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 ■:i> 't'Sj;: 
 
 m- 
 
 PLANE CURVES OF THE EIGHTH ORDER 
 
 WITH TWO REAL FOUR-FOLD POINTS 
 HAVING DISTINCT TANGENTS AND 
 WITH NO OTHER POINT SINGULARITIES 
 
 BY 
 
 ELIZABETH BUCHANAN COWLEY 
 
 Submitted in Partial Fulfilment of the Requirements for 
 
 THE Degree of Doctor of Philosophy, in the Faculty 
 
 OF Pure Science, Columbia University 
 
 PRESS Op 
 
 The New Era Printing CoMPAur 
 Lancaster. Pa. 
 
 1908 
 
 
 ^■**.* 
 
PLANE CURVES OF THE EIGHTH ORDER 
 
 WITH TWO REAL FOUR-FOLD POINTS 
 HAVING DISTINCT TANGENTS AND 
 W^ITH NO OTHER POINT SINGULARITIES 
 
 BY 
 
 ELIZABETH BUCHANAN COWLEY 
 
 Submitted in Partial Fulfilment of the Requirements for 
 
 THE Degree of Doctor of Philosophy, in the Faculty 
 
 OF Pure Science, Columbia University 
 
 OF THE " 
 
 UNIVERsn 
 
 OF 
 
 Press ok 
 
 The New Era Printing compan- 
 
 Lancaster. Pa 
 
 190.S 
 
PLANE CURVES OF THE EIGHTH ORDER 
 
 With Two Real Four-fold Points Having Distinct Tan- 
 gents AND with No Other Point Singularities. 
 
 The purpose of this paper is the discussion of plane curves of the 
 eighth order with two four-fold points having distinct tangents and 
 with no other point singularities.* The work can be conveniently 
 arranged under four heads : 
 
 1 . Possible forms of curves. 
 
 2. Existence. 
 
 3. Classification. 
 
 4. Relation to space curves. 
 
 1. Possible Forms of Curves. 
 
 These plane curves of eighth order C^ with two four-fold points 
 and 0' are of deficiency nine. Hence the number of branches 
 cannot exceed ten.f C^ can cut the line 00' at no points except 
 and 0' . If there are odd branches, there must be an even number 
 of them, since the curve is of even order. Each odd branch passes 
 an odd number of times through one and an even number of times 
 through the other 0, for each odd branch cuts a line in an odd num- 
 ber of points. All odd branches must pass an odd number of times 
 through the same (say 0) ; for otherwise two odd branches would 
 intersect in an even number of points. As odd branches may occur, 
 it is necessary to distinguish between finite and infinite for the pres- 
 ent at least. The curves will be considered as they appear when 
 projected so as to cut the line at infinity the least possible number of 
 times. For convenience, the term loop will be used to denote a part 
 of a circuit that starts from one and returns to the same with- 
 out passing through the other ; and the term intermediate part for 
 a portion extending from one to the other. No intermediate part 
 can be a complete circuit ; a loop may or may not be a whole branch. 
 According to the scheme of classification which is adopted (see page 
 
 * No papers on this subject are known to me. 
 
 t Harnack, Mathematische Annalen, X, p. 189, 1876. 
 
 3 
 
 -j <Qp fl r^jT^ 
 
4 PLANE CURVES OF THE EIGHTH ORDER. 
 
 9), a curve which lies entirely in the finite portion of the plane is 
 not essentially changed if some of its loops or intermediate parts are 
 stretched out to cut the line at infinity an even number of times. 
 Hence it is necessary to consider only those cases in which the infi- 
 nite loops and intermediate parts cut the line at infinity only once. 
 Two such loops must intersect once at least, just as two real odd 
 algebraic curves cut in at least one real point. Hence all the infinite 
 loops must be at the same 0, for if they do not cut at an a double 
 point is introduced into the C\ 
 
 Let a = the number of infinite loops at 0. 
 h = the number of finite loops at 0. 
 c = the number of finite loops at 0', 
 d = the number of infinite intermediate parts. 
 e = the number of finite intermediate parts. 
 
 Since there are four distinct tangents at each 0, there are eight 
 directions by which a moving point may leave an 0. As an inter- 
 mediate part absorbs one direction at each 0, the number of loops at 
 equals the number at 0'. Therefore, a + b = c (1). The number 
 ez=S — (a-\-b-\-G + d)= 8 — 2(a + 6) — d. Consider a line 
 which cuts each of these finite intermediate parts and each infinite 
 loop in one real point ; i. e., a line cutting line 00' between and 0'. 
 The number of real points, 8 — 2(a -f 6) — cZ 4- a is either equal to 
 or greater than {a + d) ; for otherwise, by a homographic transfor- 
 mation, one could reduce the number of real points in which the 
 C^ is cut by the line at infinity. This last condition reduces 
 to 4: = a -\- b -\- d (2). Moreover, the number of real points in 
 which the line at infinity cuts C^ is even. Hence a + cZ is an even 
 number (3). All real positive values of a, b, c, d, e which satisfy 
 these three equations are given in the following table. The number 
 T represents the number of types. 
 
 
 a 
 
 6 
 
 c 
 
 d 
 
 e 
 
 T 
 
 A 
 
 
 
 
 
 
 
 
 
 8 
 
 1 
 
 B 
 
 
 1 
 
 1 
 
 
 6 
 
 6 
 
 C 
 
 
 2 
 
 o 
 
 
 4 
 
 30 
 
 D 
 
 
 3 
 
 3 
 
 • 
 
 2 
 
 21 
 
 E 
 
 
 4 
 
 4 
 
 
 
 
 7 
 65 
 
PLANE CURVES OF THE EIGHTH ORDER. 
 II ^ (1) 2 6 1 
 
 III 
 
 
 (2) 
 
 
 1 
 
 1 
 
 
 4 
 
 12 
 
 
 (3) 
 
 
 2 
 
 2 
 
 
 2 
 
 45 
 
 B 
 
 (1) 
 
 1 
 
 
 
 1 
 
 1 
 
 5 
 
 3 
 
 
 (2) 
 
 
 1 
 
 2 
 
 
 3 
 
 30 
 
 
 (3) 
 
 
 
 3 
 
 
 1 
 
 54 
 
 C 
 
 (1) 
 
 2 
 
 
 
 2 
 
 
 
 4 
 
 2 
 
 
 (2) 
 
 
 1 
 
 3 
 
 
 2 
 
 9 
 
 
 (3) 
 
 
 2 
 
 4 
 
 
 
 
 15 
 171 
 
 A 
 
 
 
 
 
 
 
 
 4 
 
 4 
 
 1 
 
 B 
 
 
 1 
 
 
 
 1 
 
 3 
 
 3 
 
 3 
 
 C 
 
 
 2 
 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 D 
 
 
 3 
 
 
 
 3 
 
 1 
 
 1 
 
 1 
 
 E 
 
 
 4 
 
 
 
 4 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 Total 243 
 
 Consider class I in which all the curves lie entirely in the finite 
 portion of the plane. 
 
