplicity of each, the order of the surface 
 is found to be 28. This accords with the result obtained from 
 the formula 
 
 C = f w (w — i), 
 
 when m is the order of the minimal curve. 
 
10 
 
 There are in the a:_>'-plane 
 
 3 finite straight lines of multiplicity 4 12 
 
 I straight line at infinity of multiplicity 4 4 
 
 I hypocycloid of multiplicity i, degree 4 4 
 
 I " " " 2, " 4 8 
 
 28 
 
 8. Curves of Intersection in the xz- Plane. — These are 
 determined by the conditions 
 
 {A) cos cp ^ o, {B) sin q) ^= o, (Q cos 2^ = A, 
 
 (A) is the condition for the a;-axis. 
 
 (B) leads to a curve of the fourth degree with the 
 equations 
 
 X = — i (^< + 2 p^ — 2). 
 (30) 
 
 y = ip{p- + 3). 
 
 The values of the direction cosines of the normal to the sur- 
 face at a point n, v are 
 
 (31) x = ^tA^, F = /^^^. z = i^^. 
 
 uv -\- \ uv -\- \ uv -}- I 
 
 Since n = v when sin^ = o, the normals to the surface 
 along the curve defined by (30) fall into the a-^-plane. The 
 curve is therefore a geodesic. 
 
 The slope of the tangent at any point is derived from 
 
 cix = — p {p- + i) itp, dz ^ 2 [p- -}- i) dp. 
 dz 
 
 The substitution of — p for p shows that the curve is 
 symmetric with respect to the .:v-axis which it cuts orthogon- 
 ally at .V = 1/2 and in an isolated double point at jv: = — 1/4. 
 In addition to this singularity, it has two imaginary cusps 
 corresponding to the roots of p- + i = o. Since it is unicursal 
 and of degree 4, it can have no other double points. 
 
 It cuts the .c-axis in four points, two real at .;• = ± 2.128 
 and two imaginary. It proceeds to infinity without maxima, 
 minima or points of inflection and becomes parallel at 
 infinity to the .r-axis. It is of the type of the ordinary 
 parabola. 
 
11 
 
 Condition (C) gives rise to a double curve of the tenth 
 degree with the equations 
 
 _^{p' + 6 ,/ + 6) {p' + 2 p2 -2) 
 4 
 
 X 
 
 (32) 
 
 2 := 
 
 V 
 
 P* 4- 4 P^ + 2. 
 
 The curve cuts the :i:-axis at the origin, an isolated point, 
 and at ^ = — 3/2 and .v = 3/4, cusps of the hypocycloids in the 
 ATj'-plane. 
 
 Since to one value of x correspond four values of p\ the 
 curve has eight branches arranged in pairs symmetric to the 
 ;t;-axis. On throwing the equation for x into the form of a 
 biquadratic in r, (r = p^) and forming the discriminant 
 
 J = 64/9 (18 .r' + 82 X* + 216 x^ + 306 -t- + 270 -r -f 81) 
 
 it is found that the only real value of x for which A is zero 
 is one that lies between — .5 and — .6 and may be conveniently 
 designated by a,,. For x < a,,, two values of r are imaginary; 
 for 0: > «o, all are real. For — 3/2 < x < 00, one of the real 
 roots is positive. This corresponds to the non-isolated part 
 of the curve. 
 
 When with these results are combined the facts presented 
 in the following table, showing the comparative values of ^ 
 a; and z 
 
 — 4-7 
 + o 
 Imaginary. 
 
 3-4 
 
 Real. 
 
 —3 
 -h 
 o 
 
 2.7 . . — 1.2 
 + o - o + 
 
 Imaginary 
 
 -.58 
 
 Real. 
 
 •73 
 
 o 
 
 Real. 
 
 the conclusion is reached that the curve, so far as it is real, 
 consists of two detached parts; the first wholly isolated, asso- 
 ciated with values of r ranging from — 3.4 to — 3; the second 
 partly isolated, associated with values ranging from — .58 
 to 00 . 
 
 Further light is shed upon the course of the curve through 
 the values of dx and dz. 
 
 dx = 
 
 ^ r'^ _|_ 10 r* -f 36 r-^ + 56 r^ + 44 r + 24 
 
 dr 
 
 d2 = 
 
 v/" 
 
 / ^"+3 V/^ r^+ ior* + 36r^ + 56r-+44r-f 24 
 V r-^ -1- 4r + 2 / (r2 + 4r -f 2)^ ' 
 
 1/2 
 
 + 4r + 
 
 dx ^ ^ V r2 -I- 4r + 2 / 
 Since the equation 
 
 r^ _}- 10 r^ -f- 36 r'^ + 56 r^ -f- 44 2" + 24 = o 
 
 dr 
 
12 
 
 has but one real root and that lies between o and — 3; and 
 the equation 
 
 r^ + 4r+2=o 
 
 has the roots — 3.4 and — .58, the common factors of dx and 
 dz determine no cusps upon the real branches of the curve. 
 
 The wholly isolated part of the curve resembles a semi- 
 cubic parabola with a cusp on the ;r-axis at.T= 3/4, from which 
 point the curve extends in the positive direction to infinity. 
 The second part consists of two branches parallel to the xr-axis 
 at 2; = 00, 5- = rh 00 when they take their rise, cutting each 
 other at an angle of 120° on the .T-axis at x = — 3/2, proceed- 
 ing from this point in connection with real parts of the sur- 
 face, intersecting the geodesic in the two points ± 2.128 on 
 the ^-axis and becoming parallel to the .;i:-axis at infinity. 
 
 To recapitulate, the curves in the jr^-plane are: 
 
 I straight line (the .r-axis) of multiplicity 4 4 
 
 I geodesic of degree 4, and multiplicity i 4 
 
 I curve of degree 10, and multiplicity 2 20 
 
 28 
 
 9. Symmetry of the Surface. — It is unnecessary to make a 
 special study of the curves of intersection in the planes 
 _y = ± v^3 X. The sections of a surface by a set of planes of 
 symmetry that pass through a straight line, as the ;?-axis, and 
 are so related that the angles between consecutive planes are 
 equal, cannot be of more than two types; for any one of the 
 
 planes (i), (3), (5) is symmetric to some other of the same 
 
 group with respect to some one of the planes (2), (4), (6) 
 
 and vice versa. If «, the number of planes, is odd, the w'^ 
 plane belongs to the same group as the first, but at the same 
 time is symmetric to the second plane with respect to the 
 first, so that the two types of section furnished by consecutive 
 planes can differ only in the position of their component curves 
 with respect to the origin. 
 
 In the present case since the curves in the .I'^-plane are 
 symmetric to the a;-axis, the two types of section made by the 
 three vertical planes differ only in the relation of their curves 
 to the ^•-axis. The geodesies and double curves in all three 
 planes pass through the same two points on the ^-axis. These 
 are singular points for the surface of multiplicity three. 
 
 The determination of the planes and axes of symmetry of 
 a minimal surface by means of its plane geodesies and straight 
 lines is not necessarily exhaustive since the theorems con- 
 verse to those of Schwarz are not true. But that in the pres- 
 
13 
 
 ent instance it is complete is readily seen. If the double sur- 
 face 1-1 — 2 should possess an additional plane of symmetry, 
 the latter must bisect the angle between two of the planes 
 j/ = <?, j/ = rb^3 X since in any other position it would re- 
 quire the existence of a second set of straight lines lying on 
 the surface. The existence of one such plane would imply 
 the existence of three bisecting the angles between the planes 
 of symmetry already found, and presenting a third type of 
 section. As no such type can exist, neither can the assumed 
 planes. 
 
