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 ON THE CORRELATION OF TWO PLANES.* 
 
 BY 
 
 T. ARCHER HIRST, F.R.S. 
 
 Definition and Determination of a Correlation* 
 
 1. A correlation is said to be established between two planes, when 
 their points and right lines are so associated, that to each point in one 
 of the planes, and to each line passing through that point, respectiv'ely 
 correspond, in the other plane, one line and one point in that line. 
 
 2. It will be convenient to apply the familiar terms ijole and j^olar to 
 a point and its corresponding right line. 
 
 3. It has been shown by Chaslesf and others, that such a correlation 
 
 may be established, and that in one way only, when four points, in one 
 plane, and their polars, in the other plane, are given ; provided always 
 that no three of the points are collinear, and no three polars concur¬ 
 rent. In fact, if Dj be the four points in the first plane, and 
 
 Cg 1^2 ^2 fl^^ir respective polars in the other, then, being the polar 
 of any fifth point Xj, it follow^s at once from the definition of a cor¬ 
 relation that the pencils A^ (B;^ Xj^) and B^ (Aj Hj Xj) must 
 
 be equi-anharmonic, respectively, with the rows (bo dg x^) and 
 
 %''4''^’ 2 )' Hence, by a well-known method, if be given, the 
 
 intersections of its polar x^ with Cg and and hence x.^ itself, are 
 readily found. 
 
 4. From the definition of a correlation it also follows that if the 
 polar of A| pass through A 2 , the polar of Ag will pass through Aj ; 
 and similarly, that if the pole of lie on Og, the pole of will lie in Oj. 
 Hence we may term Aj^ and Ag conjugate joints^ and and conjugate 
 lines of the correlation. 
 
 5. That in any correlation two given points, or two given right lines, 
 
 * Of the two questions, CorroLition and Ilomography, the first has heen selected 
 fjr special consideration in this paper. With suitahle modifications, however, as 
 shown in the foot-notes on pages 22 and 23, the results are all applicable to the 
 second question. 
 
 t “ Geometrie Superieuro,” art. 581. 
 
 B 
 
2 
 
 shall be conjugate to each other, is obviously to be regarded as one 
 condition ; that a given point and line shall bear to each other the re¬ 
 lation of pole and polar is, accordingly, equivalent to two conditions. 
 Hence, and from art. 3, we may infer that eight conditions are neces¬ 
 sary and sufficient for the establishment of a correlation between two 
 planes.* 
 
 6. The problem to determine a correlation between two planes which 
 shall satisfy any eight given conditions, is susceptible, in general, of a 
 finite number of solutions. The determination of this number, when 
 the conditions are of the elementary kind, single and double, described 
 in the last article,—a necessary step towards the solution of the more 
 general problem,—is one of the objects of the present paper. 
 
 Systems of Correlations. 
 
 7. Two planes may obviously be correlated in innumerable ways so 
 as to satisfy seven given conditions ; the totality of such correlations 
 constitute what may be termed a system of correlations satisfying seven 
 conditions. 
 
 For example, if the polars % ^2 three points Pj were 
 
 given, as well as two conjugate points and Ag,—a set of seven con¬ 
 
 ditions which may be conveniently indicated thus : 
 
 Pi Qi Rj All 
 
 P 2 % »2 ^2) . 
 
 then, in general, any line a^ passing through Ag may be regarded as 
 
 the polar of Ai, and a correlation established, in the manner indicated 
 in art. 3, which shall satisfy the eig^ht conditions, 
 
 Pi Qi Ri Ai 
 
 Ih 9.2 '"'2 ^2 
 
 and therefore, dfortiorf the seven given conditions (1). 
 
 By giving to all possible positions, we obtain the several cor¬ 
 relations of the system. We shall find it convenient to denote such a 
 
 system by the symbol ( ). 
 
 8. It should be observed, however, that whenever «£ passes through 
 the intersection of any two of the three given lines $'2 ’ 2 ’ 
 
 vision alluded to in art. 3 ceases to be observed, and no correlation, in 
 the ordinary sense of the term, can be established. 
 
 9. Such exceptions occur in every system of correlations. As 
 
 * This also follows immediately from the “ expression analytiqne des figures cor- 
 rclalives,” given by Chasles in the “ Geometrie Superieure,” art. 594. Expressed in 
 trilinear co-ordinates, the result there arrived at is that \.pc + + U 2 Z = 0 will he 
 
 the polar of if 
 
 A 2 P'2 ’ Vg = ^Oj + tn^i + fiy-j I I'ai + + n*y\ I + w/Ai + n"yi, 
 
 where the ratios of I, ni, n, I', n/, n', l'\ m”, n" are the eight constants of the 
 correlation. 
 
3 
 
 another example we may take the system 
 
 / Pj \ 
 
 \ P2 ^2 ^’2 ^2 / ’ 
 
 wherein and a<^ denote given conjugate lines. The several correla. 
 tioiis of this system are obviously obtained by giving to A^, the pole of 
 fl' 2 ) all possible positions on ; but whenever comes into line with 
 two of the three points the method of establishing a cor¬ 
 
 relation described in art. 3 ceases to be applicable. 
 
 10. Instead of excluding cases such as those described in the last 
 two articles, it will be found of the highest importance to admit them 
 as exceptional correlations into the system satisfying seven conditions, 
 and carefully to study their properties. The part they play in the 
 general theory of correlation will be found to be strictly analogous to 
 that played by degenerate conics in the investigations of Chasles on 
 systems of conics satisfying four conditions.* 
 
 Origin and Nature of Exceptional Correlations. 
 
 11. With a view of obtaining an insight into the nature and origin of 
 exceptional correlations, we will first consider all the exceptional forms 
 which a homographic relation between two planes may assume ; or, 
 what is equally general, all the exceptional modes of putting two 
 planes IT into perspective with each other. 
 