 A. Numbering the eight directions from each as in figure 1 it 
 is readily seen that the only way to get the eight intermediate parts 
 without introducing double points is by joining the directions having 
 corresponding numbers. Hence there is only one form possible. It 
 has four branches.* 
 
 B. If the finite loops at and 0' were on different sides of the 
 line 00' , there would be in each half plane four directions at one 
 and two at the other. These could not be connected by finite inter- 
 mediate parts alone. Hence the finite loops are on the same side of 
 00' . There are two intermediate parts in this half plane and four 
 in the other. Six forms are possible ; three with three branches, 
 two with two, and one with one branch. 
 
 C. If the two loops at are on the same of 00' , those at 0' are 
 also. This requires four intermediate parts on the other side of 00' . 
 There are three forms, one with one branch and two with two 
 branches. If the two loops at (and therefore at 0') are on differ- 
 ent sides of 00' , there are twenty-seven forms ; four with one 
 branch, thirteen with two branches, six with three, and four with 
 four. 
 
 * In this discussion, only those branches are considered which contain one or 
 both multiple points. If other branches exist, they must be simple ovals. 
 
b PLANE CURVES OF THE EIGHTH ORDER. 
 
 D. On one side of 00' there are two loops at each 0; and on 
 the other side of the line there are single loops at each and two 
 intermediate parts. Twenty-one forms are possible ; six with one 
 branch, nine with two branches, and six with three. 
 
 E. With four loops at each 0, intermediate parts are lacking. 
 There are seven forms ; two with two branches, two with three, and 
 three with four. 
 
 In class I there are sixty-five possibilities ; twelve with one 
 branch, twenty-eight with two branches, seventeen with three, and 
 eight with four. 
 
 In classes II and III, curves cannot be projected entirely into the 
 finite portion of the plane. 
 
 Class II. — A. There are no infinite loops but two infinite interme- 
 diate parts. 
 
 1. If there are no loops, there are six finite intermediate parts. 
 Duly one arrangement of these is possible. It has three branches. 
 
 2. If the two finite loops are on the same side of the line 00' , 
 only one of the four intermediate parts lies in this half plane. 
 There are six forms ; two with one branch, and four with two 
 branches. If the loops are on different sides of 00' , there are two 
 finite intermediate parts in each half plane. Six forms appear ; 
 three unipartite, two bipartite, and one tripartite. 
 
 3. If the two directions to infinity from (and therefore from 0) 
 are on different sides of 00', nine forms have one branch, ten hav^e 
 two branches, and eight three. If both directions to infinity from O 
 are in one half plane, those at 0' are in the other half plane. Six 
 forms are unipartite, nine bipartite, and three tripartite. 
 
 B. One loop and one intermediate part are infinite. 
 
 1. The one finite loop at 0' must be on the same side of 00' as 
 two of the directions to infinity from 0. Hence there are two finite 
 intermediate parts in this half plane and three in the other. There 
 are three forms, two bipartite and one unipartite. 
 
 2. If the two finite loops at 0' are on the same side of 00', the one 
 at is also in this half plane. The finite intermediate parts are in 
 the other half plane. There are six forms, one has one branch, three 
 have two branches and two three branches. If the two finite loops 
 at 0' are on different sides of 00', only one of the three finite inter- 
 
PLANE CURVES OF THE EIGHTH ORDER. 7 
 
 mediate parts is in the same half plane as the one finite loop at 0. 
 Twenty-seven forms appear, of which three can be rejected at once, 
 because it is possible to construct Hues to cut the curves in more 
 than eight points. Of the twenty-four forms remaining, six have one 
 branch, twelve have two branches, and six have three. 
 
 3. Since there are two directions to infinity from in one half plane, 
 the two finite loops at are on different sides of 00'. There are fifty- 
 four forms ; six have one branch, eighteen two branches, twenty-one 
 three, and nine four. 
 
 C There are two infinite loops. (Hence there are two directions 
 to infinity from in each half plane.) 
 
 1. If there are no finite loops at 0, there are two finite inter- 
 mediate parts on each side of 00' . Hence the two finite loops at G 
 are in different half planes. Of the six possible arrangements, four 
 are rejected because lines can be found to cut the curves in more than 
 eight points. One of the remaining forms is unipartite and the other 
 is bipartite. 
 
 2. If there is one finite loop at 0, it is in the same half plane as 
 two of the finite loops at 0', for there can be no finite intermediate 
 parts in this half plane. There are two in the other half plane. Of 
 the eighteen arrangements, nine are rejected for the reason given in 
 1. Two of the remaining forms have one branch, three have two 
 branches, three have three, and one has four. 
 
 3. Since there are four loops at 0, there can be no intermediate 
 parts. Twenty-one arrangements appear, of which six are rejected 
 for the reason given in 1. Of the fifteen curves remaining, two have 
 two branches, six have three, five have four, and two have five. It 
 is interesting to note that these are the only examples of these curves 
 with more than four branches. 
 
 Class III. — (Since a + cZ = 4, 6 = 0.) 
 
 A, Since the four infinite portions are intermediate parts, there 
 are no loops at 0. There is only one arrangement possible for the 
 four finite intermediate parts. The curve is bipartite. 
 
 B. If one of the four infinite parts is a loop, there is a finite loop 
 at 0' . It is in the half plane which has three directions to infinity 
 from 0. One of the finite intermediate parts is in this half plane. 
 One of the forms is tripartite and two are bipartite. 
 
8 PLANE CURVES OF THE EIGHTH ORDER. 
 
 C If there are two infinite loops and two infinite intermediate 
 parts, there are four directions to infinity from in one half plane. 
 The two finite loops at 0' must lie in this same half plane, and the 
 two finite intermediates in the other half plane. There are two 
 forms having three and four branches respectively. 
 
 D. If three of the infinite parts are loops, there are four directions 
 to infinity from Q in one half plane. Hence two of the three finite 
 loops at lie in this half plane. There are six arrangements, but 
 five are rejected for the reason given above. The curve has four 
 branches. 
 