 There can be no additional axes of symmetry ; for they could 
 be no other than a set of three bisecting the angles between 
 the straight lines of the surface, and in this position since the 
 surface is symmetrical to the a;j/-plane, they would with the 
 ^-axis determine three additional planes of symmetry. 
 
 It is possible to place some limitation on the nature of the 
 curves of intersection of an algebraic surface and a plane of 
 symmetry. Such curves, if neither imaginary nor multiple, 
 must be geodesies. If F (,r, j/, :;) = o represents an algebraic 
 surface referred to a plane of symmetry as the a'j'-plane, 
 F (.?•, J/, ^) contains only even powers of ;;• and dFjds contains 
 odd powers in every term. The direction cosines of the normal 
 at (:r, J, z) are : 
 
 d F dF 
 
 9 X d y 
 
 \ m '+ (1-^) '+ m "■ ^ (H) -- (Q '+ a) ■ 
 
 d F 
 
 dz 
 
 mr-i^y-m) 
 
 dF 
 Along a curve of section in the jrj/-plane, ^^ = o. Un- 
 
 less at the same time / \ 4- / \ _l / \ = o 
 
 \1{9 x) ^ \9y) + K9 ::) 
 
 that is, unless the curve is multiple, the normal to the sur- 
 face falls into the arj/-plane and coincides with the normal to 
 the curve. The latter is therefore a geodesic. 
 
 10. Curves Corresponding to the Parallels and Meridians on 
 the Sphere. — These constitute two systems cutting each other 
 orthogonally and obtained respectively by means of the 
 relations 
 
 (33) «^ ^}? ^c, 
 
14 
 
 and 
 
 (34) V ^ ft u or = k. 
 
 The A Curves. — The A curves, as they may be called, are of 
 the eighth degree, with the exception of two curves of the 
 fourth degree, namely, the geodesic hypocycloid, uv = i, and 
 the real line at infinity, uv = 0. 
 
 Since Af '^ = — As ' (-r + -l>), a£ ~ * * = — Ae ~ ' <" + ^), and on a 
 
 double surface Af '* Af ~ '* and —si(^+4>), — e ~ ^ ('^ + '^) fur- 
 
 A A 
 
 nish the same point, all of the curves of the system are 
 obtained by assigning to A values ranging from i to co . A 
 being fixed, the substitutions cp, n — qj, 7t ^ (p show the 
 curves to be symmetric with respect to the xy- and xs- 
 planes and to the .i'-axis, and therefore to all the planes and 
 axes of symmetry. 
 
 The curves are closed. With the exception of the 
 geodesic hypocycloid, each curve cuts the .i^^-plane in three 
 double points, one on each of the straight lines of the surface 
 and cuts each vertical plane of symmetry in a double point 
 on the straight line, two ordinary points on the geodesic and 
 two double points on the double curve. One curve of the 
 system cuts the ^-axis in the two singular points. No two of 
 the curves intersect. 
 
 The cp Curves. — These are of the fourth degree. They 
 include the vertical geodesies and the straight lines of the 
 surface. All the curves of the system are obtained by assign- 
 ing to cp values ranging from to ;r, and all the points of a 
 curve, by permitting \ to vary from to ^ . 
 
 Each curve is symmetric with respect to the ;v^-plane and 
 to cp and n — cp correspond curves symmetrically situated 
 with respect to the rr^-plane and intersecting upon the double 
 curve in that plane if 45° ^ 9? ^ 90°. 
 
 From the direction cosines of the tangent 
 
 — (A'^ — i) cos \(p + A"' (A* — i) COS ^cp 
 P 1/2 '- ' 
 
 — (A'"* — i) sin 4(p — A- (A^ — i) sin 2<p 2 (A" + i) cos 3(p 
 
 P 1/2 ' P z/2 
 
 where 
 
 P 1/2 = 
 ^(A8_i)24-(A«— A2)^— 2(A«— OCA"— A-)cos6<p + 4A^ (A^ + i)^ cos^3<p 
 
 it is evident that the curves cut the geodesic hypocycloid 
 orthogonally except at the cusps and become parallel to the 
 arK-plane at infinity. 
 
15 
 
 II. Model of the Surface. — By making a framework of 
 cardboard representing the non-isolated curves of inter- 
 section in the planes of symmetry, and using- this to support 
 a network of wires representing certain q) and A curves, 1 
 constructed a skeleton model of the surface. The following 
 values, multiplied throughout by two, constitute the data 
 for the construction. 
 
 Geodesic 
 
 Geodesic in 
 
 Double Curve in 
 
 Hypocycloid. 
 
 X2- 
 
 Plane 
 
 
 xs-Plane. 
 
 i> 
 
 X 
 
 y 
 
 P 
 
 X 
 
 ^ 
 
 r[=ff) 
 
 X 
 
 z 
 
 oo 
 
 —1.50 
 
 
 
 
 
 •50 
 
 
 
 
 
 — 1-5 
 
 
 
 20° 
 
 — I 
 
 32 
 
 — .02 
 
 I 
 
 ■49 
 
 .20 
 
 . I 
 
 — I 
 
 23 
 
 •45 
 
 40° 
 
 — 
 
 «5 
 
 — -15 
 
 2 
 
 .48 
 
 .40 
 
 .2 
 
 — 
 
 99 
 
 .82 
 
 60° 
 
 — 
 
 2q 
 
 — -43 
 
 3 
 
 •45 
 
 .62 
 
 •3 
 
 — 
 
 79 
 
 1. 12 
 
 80O 
 
 
 30 
 
 — .81 
 
 4 
 
 .41 
 
 .84 
 
 •4 
 
 — 
 
 59 
 
 1^39 
 
 90° 
 
 
 SO 
 
 — ^i .00 
 
 5 
 
 ■36 
 
 1.08 
 
 ■5 
 
 — 
 
 40 
 
 1.63 
 
 iioo 
 
 
 73 
 
 — 1.26 
 
 6 
 
 .29 
 
 1-34 
 
 .6 
 
 — 
 
 23 
 
 I 85 
 
 120° 
 
 
 75 
 
 — 1 .30 
 
 7 
 
 .19 
 
 1.63 
 
 •7 
 
 — 
 
 04 
 
 2.06 
 
 130° 
 
 
 73 
 
 -1.26 
 
 8 
 
 .08 
 
 1.94 
 
 .8 
 
 
 12 
 
 2.27 
 
 150° 
 
 
 62 
 
 — -93 
 
 86 
 
 
 
 2. 13 
 
 •9 
 
 
 29 
 
 2.46 
 
 170° 
 
 
 ?i 
 
 — -34 
 
 9 
 
 -.07 
 
 2.29 
 
 1 .0 
 
 
 46 
 
 2.65 
 
 180° 
 
 
 SO 
 
 I 
 
 
 
 — .25 
 
 2.70 
 
 I . I 
 
 
 64 
 
 2.83 
 
 
 
 
 I 
 
 686 
 
 -.46 
 
 3-09 
 
 1 .2 
 
 
 8-- 
 
 3.01 
 
 
 
 
 
 
 The parallels and meridians of the sphere are conformally 
 represented on the plane Z = ohy concentric circles around 
 the origin and their radii. These correspond respectively to 
 A- = (7 and cp — k. They in turn through the logarithmic 
 function are conformally represented on a second plane by 
 lines parallel to the axes with equations 
 
 X = log A and y = q). 
 