 It is clear that, so long as the centre of perspective lies in neither of 
 the two planes, the homographic relation between them is of the ordi¬ 
 nary type; we have, therefore, merely to consider the cases where the 
 centre of perspective lies, 1) in one of the two planes, and 2) in the 
 intersection of the two planes. 
 
 12. If the centre of perspective be a point 2^^ in the plane IT^, and if 
 
 cr' be the line in which the second })lane 0.' is intersected by llj, it is 
 obvious, firsts that to every point M' of H', which is situated on a\ cor¬ 
 responds, in ITj^, an indeterminate point on the line whilst to e^ery 
 
 other point of 11' corresponds the point 2^^ itself; and, secondhj, that to 
 the point 2^ corresponds a wholly indeterminate point of the plane iT, 
 whilst to every other point of llj^ corresponds the intersection M' of 
 2j^M^ and a. 
 
 With respect to the correspondence between the lines of the two planes, 
 it is equally obvious, first, that to the line o' in 11' corresponds a wholly 
 indeterminate line in 11^, whilst to every other line m' in II' corresponds 
 the line through 2^ and (m'o') ; and, secondly, that to every line in 
 Ilj, which passes through 2^, corresponds an indeterminate line in iT 
 passing through the intersection (aqo'), whilst to every other line in 
 corresponds the line o' itself. 
 
 13. If the centre of perspective 2^ lie in the intersection o' of the two 
 
 * “ Comptos reiidiis d.s seances do rAcadcinic des Sciences,” 18G4. 
 
 B 2 
 
4 
 
 planes, Tli and FI', then to 2^, regarded as a point of either plane, cor¬ 
 responds a wholly indeterminate point of the other plane ; to any other 
 point of (t\ in either plane, corresponds an indeterminte point of the 
 same line a in the other plane ; whilst to every other point, of either 
 plane, corresponds the point 2^ itself. 
 
 The lines of the two planes correspond in the following manner: to 
 cf, regarded as a line of either plane, corresponds a wholly indetermi¬ 
 nate line in the other plane ; to any other line through 2^, in either 
 plane, corresponds an indeterminate line through 2^^ in the other plane; 
 whilst to any other line, in either pl^ne, corresponds the line a itself. 
 
 14. Passing now from the perspective, to any other position of the 
 two planes, we conclude that the liomograijldc correspondence between 
 them may assume either of the folloiving two exceptional or singular forms. 
 
 First. There may be a singular point in one plane, and a singular line 
 in the other, whose respective correspondents are wholly indeterminate. 
 To each point in the singular line will then correspond an indeterminate 
 point in a determinate line passing through the singular point, whilst 
 to each such line through the singular point will correspond an indeter¬ 
 minate line passing through a determinate point of the singular line. 
 
 It is of importance to observe that between the above-mentioned 
 points on the singular line and the associated lines through the singular 
 point there must, by art. 12, always exist an equi-anharmonic, or (1,1) 
 correspondence. 
 
 Secondly. In each plane there may be a singular line, and a singular 
 point situated in that line, whose respective correspondents are wholly 
 indeterminat-e. To every other point in a singular line will then cor¬ 
 respond an indeterminate point in the other singular line, whilst to 
 every other line passing through a singular point will correspond an 
 indeterminate line through the other singular point. 
 
 15. If an exceptional homographic relation exist between two planes 
 ITj and 11', and any ordinary correlation, such as that described in art. 3, 
 be established between the latter, and a third plane fl^, it is evident 
 that between the former 11^ and this third plane fig an exceptional cor¬ 
 relation will exist; and vice versa., from any exceptional correlation, we can 
 always pass, by means of an auxiliary ordinary correlation, to an excep¬ 
 tional homographic relation. This principle enables us, readily, to deduce 
 all possible exceptional forms of correlation from the results of the last article. 
 
 IG. In so doing we must consider three cases, since the first case of 
 the last article presents two varieties. 
 
 First. The exceptional homographic relation between flj and fl' is such 
 that in I lj there is a .singular point 2^, and in 11' a singular line F. The 
 pole of cr', in the plane llo, being 2^, we .shall have, between lIi and flo, an 
 exceptional correlation ivith singiUar points 2^, whose characteristic 
 
 1 roperties, easily deducible from art. 14, may be briefly described thus : 
 
 The polar of a singular point is wholly indeterminate. The pole of every 
 
line tlirough a singular point is an indeterminate point on a determinate 
 line tlirough the other singular pointy the two lines thus associated being 
 al'ways corresponding rays of equi-anharmonic pencils. 
 
 Secondly. The exceptional homographic relation between fTj and U' is 
 such, that there is a singular line in and a singular point 2' in II'. 
 
 The polar of 2', in the plane Fig, being 0 - 2 ? we shall now have, between 
 the planes 11^ and llg, an exceptional correlation with singular lines and o-g, 
 the characteristic properties of which, as deduced from art. 14, will be 
 as follows ; 
 
 The pole of a singular line is wholly indeterminate. The polar of every 
 point in a singular line is an indeterminate line through a determinate 
 point of the other singular line, the tivo points thus associated being always 
 corresponding points of tivo equi-anharmonic rows. 
 
 Thirdly. The exceptional homographic relation between 11^ and O' is 
 such that in each of these planes there is a singular line, and a singular 
 point situated on that line. 
 
 Between the planes Oj and fig we shall now have an exceptional cor¬ 
 relation with singular lines and points (in each line a point), of which 
 the following properties are characteristic : 
 
 The pole of each singular line, as well as the polar of each singular 
 point, is ivhollij indeterminate. The polar of any point in a singular line, 
 not coincident with the singular point situated therein, is an indeterminate 
 line through the other singular point. 
 