 E. The presence of four infinite loops at necessitates four finite 
 loops at 0' . Two arrangements appear, but both are rejected for 
 the reason given above. 
 
 All forms with infinite branches have now been found. Of the 
 
 two hundred and seven, twenty-nine were rejected because in each 
 
 case a line could be found to cut the curve in more than eight real 
 
 points. One hundred and seventy-eight curves remain ; thirty-seven 
 
 have one branch, sixty-nine have two branches, fifty-three have 
 
 three, seventeen have four, and two have five. These, with the 
 
 sixty-five which are entirely in the finite portion of the plane, 
 
 include all possible forms of these curves. Of the two hundred and 
 
 forty-three, ninety-nine contain odd branches. Thirty-seven have 
 
 two odd branches, forty-three have two odd and one even, fifteen 
 
 have two odd and two even, two have two odd and three even, and 
 
 two have four odd. 
 
 2. Existence. 
 
 All possible forms of the C^ have been described ; but there is no 
 assurance that these curves actually exist. The question of their 
 existence is now before us. Let C\, C*, C*, C* be four plane 
 quartics having at and 0' nodes with distinct tangents. Then 
 Cj C* + ^^'3 ^'4 = ^ represents a curve of the eighth order with four- 
 fold points at and 0'. If X is very small the curve is nearly 
 C*Cl = 0, but with breaks at the eight points of intersection of C* 
 and C* distinct from O and 0' . By taking for C* and C* all the 
 forms of binodal quartics, by placing them in all possible positions 
 relative to one another (so long as the eight outside intersections of 
 C\ and C^ are distinct from those of C^ and C^), and by making 
 
PLANE CURVES OF THE EIGHTH ORDER. » 
 
 breaks at these eight outside points of intersection, one can obtain 
 various forms. Since all of the two hundred and forty-three forms 
 are obtainable in this way, these possible forms actually exist. 
 
 3. Classification. 
 
 Since only fifty of the two hundred and forty-three curves 
 are equivalent to others by homographic transformations, it seems 
 unwise to use homographic differences as the basis of classification. 
 By a suitable birational transformation, a C^ with two four-fold 
 points goes into another C^^ with two four-fold points ; but one could 
 not be certain that every four-fold point had separate tangents. The 
 method of classification actually used is a modification of the ideas 
 expounded by Tait,* Kirkman,t Little, | and others and applied by 
 F. Meyer, § and P. Field 1| to certain curves of the fourth and fifth 
 orders. Briefly, their scheme is as follows : The symbol for a curve 
 indicates the order in which a moving point describing the curve 
 goes through the multiple points. Calling the multiple points 
 A, B, C, D, ■•• the symbol AABBCDCD is distinguished from 
 AABCDBCD. Since the moving point may start at any point of 
 the curve, cyclic permutations of the letters in the symbol are per- 
 missible, i. e., AABCDBCD is the same as BCDBCDAA, 
 BCDAABCD, CDBCDAAB, etc. Two curves are regarded as 
 equivalent if they have the same symbol. Meyer used this method 
 for unicursal curves only. Field has used it for unicursal quintics 
 with distinct double points. In his paper on quintic curves of defi- 
 ciency one, he uses the same method when the curves are unipartite. 
 Two bipartite curves are regarded as equivalent if the symbols for 
 
 * Seven articles of the Proceedings of Edinburgh Royal Society, Vol. 9, 1877. One 
 article in the Transactions of Edin. Roy. Soc, Vol. 28, 1877. One article in the 
 Philosophical Magazine, Vol. 17, 1884. Two articles in the Trans. Edin. Roy. Soc, 
 Vol. 32, 1885. 
 
 fTwo articles in the Trans. Edin. Roy. Soc, Vol. 32, 1885. Two articles in the 
 Proc Edin. Roy. Soc, Vol. 13, 1886. 
 
 JOne article in the Trans. Edin. Roy. Soc, Vol. 35, 1889. One article in the 
 Trans. Edin. Roy. Soc, Vol. 36, 1890. 
 
 § Anwendungen der Topologie auf die Gestalten der algebraischen Curven. Dis- 
 sertation ; Magdeburg, 1878. Ueber algebraische Knoten, Proc. Edin. Roy. Soc, 
 Vol. 13, 1886. 
 
 II On the Forms of Unicursal Quintic Curves, American Journal of Mathematics, 
 Vol. 26. Quintic Curves for which p^l, American Journal of Mathematics, Vol. 27. 
 
10 PLANE CURVES OF THE EIGHTH ORDER. 
 
 the two parts of one curve are the same as those for the separate 
 parts of the other. Although the present paper employs the same 
 method for unipartite and bipartite curves and similar schemes for 
 those with more than two branches, it differs from these other papers 
 in applying it to curves with four-fold points. 
 
 The symbols for the curv'es under discussion must contain four O's 
 and four 0''s. For convenience in writing, represent by ^ and 0' 
 by B. One finds that there are seven distinct symbols for unipartite 
 curves : 
 
 1. AAAABBBB. 
 
 2. AAABBBAB. 
 
 3. AAABBABB. 
 
 4. AABBAABB. 
 
 5. AABBABAB. 
 
 6. AABABBAB. 
 
 7. ABABABAB. 
 
 For the first six symbols it is actually possible to get corresponding 
 unipartite curves. It is easily shown that it is impossible to get a 
 unipartite curve of symbol (7) without introducing additional double 
 points. The forty-nine curves of one branch are reduced to six 
 classes, denoted by the first six symbols. 
 
 In the following tables the curves corresponding to each symbol 
 are given by their numbers. In the plates only one picture is given 
 for two curves that can be obtained from one another by homographic 
 transformations. These curves are indicated by *. The number of 
 curves in the plates is therefore one hundred and ninety-three, 
 instead of two hundred and forty-three ; since fifty can be obtained 
 from others by homographic transformations. 
 
 The table for unipartite curves is : 
 
 1. AAAABBBB.— I D 4, 6, 7, 12, 15, 19. 
 
 II 5(3) 2', 16*, 18*. 
 II C(2) 2, 8. 
 
 2. AAABBBAB. — 11 A (3) 10*, 12*, 13*. 
 
 II B{2) 13, 14, 16, 18. 
 
 3. AAABBABB.— 1 CI, 10, 16, 18. 
 
 II A (3) 17*, 19*. 
 II B {2) 1, 4, 11. 
 
PLANE CURVES OF THE EIGHTH ORDER. 11 
 
 4. AABBAABB.— I C 3. 
 
 II A (3) 1*, 6, 22*. 
 II (7(1) 1. 
 