 The A and <p curves chosen for the construction of the 
 
 It 
 
 model correspond to. lines at intervals of — , 
 
 The following 
 
 table presents the points of intersection of the two sets of 
 curves. On account of the symmetry of the surface it is un- 
 necessary to give log A and cp values greater than 4^24. 
 
 The curve A =^ 1.515 is introduced because it is the A-curve 
 that passes through the multiple points on the ^--axis. 
 
16 
 
 
 
 
 Tt 
 
 2Tt 
 
 3'^ 
 
 47ir 
 
 X 
 
 
 q} = 
 
 ^==:77 
 
 ^ — :r. 
 
 <?> = — 
 
 = 'P — 
 
 
 
 
 24 
 
 24 
 
 24 
 
 24 
 
 
 X 
 
 •5 
 
 •53 
 
 .62 
 
 •71 
 
 •75 
 
 e°=i 
 
 y 
 
 
 
 — -50 
 
 —0.94 
 
 — 1 .21 
 
 — 1.30 
 
 TT 
 
 z 
 
 
 
 
 
 
 
 
 
 
 
 £ 24 
 
 X 
 
 .46 
 
 .50 
 
 .61 
 
 •73 
 
 .80 
 
 = 1. 14 
 
 y 
 
 
 
 — ^55 
 
 — 1 .01 
 
 — 1.30 
 
 — 1^39 
 
 
 2 
 
 •54 
 
 •49 
 
 •38 
 
 .20 
 
 
 
 27r 
 
 
 
 
 
 
 
 £ ^r 
 
 X 
 
 •34 
 
 •38 
 
 •59 
 
 .81 
 
 •97 
 
 = 1.299 
 
 y 
 
 
 
 - .69 
 
 —1.26 
 
 —1. 61 
 
 — 1.68 
 
 
 z 
 
 1. 16 
 
 1 .06 
 
 •81 
 
 •44 
 
 
 
 Sf 
 
 
 
 
 
 
 
 £ 24" 
 
 X 
 
 .05 
 
 .18 
 
 •51 
 
 •94 
 
 1.29 
 
 = 1.48 
 
 y 
 
 
 
 - .98 
 
 -1.76 
 
 —2.21 
 
 —2.25 
 
 
 z 
 
 1 .96 
 
 1. 81 
 
 1-39 
 
 •75 
 
 
 
 
 X 
 
 
 
 .14 
 
 .50 
 
 •97 
 
 1^37 
 
 A = 1.515 
 
 V 
 
 
 
 — 1 .04 
 
 —1.86 
 
 —2-33 
 
 —2.37 
 
 
 ly 
 
 2.13 
 
 1.97 
 
 1. 51 
 
 .81 
 
 
 
 1!L 
 
 
 
 
 
 
 
 f 24 
 
 X 
 
 -.46 
 
 — .24 
 
 •36 
 
 ^•13 
 
 1.83 
 
 = 1.686 
 
 y 
 
 
 
 —1.44 
 
 -2.58 
 
 — 3^19 
 
 -3.16 
 
 
 s 
 
 3-07 
 
 2.83 
 
 2.12 
 
 1. 17 
 
 
 
 I am indebted to the kindness of Professor Hallock of 
 Columbia University for the model of which two views are 
 presented in Plate I. Professor Hallock formed three shells 
 of paper upon a plaster impression of a clay model that I had 
 made of the large concavity shown in Fig. i. By joining 
 these shells and supplying minor parts above and below the 
 singular points on the axis of 2, he completed the model 
 depicted. 
 
 II. The Conjugate Surface /t = 2. 
 
 12. General Relations Between a Mini)nal Surface and Its 
 Conjugate. — By virtue of these, many of the properties of the 
 surface now to be treated can be inferred at once from those 
 of the surface just discussed. 
 
 As the equations of the conjugate are derived from those 
 of the original surface by the substitution of iF{ii)^ — ?'i^(T') 
 for F{ti), F^ {v) or of if {u), — if {v) for f{u),f (»), the conju- 
 gate of a double surface cannot itself be double ; for if F[u) 
 satisfies the functional relation for double surfaces, iF{u) 
 does not. 
 
 Corresponding curves on a surface and its conjugate, that 
 is, curves arising from the same relation between u and v, 
 
17 
 
 have the same spherical image. Corresponding elements are 
 orthogonal. Lines of curvature on the one surface correspond 
 to asymptotic lines on the other and vice versa. 
 
 The origin is a conical point on the conjugate of a double 
 surface. It is also a point of symmetry. From the charac- 
 teristic property already noted that each point corresponds 
 to two sets of values of the parameters, u.,v and — j)"', — 7/~\ it 
 follows that the co-ordinates of a double surface may be 
 represented by equations 
 
 X = (p (ii) -f- cp^ {v) = (p{— 7 -') -f- ^1 (— u-^) 
 
 (35) J= <^ («) + (^i M =<^{- ^~') + ^1 (- «-') 
 2 =x {u) 4- Xi (^) =x{— ^~') + Xi (— u~^) 
 
 Evidently q? (u) = (pi ( — n ~^), qji [v) = cp ( — v ^^) and similar 
 relations connect c, Gi and j, X\ so that the coordinates of 
 the conjugate surface become 
 
 X = i [<p {ii) — <Pi [v]] = — i[<p (— 7/-^) — ^] (— //-')] 
 
 (36) y = z['^ («) - <?! (^')] =-n6 (- V-') - 6, (- «-')] 
 z = z[X («) - X, {^')^ =-nx (- v-^) - X, (- ?^-')] 
 
 These equations show that to each point of the double sur- 
 face correspond on the conjugate two points symmetrically 
 situated with respect to the origin. 
 
 To one curve upon a double surface may correspond two 
 upon the conjugate. The values //, v and — v ~\ — 21 ~ "^ 
 determine upon the unit sphere points (;?■, J, z), ( — .r, — j, — z). 
 The complete spherical image of a curve on the double sur- 
 face must therefore be symmetric with respect to the origin, 
 but it inay consist of one curve, as for instance a meridian, 
 the two halves of which are given by the first and second sets 
 of values of the parameter respectively, or of two equal inde- 
 pendent symmetrically situated curves, as, for instance, two 
 parallels. In the first case, the spherical image will represent 
 a single curve on the conjugate surface ; in the second case, two. 
 
 Since the two generating minimal curves of a conjugate 
 surface are derived from those of the original by a trans- 
 formation that simply multiplies the distance of each point 
 from the origin by a constant factor, / in the case of one curve, 
 — /in that of the other, it follows that finite points remain 
 finite ; infinitely distant points remain infinitely distant ; the 
 multiplicity of points is unchanged ; stationary points corre- 
 spond to stationary points ; coincident points of the two 
 curves at infinity are unaltered, as are also the order, class 
 
18 
 
 and rank of the curves, the multiplicity of the imag-inary 
 circle at infinity upon their tangential surfaces and the rela- 
 tion of the plane of osculation and the tangent at infinitely 
 distant points to the infinitely distant plane and circle. 
 