 An exceptional correlation with singular points is determined when 
 the positions of those points, and of three pairs of conjugate lines pas¬ 
 sing through them are known. In like manner, an exceptional cor¬ 
 relation with singular lines requires, for its determination, a knowledge 
 of the positions of those lines, as well as of three pairs of conjugate 
 points situated thereon. Seven arbitrary conditions, of the kind de¬ 
 scribed in art. 5, are necessary and sufficient, as we shall see, to deter¬ 
 mine an exceptional correlation with singular points, or with singular 
 lines. An exceptional correlation of the third kind, hosvever, that is to 
 say, one which possesses singular lines and singular points situated 
 therein, cannot, in general, be made to satisfy more than six such con¬ 
 ditions. Hence it is that such exceptional correlations do not present 
 themselves in the present paper; they have been described, solely, for 
 the sake of completeness.* 
 
 * Tt is well known, and has recently been again demonstrated by Schroter, 
 (“ Journal f'iir die reine und angewaiidte Mathematik,” vol. 77, p. 105,) that two 
 correlated planes can always be made to coincide in such a manner, that to each 
 point the same line shall correspond, no matter to which of the two coincident 
 planes that point be ascribed. The point and line, in fact, then become pole and 
 polar relative to a conic. Now this conic degenerates to a line-pair, when the cor¬ 
 relation has singular points ; to a point-pair, when it has singular lines; and to a 
 line-pair-point, when it has singular points and lines. Hence the connection be¬ 
 tween degenerate conics and exceptional correlations, alluded to hi art. 10. 
 
6 
 
 17. The following properties of the two first kinds of exceptional 
 correlations will be frequently referred to ; they are easily deduced from 
 those already described. 
 
 a. The pole of every line, in either plane, which does not pass 
 through the singular point of that plane, coincides with the singular 
 point of the other plane. 
 
 b. The pole of every point, in either plane, which does not coincide 
 with the singular point of that plane, is a determinate line passing 
 through the singular point of the other plane. 
 
 c. The polar of every point, in either plane, not situated on a singu¬ 
 lar line of that plane, coincides with the singular line of the other 
 plane. 
 
 d. The pole of every line, in either plane, which does not coincide 
 with the singular line in that plane, is a determinate point in the sin¬ 
 gular line of the other plane. 
 
 Ixelations between the Characteristics and Singidarities of any 
 
 System of Correlations. 
 
 18. being an arbitrary point in one of two correlated planes, 
 its polars, in the several correlations of any system, envelope a curve, 
 w'hose class fx indicates the number of correlations of the system, in 
 each of which the polar of passes through an arbitrary point Mg of 
 the second plane. But since, in each of these correlations, (and in 
 these solely) the polar of Mg will pass through M^ (art. 4), p will also 
 be the class of the curve enveloped by the polars of an arbitrary point 
 Mg in the second plane. 
 
 Regarded as the number of correlations of the system, in each of which 
 two arbitrary points M^ and Mg are conjugate to each other, p may be 
 appropriately termed the class of the system of correlations. 
 
 19. In like manner, the poles of an arbitrary line, in either plane, 
 lie on a curve whose order v may be termed the order of the system of 
 correlations., since it indicates the number of correlations of the system 
 in each of which two arbitrary lines and ??ig are conjugate to each 
 other. 
 
 20. Between the characteristics p and r of a system of correlations, 
 and the number of exceptional correlations which that system includes, 
 relations exist precisely like those which have been established by 
 Chasles between the characteristics of a system of conics satisfying four 
 given conditions, and the number of degenerate conics included in the 
 system. 
 
 In fact, if TT denote the number of exceptional correlations with sin¬ 
 gular points, and X the number of exceptional correlations with singular 
 lines included in the system of correlatiuiis whose characteristics are 
 
7 
 
 fx and V, then 
 Hence we deduce 
 and also 
 
 yu = 2v — TTI 
 
 V = 2fx —X ) 
 
 \ + TT = + V 
 
 X — TT = 3 (/U — y) 
 
 Sju = 2X + 7r 7 
 = X + 27r 7 
 
 ( 1 ). 
 
 ( 2 ), 
 
 (3). 
 
 21. Since each of the relations (1) follows from the other by the 
 Principle of Duality, a demonstration of the first will be sufficient. 
 
 If and \ be any two lines in the first plane, and Mg any point in 
 the second plane, we have to determine the number of correlations of 
 the system (ft, v) in each of which the polar AgDg of the intersection 
 passe's through Mg. By hypothesis, there are v correlations of 
 the system in each of which the pole Ag of lies in an arbitrary ray 
 of the pencil whose centre is Mg. If the pole Bg of in each of 
 these V correlations, be joined to Mg by a ray m^, then to each ray 
 will correspond v rays 
 
 In like manner, to each ray will correspond v rays Wg, passing, 
 respectively, through the poles of a-^ in the several correlations in which 
 the poles of lie in m\. Hence there is a (r, v) correspondence between 
 the rays and m^, and consequently there are 2r rays, with each of 
 which an ^;^g and its corresponding Wa coincide. 
 
 Although each of these 2v rays passes through the poles of tq and 
 in one and the same correlation, they are not all polars of For 
 
 instance, if and 2g be the singular points of one of the tt exceptional 
 correlations included in the system, the poles of the arhltmrij lines 
 and will coincide with Sg (art. 17. a.), and by so doing cause and 
 'W ^2 to coincide with MgSg. The polar of in this exceptional cor- 
 
 • relation, however, will not in general coincide with this arhitranj line 
 but with that ray of the pencil (Sg) which corresponds equi- 
 anharmonically with the ray, through of tlie pencil (Sj), 
 
 (arts. 16. first case, and 17. b.) 
 
 Exceptional correlations with singular points being, clearly, the only 
 ones in which the poles of a-^ and hi coincide, it follows, as indicated by 
 the first of the relations (1) in art. 20, that the number /x, of correla¬ 
 tions in which the polar of (ui?>i) passes through the arbitrary point Mg 
 is less than 2y by the number tt of exceptional correlations in the sys¬ 
 tem which possess singular points. 
 
 22. The number of exceptional correlations in a system satisfying 
 seven given conditions being directly determined, the relations (1) 
 of art. 20, written in the form (3), will enable us to deduce at once the 
 characteristics of the system. 
 