 5. AABBABAB. — ll A (2) 4, 5, 10, 
 
 II B (1) 2. 
 
 6. AABABBAB.— I B 4. 
 
 II ^ (2) 7, 9. 
 
 There are ninety-seven forms of bipartite curves and they fall into 
 seventeen classes. A dash is used to separate the symbols for the 
 two branches. 
 
 1. A-AABBABB.— II B (2) 2, 3'. 
 
 2. A-AAABBBB.— II B (3) 10", 24", 26*. 
 
 II C (2) 5, 9. 
 
 3. A-AABBBAB.— II B (2) 21, 22, 24, 26. 
 
 4. ^-^^^^5^5.-111 5 r. 
 
 5. AAA-ABBBB.— II ^ (3) 5*, 19*, 21*. 
 
 6. AAB-AABBB.— II A (3) 2*, 5*, 23", 25*. 
 
 11^(2)12,19,1'. 
 II (7(1) 2. 
 
 7. AAB-ABABB.— II ^ (2) 6, 8. 
 
 II B{1) 3. 
 
 8. AA-ABABBB.— II ^ (3) 8*, IT, 14*. 
 
 II B (2) 15, 17. 
 
 9. AA-AABBBB.— I Z> 2, 5, 8, 11, 13, 16, 17, 18, 21. 
 
 11^(3) r,3*, 17*. 
 11(7(2)1. 
 
 10. AA-ABBABB.— I (7 5, 8, 11, 17. 
 
 II A (3) 18, 20, 21. 
 
 11. AB-AABBAB.— II A (2) 2, 12. 
 
 I ^ 5, 6. 
 115(1) 1. 
 
 12. AB-AAABBB.— I (7 19, 24, 27. 
 
 II B (2) 5, 8. 
 IS. AAAA-BBBB.— 1 E 2\ 
 
 II (7 (3) 7*. 
 
12 PLANE CURVES OF THE EIGHTH ORDER. 
 
 14. AAAB-ABBB.— I (7 20, 23, 28. 
 
 II B (2) 6, 10. 
 
 15. AABB-ABAB.— II A (2) 1, 3. 
 
 16. ^^^^-^^^^. — III A 1. 
 
 17. AABB-AABB.— I (7 1, 2, 4, 6, 15. 
 
 II A (3) 16*. 
 
 There are seventy forms of curves with three branches, and they 
 are in sixteen classes. 
 
 1. A-AA-ABBBB.— II B (3) 8, 13, 15, 22*, 27*, 29*. 
 
 2. A-BB-AAABB.— II B (3) 9*, 11*, 25*. 
 
 II C (2) 4. 
 
 3. A-BB-ABAAB.— II B (2) 23, 25. 
 
 4. A-ABB-AABB.— II 5 (2) 3, 20, 27, 2'. 
 
 5. ^- J 5^-^ 5^ 5. — Ill B 2. 
 
 6. A-AAA-BBBB.— II (7(3) 4*, 5*. 
 
 7. AAA-ABB-BB.— II ^ (3) 4*, 6*, 20*. 
 
 8. AAB-AAB-BB.— 11 A (3) 3*, 4*, 24, 26, 27. 
 
 9. A-A-AABBBB.— II 6^(2) 3, 7. 
 
 10. A-A-ABBABB. — IU (7 1. 
 
 11. AA-AB-ABBB.— I C 12, 14, 21, 22, 25, 26. 
 
 II B (2) 7, 9. 
 
 12. AA-BB-AABB.— I D 1, 3, 9, 10, 14, 20. 
 
 13. AA-BB-ABAB.— II A (3) 7, 9, 15*. 
 
 14. AB-AB-ABAB.— II .4 (1) 1. 
 
 15. AB-AB-AABB.— I B 1, 2, 3. 
 
 II A (2) 11. 
 
 IQ. AA-AA-BBBB.— 1 E 4, Q. 
 
 II C (3) 6*. 
 
 There are twenty-five forms with four branches and they are in 
 nine classes. 
 
 1. AB-AB-AB-AB.— 1 A U 
 
 2. AA-BB-AB-AB.— I C9, 13, 29, 30. 
 
PLANE CURVES OF THE EIGHTH ORDER. 13 
 
 3. AA-BB-AA-BB.— IE 1,2,, 5. 
 
 4. A-ABB-AA-BB.— TI B (3) T, 12*, 14*, 23, 28, 30. 
 
 5. A-A-AABB-BB.— II C(2) 6. 
 
 6. A-AAA-BB-BB.— II C (3) 3*. 
 
 7. A-A-AA-BBBB.— II C (3) 2, 9*. 
 
 8. ^-^-^5^-^55.-111 C2. 
 
 9. ^-^-^-^5555. — IIIi>l. 
 
 There are ouly two curves with five branches and they have the 
 same symbol : A-A-AA-BB-BB. These curves are II C(3) 1, 8. 
 
 In the plates different kinds of lines are used to distinguish the 
 
 different branches : heavy lines , dots . . . . , dashes , dots 
 
 separated by dashes . . _ ., and two dots separated by a dash 
 
 4. Relation to Space Curves. 
 
 If the complete intersection C^ of two surfaces S"^ and iS* of 
 second and fourth order be projected on a plane tt from a point A 
 on aS'^ but not on C^, the projection is a C^ with four-fold points at 
 O and O' , the points in which the generators of S'^ through A meet 
 the plane tt. If /S^ and S^ do not touch (i. e., if there are no point 
 singularities on C^), there are no point singularities on C^, except 
 the two four-fold points. Conversely, such a Cp can always be 
 considered as the projection of some C^ from some point on S"^ but 
 not on *S'^* 
 
 If and O' are real points, /S" is a hyperboloid of one sheet. 
 
 If the generators of S^ through O and 0' are taken as edges of the 
 
 tetrahedron of reference (A, 0, 0' , F), the equation of S^ is 
 
 a?jX^ — a'2.r3 = (1). Every point on C'^ satisfies equation (1) and 
 
 the equation of S*. By taking as triangle of reference in tt 00' A' 
 
 (where A' is the intersection of ^i^ with tt), a correspondence can 
 
 be set up between the points x of S* and the points ?/ of tt by the 
 
 equations rx^ = y\, rx^ = y,y,_, rx^ = y,y^, rx^ =y^^ (2). Since a 
 
 point X of CI is connected with a point y of (7^ by equations (2), 
 
 the equation of C^ expressed in .r's gives the equation of S^. The 
 
 * See Clebsch, Vorlesungen ueber Geometrie, Vol. II, pp. 414 sq. for the general 
 theory. 
 