 As it is upon these properties that the determination of 
 the infinitely distant elements and of the class and order of a 
 minimal surface depends, it follows that a surface and its 
 conjugate must have the same infinitely distant elements, 
 class and order unless the original surface is double. In this 
 case the class and order of the latter will be and the multiplic- 
 ity of the lines at infinity may be, only one-half of the corre- 
 sponding values for the conjugate surface. This is due to 
 reductions of the type already encountered, resulting from the 
 double generation of the double surface. 
 
 13. Equations of the Stir face. — From these general consid- 
 erations we turn to the discussion of the surface conjugate to 
 the primitive double surface of the seventh class. Obviously 
 it is related to the surface iF-i (?/) as the surface ^2 i") is 
 related to that defined by F2 (;/). It may be called the 
 primitive conjugate surface /< = 2. 
 
 Its equations in a form corresponding to (9) are 
 
 x= z [L — Zj) 
 
 (37) y=z{M-Af,) 
 
 ^=z(A^— vVi). 
 
 and in that corresponding to (22) are 
 
 X — ^ {A.* — A~^) sin 4(p — ^ (A" — X~-} sin 2q). 
 
 (38) .y = — T (A^ — A"*) cos 4<P — i (A- — A--) cos 2cp. 
 5r = — t (A-' + A-3) sin 3<p. 
 
 14. Pla]ic Lines of Curvature and Straight Lines of tJie Sur- 
 face. — To the geodesic in the .?j-plane of the double surface 
 must correspond an asymptotic line on the conjugate. This 
 must be a straight line perpendicular to the .a;j-plane since its 
 spherical image is a great circle parallel to that plane. Also 
 since every point of the geodesic is represented on the straight 
 line by two points symmetric with respect to the origin, the 
 line must pass through the origin. It is therefore the axis of z. 
 
 Similarly to the geodesies in the planes _;j' = o,y^ ± ■\/~^ x 
 correspond three straight lines in the .rj/- plane, a-=(?, x^T- VTy- 
 To the straight lines y = o, z = 0; y = ± y^~7 x, z = corre- 
 spond three geodesies in the planes x = 0, x = ^ y' V y. The 
 
19 
 
 conjugate surface is therefore characterized by four straight 
 lines, among which are the y- and 2;-axes, and three plane 
 geodesies, one of which lies in theji^z-plane. 
 
 The surface can have no other straight lines, nor plane 
 geodesies, nor any plane lines of curvature other than geo- 
 desies since these would imply the existence upon the double 
 surface of plane geodesies or straight lines additional to 
 those already found or of asymptotic lines represented on the 
 sphere by small circles. 
 
 15, T/ie Minimal Curves. — The properties of these require 
 no special discussion, with the exception of the relation of con- 
 jugate pairs of stationary points to the surface. 
 
 Such points are obtained by putting conjugate values of 
 It and V derived from 
 
 «" = — I and 7/^ = — I 
 
 into the equations of the minimal curves {a.') and {ft'). These 
 become for a pair of values, ti and u~^. ' 
 
 [a') X = 2iL, y = ziM, z = 2iN, 
 
 (ft') X = — 2/Zi, / = — 2/il/i, z =— 2lA\, 
 
 = — 2lL, = — 2zM, = 2l'N. 
 
 The conjugate points as was to be anticipated are sym- 
 metrically disposed to the axis of z. 
 
 Equations (37) when a and u~^ are put for ?/ and v become 
 
 ,6 
 
 X ^=y = o 
 
 (39) ^-^' "' 
 
 3 'i 
 
 Under the condition ?/' = — i, this value of z becomes ± 4/3. 
 The pairs of points under discussion are therefore connected 
 with two points on the ;j-axis. Since in the case of the double 
 surface, such pairs are connected with the cusps of the 
 geodesic hypocycloid from which the straight lines of the 
 surface spring, we may expect in the case of the conjugate 
 surface to find them connected with points on the ^-axis from 
 which spring the geodesies. It is to be expected that the 
 relation of tangency between the straight lines and the hypo- 
 cycloid will be preserved between the geodesies and the axis 
 of z. 
 
 16. The Infinitely Distant Elements. — These lie upon the 
 same straight lines as in the case of the double surface, but 
 the multiplicity of the real line in the a;j-plane is doubled. 
 
 17. Class and Order of the Surface. — These are fourteen 
 and fifty-six respectively. 
 
20 
 
 1 8. Curves of Intersection in the ,i"j/-plane. — These are deter- 
 mined by the equations 
 
 {A) A- 4- I = o, {B) sin 3^ = o. 
 
 Under condition {A) 
 
 X^ _ X-' = A- — X-- = A^ + A^ 
 
 By substituting 
 
 A ^ cos -^^—2 — 7t -\- 1 sm -^—^ — ^, [g an integer) 
 
 and reducing, A^ -|- A* may be thrown into the form 
 
 zcg = 2z sin — n, [g = o, i, 2). 
 
 It — ip 
 Putting v = , the equations of the section become 
 
 X = ic,j ( — 1/4 sin 2^ 4- 1/2 sin^) 
 
 (40) 
 
 y = ic,j (1/4 cos 2^ — 1/2 cos^). 
 
 These represent three imaginary hypocycloids with ver- 
 tices on the j-axis as initial points. Of these curves the origin is 
 one, appearing as a limiting case. It corresponds to the value 
 A2 = — T. For this x = y = z = irrespective of the value 
 of cp. The origin accordingly is a conical point upon the 
 surface. It appears upon every qt curve, but in general as 
 an isolated point. It is of infinite multiplicity, but since in 
 this case it appears as a hypocycloid it may be regarded as of 
 the same degree as the other two. , 
 
 These may be shown by means of the relations uv = — a?, 
 nv — — 09^ to be of degree eight and multiplicity one. 
 
 Under condition {B). 
 
 sin 4<p cos 4(p 
 
 sin 2<p cos icp 
 
 and equations (38) reduce to 
 
 , . ^^^=_J^= I^(A«^2A«-2A2-I) 
 
 (41) sin 7.cp cos zq> 4A* ^ 
 5 = o. 
 
 These are the equations of three straight lines of multiplicity 
 eight. From the equation 
 
 (42) A"* + 2 A'' + 4 { — ^ — ^ A* _ A- — I = o 
 
 \ cos 2<p / 
 
21 ,,>-^:\^".'-^' r 
 
 it is clear that each line throughout itkcpursecis co^n^ted 
 with two real nappes of the surface. "^"^'^^t.sdLEI''-"" 
 
 The degree of the surface may be estimated from the 
 degrees of the curves of intersection. These are in the vy- 
 plane 
 
 3 finite straight lines of multiplicity 8 24 
 
 1 straight line at infinity of multiplicity 8 8 
 
 2 imaginary hypocycloids of multiplicity i, degree 8 16 
 
 The origin 8 
 
 56 
 19. Curves of Intersection in the yz- Plane. — These satisfy 
 the conditions. 
 
 [A] A^ — 1=0, [E) sin 2<p = o, (C) cos 2<p = -rv— — ■. 
 