 I propose to apply this method to the determination of the charac¬ 
 teristics of the fundamental systems, that is to say, of those systems 
 which satisfy elementary conditions of the following types : 
 
8 
 
 A given point shall have a given polar (2 conditions). 
 
 A given line shall have a given pole (2 conditions). 
 
 Two given points shall be conjugate (1 condition). 
 
 Two given lines shall be conjugate (1 condition). 
 
 The characteristics of the fundamental systems being thus deter¬ 
 mined, it will be easy to deduce the number of correlations which 
 satisfy any eight given elementary conditions. 
 
 All that will be then requisite in order to enable us to ascend from 
 elementary conditions to others of a more complex character, and thus 
 to solve the problem of the correlation of two planes in all its generality, 
 wdll be a more intimate knowledge of the properties of the curves of 
 the order r, and of the curves of the class /j. which, as we have seen 
 (arts. 18 and 19), are associated with each system of correlations. 
 
 Enumeration and Glassification of the Fundamental Systems 
 
 of Correlations. 
 
 23. a, /I, y, ^ being integers satisfying the condition 
 
 2a-h2/3 + y-(-^ = 7, 
 
 Ave shall term (a/3yc) the signature of the system of correlations 
 satisfying the following conditions : 
 
 a points in one plane have given polars in the other. 
 
 (d right lines in the first plane have given poles in the second. 
 
 y points and ^ lines in each plane have given conjugates in the 
 other plane. 
 
 It is obvious that the systems of correlations whose signatures are 
 (a /3 y ^) and (/3 a y ^) are identical. The first symbol, in fact, being 
 regarded as descriptive of the data in the one plane, the second will 
 indicate the corresponding data in the correlated plane. 
 
 Now the total number of integral solutions of the equation 
 
 2«2/3-|-y-)-— 7 
 
 can be readily proved to be 52 ; hence there are twenty-six distinct 
 fimdamental systems of correlations. 
 
 24. These systems may be arranged in six groups as follows : 
 
 Group. 
 
 Sig 
 
 nature. 
 
 I. 
 
 . (3010) 
 
 (0301) 
 
 II. 
 
 . (2110) 
 
 (1201) 
 
 Ill. 
 
 C(2030) 
 
 (0203) 
 
 . 1 (2021) 
 
 (0212) 
 
 IV. 
 
 i (1130) 
 
 (1103) 
 
 .1(1121) 
 
 (1112) 
 
 
 r (1050) 
 
 (0105) 
 
 V. 
 
 . \ (1041) 
 
 (0114) 
 
 
 ((1032) 
 
 (0123) 
 
9 
 
 f(0070) (0007) 
 
 VT J (0001) (0016) 
 
 .(0052) (0025) 
 
 ^(0043) (0034) 
 
 25. In each group, the systems are arranged in two columns, such 
 
 tliat the data of each system in one column are correlative to the data 
 of the system opposite thereto in the other column. Now to every 
 correlation of the system (a [d y o) will obviously correspond, by the 
 Principle of Duality, a correlation of the system (/> a ^ y) ; moreover, the 
 class and order of the first system will be equal, respectively, to the 
 order and class of the second system; and for every exceptional cor¬ 
 relation with singular points in the former there will be an exceptional 
 correlation with singular lines in the latter, and vice versa. It will 
 suffice, therefore, to consider the systems contained in one only of the 
 two preceding columns. 
 
 Number and Nature o f Exceptional Correlations in the 
 
 Fundamental Systems. 
 
 26. I proceed first to determine, directly, the number and nature of 
 the exceptional correlations included in each of the thirteen funda- 
 
 , mental systems whose signatures stand in the first column of the pre¬ 
 ceding article. In doing so, two associated singular points of any 
 exceptional correlation will always be indicated by the symbols and 
 Sg, and two associated singular lines by and o-g. 
 
 27. Throughout this investigation it will be assumed that, between 
 the positions of the given points and lines, no special relation whatever 
 exists. 
 
 28. Let the symbol of the system (3010) be (SdO) 
 
 / Pj Qj Rj Aj \ 
 
 Vi's '•s Aj/' 
 
 In this system there can be no exceptional correlation with singular 
 lines. To prove this, I observe, first, that could not coincide with 
 any one of the three lines p^ ^2? 5 ^*or if it did, then and 
 
 Ri, as poles of and fg, would lie on (art. 17. d.) ; and Aj not being 
 situated thereon (art. 27), would have (t^ for its polar, whereas by hypo¬ 
 thesis this polar should pass through Ag; and secondly, that if there 
 were a singular line o-g not coincident with any one of the three lines 
 
 9.2 '^”25 then the poles Pj^ of these lines would all lie on o-j 
 
 (art. 17. d.), which is inconsistent with the assumed generality of the 
 data by which the system is defined (art. 27). 
 
 Again, if there be a singular point in the system, it must coincide 7 r =3 
 with one of the three points Pj Q| R^; for if it did not, p^ q<^ r^ would 
 concur in iSo (art. 17. b.), which is not the case. Now if were coinci- 
 
10 
 
 dent with P^, its polar, being perfectly arbitrary (art. 16.), might ob¬ 
 viously be regarded as coincident with Moreover, and rg, as polars 
 of and would then intersect in Sg 17- b.), and the polar of 
 
 Aj would be manner all the'seven conditions would be 
 
 fulfilled, and the exceptional correlation would be perfectly and uniquely 
 determined (art. 16) ; the polar of any point Mj, for instance, would 
 be theray ??i 2 , which passes through (^ 3 ^ 3 ) and satisfies the anharmonic 
 relation (Qj R-^ Aj M^) = (^3 ^ 3 ) (^3 A 3 
 
 In like manner, 2^ might coincide with if Sg were coincident with 
 (’’ 2 P 2 ) 5 ^^5 lastly, might coincide with R;,^, if 22 were coincident with 
 Hence we conclude that m the system under consideration there 
 are hut three exceptional correlations^ and that each of these has singular 
 points. We have, consequently, by art. 20. (3), the values 
 
 \ = 0, 9r = 3, = 1 , V = 2. 
 