14 PLANE CURVES OF THE EIGHTH ORDER. 
 
 equation of CI is Zc,j,2/f "'"'2/22/3 = ^ («+ {& or c}^4). In full 
 this is : a,y\ + y\{ajj^ + a^^) + y\{a^l + a^l + a^^y^) + yl{a^yl 
 
 + «82/3 + %ylh + «103/22/D + 2/l(«ll2/2 + «123/3 + «132/22/3 + ^uMs 
 
 + «i5 2/j2/D + y\{(^ny\yz + ^172/22/1 + «i3 2/2 2/I + (f'x,y\yl) + 2/1 Ko 2/22/3 
 
 + «2l2/22/3 + "222/22/3) + 2/,(«232/22/3 + «242/22/3) + ^252/22/3 = ^ (3)' 
 
 In obtaining the equation of S*' from (3) by means of equations 
 (2), ten terms are ambiguous : a^, a^, a^^, a^^, a^^, a^^ (twice), a,g, a^^, 
 «22- For y\y^y^ can be rendered x\x^ or x\x^x^, y\yly^ as x]x^x^ or 
 a^jar^Xg, etc. The equation of S*^ may be written : a^x\ -{- a^x\x^ 
 
 ^~ (.t'o't^i t^^o ~T~ ^4*^1 *^o ^1 ^c*^! *^o ~| fi 1 4 ^1 (*■- *C. •*/, ^~ Lt'QU/,*tr„ — ^ Ct-Q JT. ily-io . 
 
 """ ^10^1**' 2^3 "T* '^11^2 "I" ^12^3 ' '^13'^'2^3 I "l4**'2*3 "T '^15'^2'*'3 "^ "l6'*'2*'^4 
 ~t~ ^'17**^3'^4 I* ^ia'^i**^2 4 "• 19 1 3 4 "^ 20 2 4 "* 21 3 4 •" 22 1 4 
 
 "T" ^i^^i I "24?'3*4 ' ^25^4 ^ ^' 
 
 A ten-fold infinity of aS^'s can be passed through a given C^. 
 Since there are thirty-four non-homogeneous constants in the general 
 equation of an *S^*, twenty-four conditions are put on >S'* by making 
 it contain a given C\. 
 
 It is always possible to pass four cones through a C\ of the first 
 species, for the single infinity of aS'-'s that can be passed through C* 
 form a pencil of conicoids. Hence, any such C* can be considered 
 in four different ways as the intersection of a cone and another coni- 
 coid. In the case of a C\, where the 8^ has ten degrees of freedom, 
 a cone is not necessarily contained among the ten-fold infinity of 
 aS'-'s ; for it requires more than ten conditions to make certain that 
 the 8^ is a cone of the fourth order. If there is a cone among the 
 *S*'s that contain C\, the following theorems hold : 
 
 Theorem 1. The curve C\ is the envelope of the conies in which 
 the planes tangent to 8^ cut /S^ 
 
 Proof. — Any plane through a generator of S^ cuts S^ in three 
 other generators. Each of these lines has two points on (7^. If two 
 generators are coincident (i. e., if the plane is tangent to S^), two 
 points coincide with two others and the conic is doubly tangent to 
 C^y. Consecutive tangent planes contain consecutive points on C^. 
 
 Corollary. — The curve C^ is the envelope of the projections of 
 the conies in which planes tangent to 8* cut 8'^. 
 
 Theorem 2. — The surface 8^ can be pictured by a single infinity 
 of pencils of conies, each containing C, the conic in which tt cuts 8"^. 
 
PLANE CURVES OF THE EIGHTH ORDER. 15 
 
 To each generator of S* there corresponds a pencil whose base points 
 are and 0' and the two points Q and Q' in which C is cut by the 
 polar reciprocal with respect to S^ of the generator of *^S'^ 
 
 Proof. — Take as tt the plane of projection, the polar plane with 
 respect to S^ of P the vertex of S*. Call the pole of 00' with re- 
 spect to C, A'. Use as the center of projection, A, one of the two 
 points in which A'P cuts S^. Consider as the picture of any point 
 X on aS'*, the projection of the conic in which 8^ is cut by the polar 
 plane of x with reference to S^. Since each conic on S^ cuts the two 
 generators through A (AO and A 0'), each conic on S^ is projected 
 into a conic on tt through and 0'. The picture of each point on 
 S^ is a conic in tt through and O'. The converse is not true. 
 Since there are a double infinity of points on /S* and a triple infinity 
 of conies in tt through O and O', there must be another condition on 
 the conies which picture points on S*. It will be shown that each 
 generator of ;8'* is pictured by a pencil of conies whose base points 
 are O and 0' and the two points Q and Q' in which C is cut by the 
 polar reciprocal with respect to S^ of the generator of S^. The pic- 
 ture of P is C, the conic in which tt cuts S^. Each generator of S* 
 cuts a, the plane tangent to S^ at A, in a point T whose polar plane 
 T contains A. But every plane r through A cuts /6'^ in a conic 
 which projects into a line pair composed of 00' and the line QQ' in 
 which the plane t cuts tt. Since QQ' is conjugate to PT with ref- 
 erence to S', the polar plane of any point on PI contains QQ . Since 
 Q and Q' are on tt as well as on S', the pictures of the points on 
 PT contain Q and Q'. Therefore a generator PT is pictured by a 
 pencil 0, O', Q, Q'. Also, a pencil with two base points at and 
 0' and two others at the two points in which Ois cut by a line con- 
 jugate to a generator of S^ is the picture of that generator of S^. 
 
 Corollary 1. — The picture of the C* in which S^ is cut by a the 
 plane tangent to S'^ at JL is a curve of the fourth class D. 
 
 As point T moves along the curve of the fourth order C^ in which 
 a cuts S*, the line QQ' envelopes a curve of the fourth class D. 
 
 Corollary 2. — The curve D is the polar reciprocal with respect to 
 C of D', the O* in which 8* cuts tt. 
 
 The polar reciprocal of a point X of tt with respect to C is the 
 intersection of tt with the polar plane of X with reference to 8"^. 
 
16 PLANE CURVES OF THE EIGHTH ORDER. 
 
 If A'' lies on U, the polar plane of A" contains the line Q Q' which 
 is conjugate to PAT. Hence AT has this line QQ' as its polar with 
 reference to C. Conversely, every QQ' is the polar of some point 
 X on D' with reference to C 
 
 Corollary 3. — The polar reciprocal QQ' of a generator PT of *S'* 
 is pictured by a pencil of conies through O and O and B and B', 
 the projections of the two points B^ and B'^, in which PT cuts S^. . 
 The two pencils 0, O, Q, Q' and O, 0', B, B' are said to be recip- 
 rocal to each other. 
 