 A -f- I 
 
 Condition (yi) breaks up into two; {a) A^ = — i, which gives 
 the origin and {b) A^ = i. The latter leads to the equations of 
 the 2-axis. 
 
 .r ^_y = o 
 
 (43) . . , 
 
 ^^ = — ^ sm 3<p. 
 
 Between the points z = ± 4/;^ the line is the intersection of 
 six real nappes of the surface since each of its points corre- 
 sponds to six real values of cp. Beyond the limits ± 4/3 it is 
 isolated. 
 
 Condition (B) also breaks up into two ; (a) cp ^ 0, the con- 
 dition for the t'-axis and (d) 99 = tt/i from which 
 
 X = o 
 
 _y =.— 
 
 A«_2 A''— 2 A-— I (A^_ i)'^(A^4- I) 
 
 (44) -" 4A^ 4A^ 
 
 A^J _ . (A-^+ I) (A^-A-+ I) 
 
 ^= t 
 
 A.^ ~~ ' A^ 
 
 When q) = 7r/2, ?/ = — v, and A' = o. The curve (44) is 
 therefore the geodesic in the jF^-plane. It is of multiplicity 
 one ana degree eight. It is symmetric with respect to the 
 axes. Aside from the origin which appears as an isolated 
 point, it has on the axis of j two imaginary double points, and 
 on the axis of z, two real points ± 4/3. 
 
 On throwing the equation for y into the form 
 
 (45) r^ — 2 r'^ + 4j(/ r- + 2 r — I = o, (r'- = A) 
 
 and forming the discriminant 
 
 J=-/(2 7 + i6/) 
 
it is found that J has only one real positive root. It follows 
 that to a given value of j/ correspond only two real values of z. 
 These are equal and of opposite sign. The curve consists of 
 two detached parts symmetrically disposed to the jF-axis. 
 The value of the derivative shows that each of these resem- 
 bles a semicubical parabola with a cusp on the 2-axis at + 
 4/3, or — 4/3. 
 
 dz= 2 A-*(A"-^— I) {1} + A^4-l)^/A, ^ = — A-^(A'' — i)-(A^ + A--|-l)^/A 
 
 dz 2 A 
 
 Ify ^ "~ A^ — i' 
 
 The curve has four imaginary cusps corresponding to 
 the roots of A^ + A- + i = o. It becomes parallel to the axis 
 of J at infinity. 
 
 From condition {C) result the equations 
 
 .r = o 
 
 12 ,\3 n2 1 T^3 
 
 y = 
 
 (A^-ir(A^ + ir 
 
 (46) ^ 4 A^ (A^ + I) 
 
 ^ =. _ ^/T (A^ + i)^ (A^ - A^ + if' 
 ^^ A^ (A* 4- 1)3/2 
 
 The curve is double and symmetric with respect to the 
 axes. It is of degree fourteen. On writing the equation for 
 y in the form 
 
 ^' - (3 + \y) r~ + (3 - 4 j) r - I = o. (r = A^) 
 and forming the discriminant 
 
 A = — 64/27/ (4/ — 9) 
 
 it is seen that the twelve values of A corresponding to a given 
 value oiy are imaginary, with the exception of two that are 
 equal and of opposite sign. 
 
 The curve resembles the geodesic. It cuts the axes in the 
 same points; consists of two non-intersecting branches sym- 
 metric to the axis of >' ; has cusps on the ;?-axis at ± 4/3 and 
 becomes parallel to the y-axis at infinity. It differs from the 
 geodesic in the position of its imaginary cusps, determined 
 from the common factors of dz and dy 
 
 (A^-f i)Ma^- 1) (A'+4A^ -I- I) 
 A* (A* + 1)2 
 
 dz ,_ A /A^ — A^-fiV/^ 
 
 A (r' — X'^\\ 
 V 2 A Viri V A* + I / 
 
 dj/ 1" ^ A^ — I \ A* + 
 
23 
 
 There are in the li-plane : 
 
 I straight line (s--axis) of multiplicity 6 6 
 
 I " " (jz-axis) " " 8 8 
 
 I geodesic of degree 8 " " i 8 
 
 I curve " " 14" " 2 28 
 
 Origin ? 
 
 In this case the origin enters through a condition similar 
 in form to that for the axis of z. If it is allowable to regard 
 it here as of the same degree as the 2-axis it brings the sum 
 of the degrees of the curves of intersection to the desired 
 figure, fifty-six. 
 
 20. Svjuuictry of the Surfaces. — Since the curves in the yz- 
 plane are symmetric with respect to the axes, the three planes 
 of the geodesies can yield but one type of section. This sug- 
 gests that the surface possesses a second set of planes of sym- 
 metry. On substituting for Ji and v in equations (37), the 
 values — //, — V \ — ■?', — ti ; — it "', — i> ~\ it is found that 
 the surface is symmetrical with respect to all of the co- 
 ordinate planes and consequently to the axes. The ^-^-plane 
 and .r-axis, then, belong to a second set of planes and axes of 
 symmetry not discoverable throtigh the theorems of Schwarz. 
 
 21. Curves of Intersection in the xz-Plane. — The conditions 
 to be satisfied are 
 
 As before, {A) gives the origin and the 2;-axis. i^B) leads to 
 a curve with the equations 
 
 X 
 
 _ (A^— i)(A'+A^+i±aV2A«-|-5A^+2)'/''(— 3A-zh-/2A^4-5A*-f2) 
 (47) "^ 4-/'^>'tM'^'4-i) 
 
 (A"+i)(2A''+A-4-2+'/2AH-5A^"+2)^^^(A^— A-+I±-v/2A»+5A^+2) 
 
 ^~ 3A-\A^ + 1)3/2 
 
 For convenience of reference these may be expressed in 
 the form 
 
 _ (A^— i) A^'^ D 
 
 ■^ ~ 4 V^A* (A* + I) 
 
 (48) 
 
 ^ ^ (A« + I) C'^ D 
 
 3 A^ (A^-+ 1)3/2" 
 From the derivation of equations (47), it is clear that 
 
 (4Q) , — = sin 2<p, ,., . , c = sin (p. 
 
 ^^^^ -/ 2 (A* + I) 4('^' + I) 
 
24 
 
 The radical M = -/ 2A* 4- '^K^ + 2 is real for every real 
 value of X and is less in absolute value than (2A,* -f A^ + 2) or 
 than X -'^ {X^ -{. X* -\- 1); x and z are therefore real for all real 
 values of A. Since z is imaginary for all imaginary values of 
 A, except those for which it is zero or infinite, the curve has 
 no isolated branches. 
 
 On account of the possible combinations of the radicals 
 involved, to a given value of A correspond eight points. 
 These fall into two groups of four each, according as M is 
 taken with the upper or the lower signs in (47). The four 
 points of a group are symmetrically situated in the four 
 quadrants ; rt A.„ it i/X„ give the same points. Apparently 
 the curve is of multiplicity four, but the consideration of 
 equations (49) shows that such is not the case. When A is 
 fixed, the possible combinations of the signs of A^^ and 6 
 indicate four related values of ^, namely, ^, tt — q), tt -{■ q), 
 27r — cp. The change of A to — A gives rise to no new values 
 of u and v. The negative values of a are therefore not to be 
 counted in estimating the multiplicity of the curve. 
 