 110 ) 29. The next system (2110) may be denoted by 
 
 /Pi Qi ’1 ^i\ 
 
 \P 2 ?2 ^2 ^ 2 / 
 
 ;^=i Here cannot be coincident with o-^; for if it weve^p^ and ^ 3 , as polars 
 of points not situated on cr^, would be coincident with cr^ (art. 17. c.) ; hence 
 R2 must lie on (art. 17. d.) But if so, then, by art. 27, neither ^^2 nor 
 can coincide with o-g, and as a consequence, Pj and Qj must lie on Again 
 Aj^, not being on its polar, which by hypothesis passes through 
 must be coincident with Hence the only possible correlation with 
 singular lines is that in which these lines coincide, respectively, with Qi 
 and ^ 2 ^ 3 ; that this correlation satisfies the seven given conditions, 
 and is precisely determined by them, is readily seen. 
 
 77=1 Passing to possible exceptional correlations with singular points, it 
 is obvious that R 2 cannot coincide with S 2 ; for if it did, P^ and Qp the 
 poles of two lines p 2 and g '2 which do not pass through R 2 , would be 
 coincident in 2 ^ (art. 17. a.), which is not the case. Hence we infer, by 
 ai’t. 17. b., that must pass through 2 j, and as a consequence of this, 
 —or rather of the fact that neither Pj nor can, under these circum¬ 
 stances, coincide with 2 j,—that p^ and q^ must intersect in 23 . 
 
 That there is one and only one exceptional correlation which has a 
 singular point 23 coincident luith (Pgf/o), cind its associate 2 ^ situated 
 in ?q, and that this correlation is precisely determined by the seven 
 given conditions, is obvious on remarking that the anharmonic relation 
 
 ^i(PiQHi^i) = (1^222)0^2^2^2^2) 
 
 is the only remaining condition which 2 ^ has to fulfil (art. 16.). The 
 point 2 j, in fact, is the intersection, with of the line which connects Aj 
 with the only point A on P^Qi which satisfies the relation 
 
 PlQl (PiQHi-^) = (p2Q2^2‘^2)‘ 
 
11 
 
 For the system now under consideration, the equations (3) of art. 20 
 furnish the following values : — 
 
 X = 1, TT = 1, /a = 1, V = 1. 
 
 30. The next system to be considered has the signature (2030), and (2030) 
 
 / Pi Qi -^1 Pi Pi 
 
 \P2 
 
 the symbol 
 
 ^2 
 
 CJ* 
 
 Neither y )2 nor y 2 can here coincide with 0-2 5 either did, the a=o 
 
 poiars of A 2 B 2 C 2 would be coincident with o-^ (art. 17. c.), and therefore 
 could not pass, respectively, through Hence we infer that if 
 
 cr^ exist, and Qj must lie on it (art. 17. d.). But if were coinci¬ 
 dent with Pj the poiars of Aj C;^ would be coincident with cr^ 
 
 (art. 17- c.), which is inconsistent with the condition of their passing, 
 respectively, through A 2 B 2 C 2 . Hence we conclude that there are no 
 exceptional correlations ivith singular lines in the system. 
 
 Again, if there be a singular point Sg, it must lie in one, at least, of 77=3 
 the two lines p<^ and ^ 2 » for otherwise Pj and would be coincident in 
 (art. 17. a.). If ^2 were on p^, but not on q^, its associate Sj would coin¬ 
 cide with Qj (art. 17.), and ^2 itself would, by art. 16., simply have 
 to satisfy the condition 
 
 Qi(PiAiBA) = SiiteAACa).(1). 
 
 This is clearly possible, and in one way only (art. 29). 
 
 In like manner, there is a second exceptional correlation for which 2 ^ 
 is coincident with P;^, and ^2 lies on so as to satisfy the condition 
 
 V, (QABA) = Sj (2ABA).(2). 
 
 In every other exceptional correlation, ^2 must be coincident with 
 (p 222)5 a,nd 2 ^ must satisfy the hornographic relation 
 
 ^1 (PlQlAiB^C^) = (^ 2 ^ 2 ) (P2!l2^2^2^2)- 
 
 Now it is well known that there is, in general, one and only one posi¬ 
 tion for Sj consistent with this relation.* It is, in fact, the fourth 
 intersection of two conics (S^') and (S/') circumscribed to the triangle 
 AiBiCi, one of which (S/) passes through Pj, and is determined by the 
 anharmonic relation 
 
 (Sj) (P^A^^Bj^Cj) = (P2^2')(P2-^2^2^2)y 
 and the other (S/") passes through Qj in such a manner that 
 
 (S/') (Q.AiBiC,) = (^ 2 ^ 2 ) (( 72 A 2 B 2 C 2 ). 
 
 The only exceptional case which can present itself arises when tlie 
 above two conics (S/) (S/') happen to coincide. This case, however, is 
 excluded by art. 27, since it implies the existence of a special relation 
 between the positions of the given lines and points. 
 
 * See Sturm’s excellent paper on the Problem der Projectivitdt und seine Anivendung 
 auf die Fldchen ztveiten Grades., in the “ Mathematische Annalen,” vol. i. p. 5.33, wherein 
 many of the auxiliary thGorem.s employed in tliis paper arc clahoratcly demonstrated. 
 
 LIBRARY _ 
 
 Uri'^RSITY OF IfPNO!?? 
 
12 
 
 We conclude, then, that the system under consideration contains three 
 exceptional correlations with sinrjular points^ and accordingly we have, as 
 
 in art. 28, X = 0, tt = 3, ^ = 1, v = 2. 
 