 Since the conies in which the polar planes of the points of QQ' cut 
 S^ contain B^ and B'^, their projections contain B and B'. The pro- 
 jection of every conic on S"^ contains O and 0'. 
 
 Theorem 3. — The locus of points of contact of tangents from a 
 ^oint on QQ' to conies of the pencil determined by O, O', Q, Q' is a 
 •conic of the reciprocal pencil. This point is the pole of 00' with 
 reference to this conic. 
 
 Proof. — Take any definite point Q^ on QQ'. The locus of points 
 of tangency of tangents to aS'^ from Q^ is the conic K' in which S^ is 
 cut by the polar plane of ^j. K' projects into a conic K of the 
 pencil 0, O', B, B', and these tangents into lines from Q^ to points 
 of K. These lines are the tangents from Q^ to the conies of the 
 pencil 0, O, Q, Q' and the points of tangency are on K. For, each 
 plane through QQ' contains two of the tangents to aS'^, which are 
 tangent to the section oi S^ in this plane. This conic projects into 
 a conic of the pencil, with the projections of these two lines as tan- 
 gents from Q^ and the points of tangency on K. Conversely, every 
 line from a point Q^ to a point on K is a tangent to some conic of 
 the pencil O, O', Q, Q'. 
 
 Corollary 1. — Conversely, every conic of the reciprocal pencil is 
 the locus of the points of tangency of the tangents from some })oint 
 on QQ' to the conies of the pencil O, 0', Q, Q'. The pole of 00' 
 with reference to any conic of the pencil O, 0', B, B' lies on QQ'. 
 Corollary 2. — The line QQ' is the locus of the poles of 00' with 
 respect to the conies of the pencil O, O, B, B' . The line BB' is 
 the locus of poles for the pencil O, O', Q, Q'. 
 
 Theorem 4. — The curve D is the locus of the poles with respect 
 to S'^ of the planes tangent to S*'. 
 
PLANE CURVES OF THE EIGHTH ORDER. 17 
 
 Proof. — Since D lies in tt, tlie polar plane of any point of D con- 
 tains P. The polar of any point on D with respect to C is a tan- 
 gent to D' , since D is the polar reciprocal of D' . Since the polar 
 plane of any point on D contains P and a tangent to Z)', such a 
 plane is tangent to S^. Conversely, every plane tangent io S^ has 
 its pole on D. 
 
 Corollary. — The curve D is the locus of the poles of 00' with 
 reference to those conies in tt which are the projections of sections of 
 S'^ cut out by planes tangent to S*. 
 
 The polar plane of any point Q^ on line QQ' cuts S"^ in a conic 
 whose projection contains 0, 0', B, B' . Q^ is the pole of 00' 
 with reference to this conic. When and only when Q^ is on D, the 
 polar plane is tangent to S*. 
 
 Theorem 5. — If conies of the pencils O, O' , B, B' and O, 0' , 
 Q, Q' intersect at the point P, the tangents are harmonically sepa- 
 rated by POand PO'. 
 
 Proof. — Let Qp be the pole of 00' with reference to the first 
 conic and B^ the pole for the second. By theorem 3, the tangent 
 to the first conic at P contains B^ and the tangent to the second at 
 P contains Q^. Call the point in which PQ,^ cuts 00', ^and let 
 -S'' be the point of intersection of PB^^ and 00' . The pole of PQ^^ 
 lies on 00'. Since PIC' is the tangent to the first conic at P, the 
 pole of PQ(, is on PK' . Therefore K' is the pole of PQ^ with 
 respect to the first conic. Hence O and O' are harmonically sepa- 
 rated by ^and K' . But the cross ratio of this range equals that of 
 the pencil determined by the tangents at i? and the lines PO and 
 PO'. 
 
 Theorem 6. — The curve C\ is the envelope of conies through O 
 and O' which possess the two following properties : 1) the pole of 
 00' with respect to the variable conic lies on a certain curve of the 
 fourth class D ; 2) the variable conic and a certain fixed conic ( C) 
 through O and O' are harmonically separated at a point of intersec- 
 tion by the lines to and O' . 
 
 Proof. — By the corollary to theorem 1, C^ is the envelope of the 
 projections of the conies in which the planes tangent to S*' cut S'^. 
 It remains to show that such conies possess the two properties in 
 question. The poles of 00' with respect to such conies are on D 
 
18 PLANE CURVES OF THE EIGHTH ORDER. 
 
 (corollary to theorem 4). To obtain the second condition, recall that 
 such conies belong to the pencil O, O', B, B' and that the fixed 
 conic C belongs to every pencil 0, O', Q, Q' . Then by theorem 5, 
 the conies in question fulfil the second condition. The converse 
 obviously follows and completes the demonstration. 
 
 Theorem 7. — The number of conies of this system which touch 
 any line of tt equals twice the order of D. 
 
 Proof, — Any line in the plane determined by point A and any 
 line a; of TT projects into the line x. Therefore, any conic K on S^ 
 which touches any line y of plane Ax projects into a conic tangent to 
 X. Since y is tangent to K, it is tangent to S'^ ; and since y lies in 
 plane Ax, it is tangent to G, the conic in which the plane Ax cuts 
 S^. Since the only conies which belong to the system are projec- 
 tions of conies on S"^ cut out by planes tangent to S*, the question 
 becomes, How many tangents to G lie in planes tangent to S^'i 
 Since a plane tangent to S* is determined by P and a line tangent 
 to a plane section of S*, the question is. How many common tan- 
 gents are there to G and the C* in which plane Ax cuts S^'i The 
 class of this (7* equals the class of Z>', which is the same as the 
 order of D. Since G is of the second class, the number of such 
 tangents is 2m, where m is the order of D. 
 
 Theorem 8. — The number of these enveloping conies which pass 
 through any point of tt is m. These conies have a second point in 
 common. 
 
 Proof. — Take any point Y^ in tt and draw AY^ Join P to Y[, 
 the point in which AY^ cuts S^, Call the point in which PY[ cuts 
 TT, Y^. Through Y^, m tangents to JD' can be drawn, for D and Z>' 
 are reciprocals. The m planes determined by P and these tangents 
 are tangent to S*. Each plane cuts S^ in a conic which projects 
 into a conic of the enveloping system. Since each plane contains 
 the line PY^^Y[, the conic on S^ has Y[ and the projection on tt 
 passes through Y^. These conies through Y^ contain Y^ the projec- 
 tion of Fj the second point in which PY^ cuts S^. 
 