 To discover the nature of the curve, it suffices to consider 
 the change in the co-ordinates as A ranges from i to =0 . 
 When X = 1^ X — and z = ±. 4/3. An examination of the 
 factors entering into the numerators and denominators of x 
 and z shows that they all increase in absolute value with 
 A (A > i), whether Mhe taken with the upper or the lower 
 signs throughout. It follows that the curve consists of two 
 real branches in each quadrant, taking their rise on the axis 
 of 2 at 4- 4/3 or — 4/3 and proceeding to infinity. 
 
 22. Curves Corresponding to the Parallels anei Meridians. 
 
 The A Curves. — These are of the eighth degree, with the 
 exception of the axis of z, which is of multiplicity six. To a 
 curve A on the double surface correspond two on the conju- 
 gate, furnished by A and i/A. Each of these curves is sym- 
 metric with respect to the xy- and j^-planes, and the two are 
 symmetric to each other with respect to the .r^-plane. They 
 accordingly intersect in this plane on the double curve (47). 
 
 A given curve of the system cuts the -ty^-plane in three 
 double points one on each of the straight lines x = 0, x = 
 + '^ Z y '■> the _>/;-plane, in a double point on the _>/-axis, two 
 double points on the double curve and two ordinary points 
 on the geodesic ; and the .t-^^-plane, in eight ordinary points, 
 two in each quadrant, one on each branch of the double curve. 
 
 The qj Curves. — These are of degree eight. When q) is fixed, 
 the substitutions ± A, ± i/A show the ciirve to be symmetric to 
 the xy-'pl&nQ and the origin. Since, except in the case of the 
 
25 
 
 straight lines, the intersections of the curves with the xy- 
 plane are imaginary or isolated, each curve consists of two 
 non-intersecting branches. Curves cp and -t — q) are sym- 
 metric to each other with respect to the xz- and v^-planes 
 and intersect on the double curves in those planes. Through 
 each point on the >'-axis between ± 4/3 pass six (^-curves. 
 The direction cosines of the tangent are 
 
 [X^ -f- i) sin 4(p — A^ (A.'"^ -|- i) sin 2<p 
 
 (A^-hj) cos 4(p + X^ (A^ 4- I) cos 2cp 
 
 P-l/i 
 
 2 A (A"* — i) sin 3^ 
 
 where 
 
 P^h 
 
 V(A^-|-i)2-f A* (A^+i)-'^ 2 A^ (A'*^ + I) (A'^-f i) cos 6<p-f- 4 A^ (A^— i)- sin 3<p 
 
 These values show that the ^-curves, with the exception of 
 the geodesies, cut the axis of z orthogonally, and all become 
 parallel to the .t'r-plane at infinity. 
 
 23. Model of the Surface. — The following values multi- 
 plied throughout by two furnished the data for a skeleton 
 model of the surface, made on the same plan as that of the 
 double surface. 
 
 Geodesic in yz- 
 
 Plane. 
 
 Double Curve in j^'-Plane. 
 
 A 
 
 y 
 
 s 
 
 A 
 
 y 
 
 z 
 
 I 
 
 
 
 1-34 
 
 I 
 
 
 
 1-34 
 
 1.2 
 
 — •03 
 
 1.54 
 
 1. 14 
 
 .02 
 
 1.44 
 
 1.4 
 
 — .17 
 
 2.07 
 
 1.299 
 
 .14 
 
 1.76 
 
 1.6 
 
 — •51 
 
 2.89 
 
 1.48 
 
 ■49 
 
 2.32 
 
 1.686 
 
 — •75 
 
 3^34 
 
 1.686 
 
 1.22 
 
 3^19 
 
 Double Curve in xs'-Plane. 
 
 Branch I. 
 {M taken with upper signs.) 
 
 Branch II. 
 {M taken with lower signs.) 
 
 A 
 
 X 
 
 z 
 
 A 
 
 X 
 
 S 
 
 I 
 1. 14 
 1.299 
 1.48 
 1.686 
 
 
 
 .01 
 
 .09 
 
 •32 
 .84 
 
 1.34 
 1.44 
 
 1.76 
 2.36 
 
 3^i8 
 
 I 
 
 1. 14 
 1.299 
 1.48 
 1.686 
 
 
 .12 
 
 •49 
 1. 16 
 
 2.26 
 
 1.34 
 1.40 
 1.60 
 
 1-95 
 2.36 
 
26 
 
 Intersections of Certain A- and ^j-Ccjrves. 
 
 
 
 
 n 
 
 in 
 
 1"^ 
 
 47r 
 
 A 
 
 
 qj =z 
 
 ^ = T7 
 
 ^ = ^7 
 
 (P=- — 
 
 <P = — 
 
 
 
 
 24 
 
 24 
 
 24 
 
 24 
 
 
 .1- 
 
 
 
 
 
 
 
 
 
 
 
 f° = i 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 IT 
 
 z 
 
 
 
 — -51 
 
 — .94 
 
 —1.23 
 
 — '•33 
 
 e 24 
 
 X 
 
 
 
 .07 
 
 .10 
 
 .09 
 
 .01 
 
 = 1. 14 
 
 y 
 
 — .54 
 
 — -49 
 
 — -37 
 
 — .19 
 
 
 
 
 z 
 
 
 
 — -55 
 
 — 1.02 
 
 — 1-33 
 
 — 1.44 
 
 2jr 
 
 
 
 
 
 
 
 s 24 
 
 X 
 
 
 
 •17 
 
 .27 
 
 .24 
 
 .07 
 
 = 1.299 
 
 y 
 
 — 1. 12 
 
 — 1.07 
 
 - -79 
 
 — -39 
 
 .04 
 
 
 z 
 
 
 
 — .68 
 
 — 1.25 
 
 — 1.63 
 
 —1.77 
 
 ^ 
 
 
 
 
 
 
 
 E 24 
 
 X 
 
 
 
 •35 
 
 .56 
 
 •53 
 
 .24 
 
 = 1.48 
 
 y 
 
 — 2.02 
 
 -1.83 
 
 —1.32 
 
 — .61 
 
 .14 
 
 
 z 
 
 
 
 — -91 
 
 — 1.68 
 
 — 2.19 
 
 —2.37 
 
 47r 
 
 
 
 
 
 
 
 £ 24 
 
 X 
 
 
 
 .68 
 
 1. 10 
 
 I. II 
 
 .65 
 
 = 1.686 
 
 y 
 
 —3-24 
 
 —2.93 
 
 —2.08 
 
 — .88 
 
 •37 
 
 
 z 
 
 
 
 —1.28 
 
 -2.36 
 
 —3-09 
 
 —3-34 
 
 My sincere thanks are due to Professor F. D. Sherman, of 
 Columbia University, who from the wire model made the 
 drawing- of the surface reproduced in Plate II. 
 
 III. Properties of the Primitive Surfaces in General. 
 
 When the methods used in the preceding pages are ap- 
 plied to the discussion of the primitive surfaces double and 
 conjugate for a general value of /^, results are obtained that 
 are presented below in tabular form. By reference to page 4 
 the results due to the paper already cited may be dis- 
 tinguished. In treating the conjugate surfaces, only those 
 points in which they differ from the corresponding double 
 surfaces will be noted. 
 