 ( 2021 ) 
 
 A = 1 
 
 frr: l 
 
 31. We proceed to the system (2021) whose symbol is 
 
 /Pi Qi Aj cA 
 
 9 .^ ^2 ^2 ^ 2 / 
 
 If ^2 coincident with a singular line 0 - 2 , then would lie on 
 (art. 17. d.), and at the same time the polars of Ag and B 2 would coincide 
 with (Tj (art. 17. c.). But, since Q^, A^ and Bj are not in a line (art. 27), 
 this is obviously inconsistent with the condition that these polars should 
 pass, respectively, through Aj and Bj^. In like manner, ^3 cannot be 
 coincident with (Xg ; hence, if this singular line exist at all, Pj and 
 must lie on its associate (art. 17. d.), and the polars of Aj and Bj must 
 coincide with o-g (art. 17. c.), that is to say, o-g must pass through 
 
 Ag and Bg. 
 
 This being clearly possible, and moreover the seven conditions being 
 precisely sufficient to determine such an exceptional correlation,* we 
 conclude that the system contains one^ and 07ily one, exceptional correlation 
 ivith sinrjidar lines; the latter being coincident, respectively, with PjQi 
 and AgBg. 
 
 Again, it can be shown, as in art. 30, that if there be a singular point 
 Sg, it must lie on one, at least, of the lines p^ $' 2 ' N^ow if Sg were 
 situated on pg, but not on q^, then its associate would be coincident 
 with Qj, and the pole of Cj would be coincident with Sg (art. 17. a.) ; but 
 as this pole must be situated on Cg, (pgCo) is the only possible position 
 for 2g. In like manner, if Sg were on q^, but not on ^g, it would neces¬ 
 sarily coincide with ($' 2 ^ 2 ), and its associate with Pj. Moreover, 
 it is evident that tbe seven given conditions suffice precisely to deter¬ 
 mine one of each of the exceptional correlations here described. 
 
 The only remaining position possible for (^ 272)5 which case 
 
 the polar of Cg, necessarily a point on C|, would be coincident with 2^ 
 (art. 17. a). The sole condition to be satisfied by this point 2^ on Cj is 
 
 -1 (Pi Qi A-i Bi) (P272) (P2 72 ^2 P2) .; 
 
 hence must be one of the intersections of by the conic (S^) which 
 passes through Pj Aj B^ and satisfies the anharmonic relation 
 
 (Si) (Pj Qj Aj Bj) = (poq^) (P2 72 -^2 ^ 2 )* 
 
 Taking into account the two solutions of (I), we conclude that in the 
 system under consideration there are four exceptional correlations ivith sin- 
 (jular qjoinis. 
 
 * The pole of any line for example, would be the point M 2 , on A 2 B 2 , deter¬ 
 mined by the relation Ti Q) B’l (i m^) = ATIh M 2 ). 
 
13 
 
 Tho equations (3) of art. 20 give us, in the present case, the values 
 
 A = 1, TT = 4, fj, =. V = 3. 
 
 32. We arrive next at the system (1130), which is defined by the (iiso) 
 
 / Pj iq A| V 
 
 \ ^2 ^2 ^2 / 
 
 Since in this case, cannot coincide with cr^^, (for if it did, the polars 
 of PjAj^BjCj would be coincident with cr^, which is obviously impos¬ 
 sible,) <t 2 , if it exist, must pass through its pole P 3 , and for a like 
 reason its associate must pass through Pj (art. 17. d.). Such an ex¬ 
 ceptional correlation, however, could not possibly satisfy the conditions 
 in virtue of which the polars of A^, B^, pass, respectively, through 
 A 2 , B 2 , C 2 ; hence we conclude that the system under cunsideration in¬ 
 cludes no exceptional correloMon with singular lines. 
 
 If Pj were a singular point, P 2 , as pole of a line which does not 
 pass through that singular point, would necessarily be its associate, 
 and vice versa (art. 17. a.). Such a correlation is clearly possible, and 
 in one way only. 
 
 In any other exceptional correlation with singular points, would 
 necessarily lie on p ^, since S 2 does not coincide with Pg; and for a like 
 reason its associate ^3 would necessarily lie onp 2 (art. 17. b.). The sole 
 condition to be fulfilled by these associated singular points 2 ]^, ^3 is 
 
 ^1 (Pi ^1 ^ 1^1 ^]) — ^2 (^ 2^2 ^2 ^2 ^ 2 ) . (^)j 
 
 a relation which can be satisfied in two, and only two, ways. To prove 
 this, I observe that, as shown in art. 29, to each point on 2^\ cor¬ 
 responds one, and only one, point 83 on p^ such that 
 
 — ^2 (^ 2 ^ 2 -^ 2 ^ 2 ^ 5 
 
 whilst to this point 83 corresponds one, and only one, point 8 ^ on p)i 
 such that 81 ( 2^1 Pj Aj C^) = 83 (P 2 P 2 ^2 ^ 2 )- 
 
 Hence we may say that to each point 8 ^ corresponds one, and only 
 one, point 81; for a like reason, to each point 81 corresponds one, and 
 only one, point 8 ;^. By a well-known theorem, therefore, there are two 
 points on p-^ in each of which 8 ^ and 81 coincide. 
 
 Each of these points, considered as a position of 2 ^, together with its 
 corresponding 83 , considered as a position of 23 , will clearly give a 
 solution of the relation (1). Hence we conclude that the system under 
 consideration includes three excep)Uonal correlations with singular points. 
 
 For the system now under consideration, therefore, we have, as in 
 arts. 28 and 30, the following values : 
 
 A = 0, TT = 3, ^=1,, r = 2. 
 
 33. Next, in the order of sequence adopted in art. 24, eomes the ( 1121 ) 
 system ( 1 P 21 ), whose S 3 'mbol is 
 
 Pi ) 
 
 p 2 ^2 '^2 ^2 
 
14 
 
 ;^_2 It can be shown here, precisely as in the preceding case, that if there 
 be an exceptional correlation with singular lines, o-j^must pass through P^, 
 and its associate o-g through Pg. But must also pass either through 
 Aj or ; for if it did not do so, the polars of these points would coincide 
 with o-g (art. 17. c.), which is impossible, because cr^ could not pass at one 
 and the same time through Pg, A^, and B 2 (art. 27). It is obvious, how¬ 
 ever, that P^Aj and P 3 B 2 , as well as P^ B^^ and P 2 A 2 , will be asso¬ 
 ciated singular lines of an exceptional correlation satisfying the given 
 conditions. Hence the system includes two, and only two, exceptional cor¬ 
 relations with singular lines. 
 