 Corollary 1. — If Y^ is on C^, Y^ is on B'. 
 
 Corollary 2. — If Y^ is on C (not at O or 0'), Y[ coincides with 
 Yy and therefore J'^ coincides with Y^ 
 
 Corollary 3. — The eight points of intersection of C and B' are 
 
 one;. 
 
PLANE CURVES OF THE EIGHTH ORDER. 19 
 
 Theorem 9. — The tangents to C^ at O or O' are the tangents to 
 D from the same point. 
 
 Proof. — If J^j is at O, the whole line AY^ lies on >S^ and there- 
 fore contains four points Y[ on (7^. The four lines joining these 
 points to P lie in plane PA O and cut tt in four points Y^ on line 
 A'O (A' is the projection of P). By the first corollary to the last 
 theorem, these four points 1^ on A'O are on D'. The polars of 
 these points are tangents to D from O. Of the m tangents to D' 
 from any one of these four points l^^, the tangent to D' at 1^ counts 
 as two. Each of the m planes determined by P and one of these 
 tangents cuts S' in a conic through Y[. The tangent to such a 
 conic at Y[ lies in the plane tangent to S^ at Y[. Since PY^ is a 
 generator of S* and Y[ is in a, the plane tangent to S^ at Y[ cuts tt 
 in a tangent to D. Since the tangent plane at Y[ contains A, every 
 line in it projects into this tangent to D from 0. In particular the 
 tangent at Y[ to the two coincident sections cut out by the plane 
 through the tangent to D' at Y^ projects into this tangent to D from 
 O. But these two coincident conies are the limit of two consecutive 
 conies of the system and hence their projection touches C^ at O. 
 Therefore, a tangent to D from O is a tangent to C* at 0. 
 
 Corollary. — If 2) contains O (or 0'), two or more tangents to C^ 
 at O coincide. 
 
 For if O is a point on D, at least two of the tangents to D from 
 O coincide. 
 
 Theorem 10. — A line joining O or O' to any one of the 2m points 
 of intersection of C and D is tangent to C^. 
 
 Proof. — The polar plane of any one of these points Y^ is tangent 
 to /S* (theorem 4) and S~. Such a plane cuts S^ in a degenerate 
 conic whose projection is the line pair Y^O, Y^O'. Since this line 
 pair belongs to the system enveloping (7^ (corollary to theorem 4), 
 it touches C^, twice. Since Y^ and Y^ O' are projections of gen- 
 erators of S"^, each contains four points of C^ distinct from O (or 
 O). One point of tangency is on OY^ and the other is on O'Y^. 
 
 Corollary. — The system of conies enveloping C\ contains 2m de- 
 generate conies consisting of a line through and another through 
 
 a. 
 
 Theorem 11. — There are m double tangents to C^ concurrent at 
 A'. 
 
20 PLANE CURVES OF THE EIGHTH ORDER. 
 
 Proof. — The polar plane of any one of the m points of intersection 
 of Z) and the line 00 is tangent to S^ (theorem 4), and contains 
 the line A A'. Hence the conic in which this plane cuts S"^ projects 
 into a degenerate conic of the system consisting of the line O' and 
 a tangent AH to D' from A . Each conic of the system touches 
 
 C^ twice. Since line 00' cuts four distinct branches of C^ at 
 
 p p 
 
 and four others at O' the line 00' can not touch C^. Hence the 
 line AH is a double tangent to C^. Since the line 00' cuts D in 
 m points, there are m such lines AH. 
 
 Theorem 12. — The curve C^ goes into itself by a quadric inver- 
 sion whose pole is A and whose conic is C 
 
 Proof. — The points B and P' are two vertices of the complete 
 quadrangle formed by O, O', Q, Q'. Since C is a conic through 
 these last four points, P and P' are harmonically separated by the 
 points of intersection of C and line PP'. The points P and P are 
 on C^ and the line PP' contains A. On each line through A there 
 are four pairs of points P, P' corresponding to the four generators 
 of S* determined by the four points in which this line cuts P'. 
 Since the points of a pair are collinear with A and conjugate to C, 
 each goes into the other by a quadric inversion whose pole is A and 
 whose conic is C Therefore, C] goes into itself by this inversion. 
 C^^ might be considered as an anallagmatic curve, if one were to 
 extend the term to include general quadric inversion, instead of 
 limiting it, as is usually done, to circular inversion. 
 
 Method of generating a C^ with two real four-fold points. Let 
 O and O' be any two real points on any conic C and let D' be any 
 general plane curve of the fourth order in the same plane as C. Let 
 Q and Q' be the two points in which C is cut by the polar with 
 reference to C of any point on P'. Draw the lines joining Q and 
 Q' to O and 0'. The double points P and P' of these line pairs 
 are on a C] with four-fold points at and O'. 
 
 A Contact Transformation. 
 
 By a contact transformation whose characteristic equation is r 
 = x^y -\- y^x -}- 3 (7 -f a;_j/ = 0, a certain class quartic is transformed 
 into a C^^ with two four-fold points. 
 
 Proof. — Consider the order cubic composed of the line z = and 
 
PLANE CURVES OF THE EIGHTH ORDER. 
 
 21 
 
 the conic C^ ■}- xy = 0, having y = tangent at ^ (1, 0, 0) and 
 X = tangent at B {0, 1, 0). The first polar of a point [x^, y^, z^) 
 with reference to the cubic z{Cz^ -{- xy) ^ (1) is x^yz -\- y^zx 
 4- z^{3Cz'^ + xy) = 0(2'). Using non-homogeneous codrdiuates, this 
 becomes r = x^y -f y^x -{■ ZC + xy = (2). This can be used as 
 the characteristic equation for a contact transformation, since* 
 
 
 
 dr 
 dx 
 
 dr 
 dy 
 
 dr 
 
 av 
 
 d\ 
 
 dxy 
 
 dxdx^ 
 
 dydx^ 
 
 dr 
 
 av 
 
 d'r 
 
 %i 
 
 dxdy^ 
 
 ^y^Vi 
 
 2/1 + 2/ ^1 + ^ 
 
 y 
 
 X 
 
 
 
 
 
 = x{y + y^x + 2xy 
 This ={= 0, because 
 of r = 0. 
 