27 
 
 Double 
 
 Priniittvc Surfaces. 
 
 IX either odd or even. 
 
 Conjugate 
 
 1 . Properties of the minimal curve : 
 Order, 2/< -f- 4; class, 2// -\-2; rank, 
 VI + 4- 
 
 Multiplicity of circle at infinity on 
 tangential surface is i. 
 
 Number of coincident points at 
 infinity, 2/« + 4. 
 
 Plane of osculation at infinitely 
 distant points falls into plane at 
 infinity, and tangent line becomes 
 tangent to imaginary circle. 
 
 Stationary points are furnished by 
 the roots of W^t^ + 2 = ( _ i)^ + '• 
 
 2. Class of the Surface, 2/< -f- 3- 
 Order of the Surface, (2//4-3)(/' + 2). 
 
 3. Infinitely distant elements lie 
 on three straight lines of multiplicity 
 [n + 2) with equations. 
 
 ^^ = o r = o 
 X -\- i y =: o r = o 
 X — i y ^ o r ^ o 
 
 4. All plane lines of curvature 
 on the surface are geodesies. 
 
 No asymptotic lines exist corre- 
 sponding to small circles on the 
 sphere. 
 
 The surface possesses (// + i) 
 geodesies lying in planes passing 
 through the axis of z. To these 
 planes the .I'S'-plane always belongs. 
 
 The surface possesses (// -f- i) 
 straight lines lying in the .r_y-plane 
 and passing through the origin. 
 
 5. There are no vertical planes of 
 symmetry, nor axes of symmetry 
 that lie in the ay/-plane, aside from 
 those determined by the geodesies 
 and straight lines mentioned above. 
 
 Class, 2(2// -1- 3). 
 
 Order, 2(2// + 3) {n + 2). 
 
 The real line is of multiplicity 
 
 2(/< + 2). 
 
 To these planes the .rr-plane does 
 not always belong. 
 
 Among the {h + 1) lines, the axis 
 of/ is always included. 
 
 5. There are (/< + i) additional 
 planes and (/< -f- O ^^^s of symmetry, 
 bisecting the angles between the 
 planes and lines of the first set. 
 The planes of the second set do not 
 contain geodesies. The lines do 
 not lie on the surface. 
 
 Also the .t;y-plane and 3-axis are 
 respectively a plane and axis of 
 symmetry. 
 
 /^ eve7t. 
 
 I. The surface possesses a geo- 
 desic in the .ry-plane. The curve 
 is a hypocycloid with (// + i) cusps. 
 It is cut by each straight line of the 
 surface in a cusp, an ordinary point 
 
 and double points. 
 
 The .rjz-plane is a plane of sym- 
 metry containing a geodesic. 
 
 I. The s-axis between the points 
 ± \li-i 4- 1 is connected with 2 (// + 1 ) 
 real nappes of the surface. Outside 
 of these limits it is isolated. 
 
 The xy-plane is a plane of sym- 
 metry, not containing a geodesic. 
 
28 
 
 2. The s'-axis does not lie on the 
 surface, and is not an axis of sym- 
 metry. 
 
 3. The (/< + vertical planes of 
 symmetry pass through the (// + i) 
 straight lines and both pass through 
 the cusps of the hypocycloid. To 
 the lines always belongs the axis of x. 
 
 4. Pairs of conjugate stationary 
 points of the minimal curve are 
 symmetric to the .ij-plane, and the 
 middle points of lines joining such 
 pairs are the cusps of the geodesic 
 hypocycloid. 
 
 5. The curves of intersection in 
 the .ry-plane are 
 
 {}i -{- i) finite 
 straight lines of 
 multiplicity 
 
 {H + 2) (//+ i)(« + 2) 
 
 I line at infinity 
 of multiplicity 
 
 (/" + 2) // + 2 
 
 /</2 isolated hy- 
 pocycloids of 
 multiplicity 2, 
 degree {/i +2).. /' {11 + 2) 
 I geodesic hy- 
 pocycloid of 
 multiplicity i, 
 degree [n + 2). ._ /< + 2 
 
 (2// + 3)(/' + 2) 
 All of the hypocycloids may be 
 derived from any one of the number 
 by factors of proportionality. 
 
 The (// + i) straight lines are 
 tangent to the geodesic at the cusps 
 and from these points outward to 
 infinity are each connected with two 
 real nappes of the surface. 
 
 6. The sections by the vertical 
 planes of symmetry are of two types 
 differing in the position of corre- 
 sponding curves with respect to the 
 axis of z. 
 
 The curves of intersection in the 
 ;r2'-plane are: 
 
 I straight line (.r-axis) of mul- 
 tiplicity ( // + 2). 
 I geodesic of multiplicity i,and 
 
 degree (/< -f 2). 
 I multiple curve. 
 The geodesic resembles an or- 
 dinary parabola. It cuts the .I'-axis 
 orthogonally at the simple point on 
 the geodesic hypocycloid. It is 
 symmetric to the jr-axis and cuts the 
 ^■-axis in two real points. Through 
 these pass all the (/< + i) vertical 
 geodesies. 
 
 2. The 5--axis lies on the surface, 
 and is an axis of symmetry. 
 
 3. The planes of the geodesies 
 and the straight lines of the surface 
 are still united in position ; but the 
 hypocycloids are imaginary. To 
 the planes always belongs the yz- 
 plane. 
 
 4. Pairs of conjugate stationary 
 points are symmetric to the axis of z 
 and furnish the points ± 4//< -|- i. 
 
 5. The curves of intersection in 
 the .ry- plane are : 
 (/< + i) finite 
 
 straight hues 
 
 of multiplicity 
 
 2(/' + 2) 2(/< + I) (/< + 2) 
 
 I Straight line 
 
 at infinity of 
 
 multipli city 
 
 2{n + 2), 2 [1.1 + 2) 
 
 fi + I imaginary 
 hypocycloids 
 
 (including the 
 
 origin) of 
 
 multiplicity I, 
 
 and degree 
 
 2{H +2) 2(/<+ !)(/< + 2) 
 
 2(2A + 3)(/' + 2) 
 
 The (/< + 1) straight lines through- 
 out their course are each connected 
 with two real nappes of the surface. 
 
 6. The sections are of two types 
 according as the cutting plane does 
 or does not contain a geodesic. The 
 j/s-plane furnishes a section of the 
 first type ; the ^^•-plane, of the 
 second. 
 
 The curves in the^y^'-plane are: 
 
 I conical point, the origin, 
 
 I straight line (the axis of s) of 
 
 multiplicity, 2 (/< + 1). 
 I geodesic of multiplicity i, and 
 
 degree 2 (/< + 2). 
 I multiple curve. 
 
 The geodesic is symmetric to the 
 axes. It has two cusps on the axis 
 of s at ± 4//< + I, and cuts the 
 _y-axis in imaginary or isolated points. 
 It consists of two branches, each 
 resembling a semi-cubical parabola. 
 The curves in the .rjr-plane are 
 I conical point, the origin. 
 
29 
 
 7- The A curves are of degree 
 2 (// -f- 2) with the exception of the 
 geodesic hypocycloid and the real 
 line at infinity. These are of degree 
 (/< -f- 2). The curves are symmetric 
 to all the planes and axes of sym- 
 metry and cut the ;rK-plane in 
 (/< + i) double points, one on each 
 of the straight lines. 
 