 7 r =2 Again, Pj cannot coincide with a singular point ; for if it did, the 
 pole of would coincide, in the associated singular point S 2 , with P 2 , 
 the pole (art. 17. a.) ; whereas, by hypothesis, the pole of Cj lies on Co. 
 Hence 22 ’ exist, must lie onp 2 (art. 17.b.), and for alike reason its 
 associate 2 ^ must lie on . Moreover, if 2 ^ were not on Cj, its associate 
 2 ^ wmuld, as pole of Ci (art. 17. a.), lie on C 2 ; and vice versa, if 22 were 
 not on Co, its associate 2 ^ would lie on Cj. Hence we conclude that the 
 system includes two exceptional correlations with singular points. In one 
 of these 2^ is at {p\C^, and its associate 22 is the sole point on p^ w^hich 
 satisfies the relation 
 
 ^ 2 ( 1 * 2 P 2 '^ 2 B 2 ) ; 
 
 in the other, 22 is at (^ 2 ^ 2 )? and its associate 2 ^ is the sole point on 
 which fulfils the condition 
 
 — (^ 2 ^ 2 ) (^ 2 P 2 ^ 2 ^ 2 )* 
 
 In the present case, therefore, we have as result, 
 
 X — 2, TT = 2, /u = 2, V = 2. 
 
 ( 1050 ) 
 
 A = 0 
 
 34. We have now reached the fifth group of fundamental systems of 
 correlations, the first in which is the system (1050) defined by the 
 
 P. 
 
 symbol 
 
 /Pi Ai Bi Cl Di El 
 
 \ P 2 ^2 ^2 ^2 ^2 ^ 
 
 ') 
 
 7r = 3 
 
 Here there can he no exceptional correlations with singular lines; for 
 since no such line (Ti could pass through more than two of the five 
 given points A^Bj^CiDjE^ (art. 27), the polars of at least three of these 
 points would be coincident in 0-3 (art. 17. c.), and therefore could not all 
 pass through the points respectively conjugate to them. 
 
 On the otlier hand, it is obvious here, as in art. 30, that P^ looidd he 
 the singular point of one exceptional correlation in the system, its 
 associate being the sole point 23 in the second plane for which 
 
 Pi(AiI3iCiDiEi) = SjCA^B^C^D^E,). 
 
 If there be any singular point 2 j^, not coincident with P|, then its 
 associate 23 must lie on p 2 (art. 17. b.), and by art. 16 the two points 
 must satisfy the homographic relation 
 
 2i(P^AiBiCiDiEi) = 23(213A 3 B 2 C 3 T) 3 E 3 ) . 
 
 ( 1 ), 
 
15 
 
 an equation of which there are two solutions. The proof of this depends 
 upon the following Theorem : 
 
 Sg being any jpoint on ^ 9 , the locus of a jpohit 8-^ such that 
 
 SiCPiAiBiCiUi) = .(2) 
 
 is a cubic which passes through and has a double point at 
 
 In fact, it has already been proved in art. 32, that on any line drawn 
 through one of these four points, say A^, there are, exclusive of the 
 point Ai whicii is arbitrary, two points which satisfy the relation 
 ( 2 ); and A^ itself obviously does so wPen Sg has the position determined, 
 as at the end of art. 29, by the relation 
 
 ~ ^2(P2^2^2^2)- 
 
 Moreover, is a double point of the cubic, because, as in art. 31 (1), 
 there are two points Sg on pg such tliat 
 
 Pi(AiBiCiDj) = Sg(A2B2CgD2). 
 
 In like manner, the locus of a point S/ such that 
 
 ^1 ~ ^ 2(^2 ^ 2^2 ^ 2 ^ 2 ) 
 
 is another cubic having a double point at P^, and also passing through 
 
 Now, exclusive of the double point P^ (which counts as four inter¬ 
 sections), and the three points Ai,Bj, Ci, none of which leads to a solution 
 of ( 1 ), these two cubics intersect in two points 2 i, each of which, with 
 its associated point Sg on pg (determined by the relation 
 
 (Pi -^1 El El) = S2 (p 2 A2 B2 C2), 
 
 and, therefore, the same for both cubics) obviously satisfies the relation 
 (1). We conclude, therefore, that the system noiu under consideration 
 includes three exceptional correlations with singular points. 
 
 We have again, therefore, the values 
 
 \ = 0, TT = 3, ^ = 1, V = 2. 
 
 35. The second system in the fifth group of art. 24 has the signature 
 (1041) and the symbol 
 
 /Pi Ai Bi Cl Di 61 \ 
 
 \ P2 A2 B2 C2 D2 62} 
 
 By a process of reasoning similar to that employed in the last two 
 articles, it may be readily shown that in the present system there are no 
 exceptional correlations which have singular lines. 
 
 If any singular point Si were coincident with Pi, the pole of e^ w^ould 
 coincide with its associate S 2 (art. 17. a.). Hence, and from art. 16, we 
 conclude that the singular point S 2 associated with Pi lies on 62 , and satis¬ 
 fies the relation Pi (Ai Bi Ci Di) = S 2 (A 2 B 2 C 2 D 2 )- 
 
 Of this there are obviously two solutions (art. 31), in other words, 
 there are two points Sg on 02 each of whose associates is comcident 
 loiih Pi. 
 
 (lan) 
 
 A = 0 
 
 7r = 6 
 
16 
 
 Again, if there were any singular point not coincident with Pj, its 
 associate 2 ^ would necessarily lie on (art. 17. b.), and satisfy the rela¬ 
 tion 2 i (Pj Aj Bj Cl Di) == 2 ^ (p 2 -^-2 B 2 C 2 Dg). 
 