 Corresponding to each pair of values (x^, y^) there is a conic which 
 contains A and B. It will now be shown that every conic is har- 
 monically separated from the fixed conic .ry — 3(7= (3) by the 
 lines to A and B. The points of intersection of (3) with a conic 
 (aside from A and B) are 
 
 »'o = 
 
 3Cx^ 
 
 - 3(7d= \/9C'-Sx^y,C 
 
 Vo = 
 
 _ - 3 (7=b V9 C- Sx^y^ C 
 
 X, 
 
 The tangent to (3) at (x^, y^ is x^y -\- y^x — 6 C = (4). The tan- 
 gent to the r conic at the same point is x[y^ + y^ + y {qi\ -\- x^ 
 + ^i2/o + *o3/i + 6 (7 = (5). The line through A and point [x^, y^ 
 is y — y^ = (6). The line joining B to the point (x^, y^) is x — cc^ 
 = (7). Take \^ so that (4) = y — y^ -^ X^(x — x^) = ; and X^ 
 so that (5) = 2/ — y^ + Xjx — x^) = 0. Then 
 
 Vo . 6 (7 - jK^y, 
 
 X, := — , '^1 = 
 
 ic: 
 
 2 > ^^2 
 
 2/1+2/0 
 
 or 
 
 \ = - 
 
 *^i + *^o 
 2a'i2/o + a^o2/i + 6 (7 + 2^,3/0 
 
 ^0(^1 + ^0) 
 
 Substituting the values of x^ and y^, 
 
 *Lie, S., and Scheffers, G., Geometrie der Beruehrungstransformationen, Vol. 
 I, p. 54. 
 
22 PLANE CURVES OF THE EIGHTH ORDER. 
 
 _ 6 (7 =F 2 1/9 a- 3aji3/i C - x,y^ 
 
 1 x\ 
 
 X = ± 2/9(7- 3a;,y^C- eg + a;,y^ ^_-^ 
 
 x' 
 
 Therefore the cross ratio of the pencil determined by lines (4), (5), 
 (6), and (7) equals — 1 . Hence the first polar of any point of the 
 plane with respect to (1) is a conic containing A and B and sepa- 
 rated harmonically from (3) by lines to A and B. Each conic has 
 its pole of AB at the point (— x^, — y^, -\-z^). As this point de- 
 scribes a general curve of the fourth class, the conies envelope a C^ 
 with quadruple points at A and B. The curve which is transformed 
 into C^ is described by the point {-{-x^, -f- 2/ij + ^J. 
 
 LIFE. 
 
 I, Elizabeth Buchanan Cowley, was born in Allegheny, Pennsyl- 
 vania, where I received my early education in the public schools. I 
 then studied at the Indiana State Normal School of Pennsylvania for 
 two years and received the degree of B.S. in July, 1893. The next 
 four years were spent in teaching in the public schools of that state. 
 I was given a life diploma by the state. In September, 1897, I en- 
 tered Vassar College and received my A.B. in June, 1901. I was 
 awarded the graduate scholarship in mathematics and astronomy for 
 the next year and obtained my A.M. in 1902. Through the courtesy 
 of Professor Whitney an opportunity was given to me to work out 
 the definitive orbit of a comet. Dr. Furness gave valuable advice 
 and Miss Whiteside assisted in calculations. In 1902 I received 
 an appointment as instructor in mathematics at Vassar College 
 and am still holding this position. The summer vacations of 1903, 
 1904, and 1905 were spent in resident study at the University 
 of Chicago, where I had twelve weeks of mathematics and physics 
 each year. I studied under Professors Bolza, Dickson, Millikan, 
 Moulton and Slaught and Doctors Gale and Jewett and the late Pro- 
 fessor Maschke. In February, 1906, I began work at Columbia 
 University, attending lectures that second semester and the entire 
 
PLANE CURVES OF THE EIGHTH ORDER. 23 
 
 year from September, 1906, to June, 1907. I also took the grad- 
 uate courses offered at the summer school in 1906. My work was 
 with Professors Kasner, Keyser, and Maclay. I am a member of 
 the American Mathematical Society, the Association of Teachers of 
 Mathematics, and Circolo Matematico di Palermo. The Astrono- 
 mische Nachrichten published my paper. The Definitive Orbit of 
 the Comet 1826II. as a separate pamphlet at Kiel, Germany, in 
 1907. I have also published shorter articles in the Bulletin of 
 the American Mathematical Society. I desire to express my grati- 
 tude to all my teachers, but especially to Professor Keyser, who is 
 able in an unusual degree to inspire his students with a hearty en- 
 thusiasm for study. To Professor Henry S. White, as head of my 
 department at Vassar College, I am indebted for encouragement, 
 criticism, and sympathy with my work. It is my pleasure also to 
 state my obligation to President Taylor and the trustees of this col- 
 lege for permission to attend lectures at Columbia University while 
 teaching at Vassar College. 
 
 UNIV 
 
 OF 
 
PLATE II. 
 
 10 
 
 11 
 
 12 
 
 13 
 
 / V. 
 
 19 
 
 20 
 
 •21 
 
 22 
 
PLATE III. 
 
 25 
 
 .. 8 
 
PLATE IV. 
 
 10 
 
 11 
 
 12 
 
 17 .. 
 
 •.•■•■c» 
 
 1 .-. ; 
 
 
 t 
 
 IE 
 
 
PLATE V. 
 
 iia:(i) 
 
 II A (2) 
 
 II A (2) 
 1 
 
 10 
 
 11 
 
PLATE VI. 
 
PLATE VII. 
 
 II A (3) 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
PLATE VIII. 
 
 II B (2) 
 12 
 
 13 
 
 14 
 
 16 
 
 10 
 
 11 
 
 19 
 
PLATE IX. 
 
 21 
 
 22 
 
 '-.^/:> 4 
 
 -^/:-. 
 
 * m • m •' 
 
 24 
 
 25 
 
 
 26 
 
 27 
 
PLATE X. 
 
 10 
 
 11 
 
 12 
 
 ^ 'J 
 
 21 
 
 14 
 
 23 
 
 16 
 
PLATE XI. 
 
 26 
 
 27 
 
 28 
 
lie (2) 
 
 PLATE XII. 
 
 Cx.-; 
 
 \ 
 
 
 lie (3) 
 8 
 
 lie (3) ^ J 
 
 :#; 
 
 ■-'M-' 
 
 \' 
 
 7\-N 
 
 \ 
 
 "■^x/y 
 
 N 
 
 III A 
 
 \ 
 
 
 IIIB 
 
 
 iiie 
 
 HID 
 
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