 8. The (p curves are of degree 
 (/< -(- 2). They are symmetric to 
 the .ry-plane, which they cut orth- 
 ogonally on the geodesic. They 
 become parallel to the .tj-plane at 
 intinity. 
 
 I straight line (the ^--axis) of multi- 
 plicity 2 (/« + !)• 
 I multiple curve. 
 
 7. The A curves are of degree 
 2 (/< -\- 2) with the exception of the 
 axis of z. They are symmetric to 
 the planes of the geodesies and the 
 straight lines of the surface. Two, 
 A and '/A, are symmetrically placed 
 with reference to the remaining 
 planes of symmetry and intersect on 
 multiple curves in these planes. 
 Both curves correspond to the one 
 curve A on the double surface. 
 
 8. The q) curves are of degree 
 i[h 4- 2). They are symmetric to 
 the origin and the ,tj-plane. Each 
 consists of two non-intersecting 
 branches symmetrically placed with 
 reference to the .ij-plane and 
 cutting the axis of 2 orthogonally 
 between ± 4/« -f- i. 2 (/< -|- i) of 
 the curves pass through each non- 
 isolated point of the axis of z, and 
 all become parallel to the x^-plane 
 at infinity. 
 
 H odd 
 
 I . The axis of z between the points 
 ± 4//< -f- I is connected with (/< + i) 
 real nappes of the surface. Outside 
 of these limits it is isolated. 
 
 2. The vertical planes of the geo- 
 desies always include the xz- and 
 /^■-planes. 
 
 The straight lines in the .ij-plane 
 bisect the angles between the planes 
 of the geodesies. 
 
 3. Pairs of conjugate stationary 
 points of the minimal curve are 
 symmetric to the z axis, and fur- 
 nish the points ;? = ± 4/// 4- i. from 
 which the geodesies spring. 
 
 4. The curves of intersection in 
 the -vy-plane are : 
 
 {n 4- i) finite 
 
 straight lines 
 
 of multiplicity 
 
 U'+ 2) [h 4- I) (/' +2) 
 
 I straight line at 
 
 infinity /< 4" 2 
 
 u -\-\ . 
 
 imagmary 
 
 hypocycloids 
 of multiplicity 
 I, and degree 
 
 2 (;u 4- 2) . . . . [u -^ i) [u 4- 2) 
 (2 n 4- 3) (/< 4- 2) 
 
 1. The surface possesses a geo- 
 desic hypocycloid in the ^j- plane. 
 This has 2(// 4- cusps. It is cut 
 by each straight line of the surface 
 in two cusps and (/< — i) double 
 points. 
 
 2. The vertical planes of the geo- 
 desies always bisect the angles be- 
 tween the (/< 4" i) straight lines. 
 
 These lines always include the x- 
 and _)/-axes. 
 
 3. Pairs of conjugate stationary 
 points are symmetric to the .ij-plane 
 and furnish the cusps of the geodesic 
 hypocycloid. 
 
 4. Curves in the .rj/-plane: 
 (/< 4- i) finite 
 
 straight lines 
 of multiplicity 
 2 (/< 4- 2)... . 2(//4- i) (/<4- 2) 
 
 I straight line 
 at infinity of 
 m u 1 1 i p 1 i c ity 
 2 (/< 4- 2) 2(/< 4- 2) 
 
 I geodesic hypo- 
 cycloid of de- 
 gree 2 (/< 4- 2) 2(/< 4- 2) 
 
 isolated 
 
 2 
 
 hypocycloids 
 
 of multiplicity 
 
 2, degree 
 
 2(/< 4-2) .... 2(/<— i) (//4-2) 
 
30 
 
 The straight lines are connected 
 with real parts of the surface 
 throughout their course. One real 
 nappe passes through each. 
 
 5. The sections by the vertical 
 planes of symmetry are of two types 
 differing in the position of corre- 
 sponding curves with respect to the 
 ;ir-axis. 
 
 The curves of intersection in the 
 _y5'-plane are : 
 
 I straight line (s'-axis) of multi- 
 plicity {/t + 2). 
 
 I geodesic of degree {fi + 2). 
 
 I multiple curve. 
 
 The geodesic is symmetric to the 
 axis of z on which it has a cusp 
 at — 4//<-)-i. Its intersections with 
 the .r-axis are imaginary. It re- 
 sembles a semi-cubical parabola. 
 
 6. The A curves are of degree 
 2 (/< -|- 2) with the exception of the 
 s'-axis, and the real line at infinity. 
 The curves are symmetric to the 
 planes and lines of symmetry and 
 cut the ,ry-plane in 2 (//-(- i) ordi- 
 nary points, two symmetrically 
 placed with reference to the origin 
 on each of the (/< -}- i) straight 
 lines. 
 
 7. The cp curves are of degree 
 (u -f- 2). They are symmetric to 
 the ^--axis, and with the exception of 
 the (/< -|- i) straight lines have no 
 real points in the ,tj-plane. /< + i of 
 the curves pass through each non- 
 isolated point of the ^--axis. All 
 become parallel to the aj-plane at 
 infinity. 
 
 I h y p o c ycloid 
 (the origin) 
 of multiplicity 
 I, degree 
 
 2(«-f-2) 
 
 2{U + 2) 
 
 2(2//4-3)(/, + 2) 
 
 The straight lines are tangent to 
 the geodesic hypocycloid at the 
 cusps and proceed outward from 
 these points in connection with two 
 real nappes. 
 
 5. The sections are of two types 
 according as the planes do or do not 
 contain geodesies. None of the 
 planes of the geodesies coincides 
 with a co-ordinate plane. But by 
 rotation of the surface one may be 
 brought into coincidence with the 
 j2-p\a.ue. The curves of intersec- 
 tion are found to be : 
 
 I conical point, the origin. 
 
 I geodesic of degree 2{u + 2). 
 
 I multiple curve. 
 
 The geodesic consists of two 
 branches, each of the same type as 
 the vertical geodesies of the double 
 surfaces /< even. These branches 
 intersect on the ^•-axis. 
 
 The curves of intersection in the 
 j'^'-plane are : 
 
 I conical point, the origin. 
 
 I straight line ( r-axis) of multi- 
 plicity 2(/< + 2). 
 
 I multiple curve. 
 
 6. The A curves are of degree 
 2{/ii -f- 2). Each is symmetric with 
 respect to the ^•-axis and to the 
 vertical planes of the geodesies. 
 Curves A and ^/A. are symmetric to 
 each other with respect to the re- 
 maining planes of symmetry, and 
 give by their intersection, multiple 
 curves in those planes. Both corre- 
 spond to a curve A on the double 
 surface. 
 
 7. The (p curves are of degree 
 2{u -f- 2). They are symmetric to 
 the origin and consist of two non- 
 intersecting branches symmetrically 
 disposed to the ^-axis. They cut 
 the geodesic hypocycloid orthogon- 
 ally and become parallel to the 
 -ij-plane at infinity. 
 
PLATE I, 
 
 FIG. 1. 
 
 FIG. 2. 
 
PLATE II. 
 
' A T TFfl 
 

 K/i-iKl^TJ&a-r^ay.'-', * 
 
 ■^"■,f? ;':;•&■;