 Now all such points 2 i, as we have proved in the preceding article, 
 
 .3 
 
 lie on a cubic Si which passes through Ai Bi Ci Dj and has a double 
 point at Pi. Hence, and from the fact that 22 must lie on 62 whenever 
 2 i is not on 61 , and vice versa, that 2 i must lie on Ci whenever 22 is not 
 on 62 (art. 17. a.), we conclude that there wcq four, and only four, possible 
 p)ositions of ^2 on p 2 . In fact, in one of them 22 coincides with {p-ief), 
 and its associate 2 i is the sole point on Si which satisfies the relation 
 2 i (Pi Ai Bi Cl Di) = {pie^{p 2 A 2 B 2 C 2 D 2 ), 
 
 whilst, ill its remaining three positions on the line ^21 22 corresponds, 
 respectively, to the three intersections 2 ^ 2 i 2 ^" of Ci and the cubic (Si ). 
 
 On the whole, therefore, there are in the system six exceptional correla¬ 
 tions luith singular points. We have consequently the values, 
 
 X = 0, TT = 6, /I = 2, r = 4. 
 
 ( 10 . 32 ) 36. The last of the systems we have to consider in Group V. has the 
 
 siernature (1032), and is defined by the symbol 
 
 (P. A, bI C, M. 
 
 ^2 ^2 '^2 ® 2 / 
 
 A =3 there be any singular line it must pass through Pj, for other¬ 
 
 wise its associate would coincide with p^, and the polars of A^ Bg Cg 
 would be coincident with (art. 17. c.), instead of passing, respectively^ 
 through Aj Bj Cj. Moreover, must likewise pass through one of 
 the points A^ B^ C^, otherwise the polars of these points would be 
 coincident with instead of passing, respectively, through Ag Bg Cg. 
 But if o-j pass through P^ and one of the points A^ B^ C;^, its associate 
 o-g. with which the polars of the other two will coincide (art. 17. c.), must 
 pass through the conjugates of the remaining two points. On the other 
 hand, it is obvious that all the seven given conditions could be satis¬ 
 fied, and in one way only, by each exceptional correlation of the above 
 kind. 
 
 Hence we conclude that in the system there are three exceptio^l corre¬ 
 lations with singular lines; the latter being P^A^ and BgCg, PiBj and 
 and PjCi and AgBg, respectively. 
 
 n=ry If Pi were a singular point, (d^e^) would be its associate, since the 
 poles of dj and would necessarily coincide with this associate (art. 17. a.), 
 and, at the same time, be situated on d^ and Co respectively. Moreover, 
 it is obvious that one, and only one, exceptional correlation, with these 
 singular points satisfies the re({uired conditions. 
 
 Since, in any other possible correlation • with singular points. Pi 
 would not coincide with 2 ^, 22 would necessarily lie on j >2 17. b.). 
 
17 
 
 Farther, if 2^ were not coincident with either of the points in which 
 is intersected by and e^, then the poles of the latter lines would coin¬ 
 cide, in with (art. 17. a.). This is clearly possible, and in one 
 
 way only (art. 29) ; the necessary and sufficient relation to be satisfied by 
 
 Sg being (d^ (Pj Cj^) = -^2 ^2 ^ 2 )- 
 
 If Sg were coincident with then 2^, as pole of would be 
 
 situated on e^, and satisfy the equation 
 
 ^1 (^1 "^1 ^1 ^ 1 ) — (Pi ^h) (P2 ^2 ®2 ^ 2 ) 5 
 of this equation there are obviously two solutions (art. 31). 
 
 In like manner, if were coincident with (then there would 
 be two positions of 2^, on rZg, each of which would satisfy the necessary 
 and sufficient condition 
 
 — (P2 ^2)(P2 "^2 ®2 ^ 2 )* 
 
 We conclude, therefore, that the system under consideration contains^ 
 on the whole, six exceptional correlations u'itli singular gjoints. 
 
 The following are the numerical values deducible from the above 
 results: \ = 3, tt = 6, p = 4, v = 5. 
 
 37. The first of the four systems in Grroup VI. has tlie signature ( 0070 ) 
 (0070), and is thus defined : 
 
 /A, B, Cl Di El Fi GA 
 \A 2 B, C2 D2 E 3 F2 Go/' 
 
 There are here, as in art. 34, iw correlations with singular lines. Arro 
 
 The necessary and sufficient conditions to be fulfilled by a pair of 7r=3 
 singular points is 
 
 2i(AiBiCir)iEiFiGi) = 22(A2B2C2D2B2F2G2) . (1), 
 
 of which equation, as Sturm and others have shown, there are three 
 solutions. This important theorem results, in fact, from the following 
 considerations : In art. 34(1) it has been shown that, irrespective of Fg 
 (which is an arbitrary point), there are on any line/g passing through 
 Fg, two points Sg, to each of which corresponds a point Si, such that 
 
 Si (Ai Bi Cl Di El Fi) = S 2 (A 2 B 2 C 2 D 2 E, FA . (2). 
 
 It is obvious, too, that Fg itself counts as one such point; since one, 
 and only one, point Si can be found to satisfy the condition 
 
 Si (A, Bi C, D, El) = Fj (A5 B, C, D, E,). 
 
 Hence, and from symmetry, we infer that the locus of the point Sg, 
 to which another point S^ can correspond so as to satisfy the condition 
 (2), is a cubic passing through the six points Ag Bg Cg Dg Eg Fg. The 
 corresponding points Sj lie, of course, on another cubic, passing through 
 Ai Bj Cj Dj Ej Fj, between whose points and those of the first cubic a 
 (1, I) correspondence obviously exists. 
 
 c 
 
18 
 
 In like manner, the locus of each of the points S^, Sg, which satisfy 
 
 the relation Sj (A;^ C;^ G^) = Sg (Ag Cg Dg Eg Gg) . (3) 
 
 is a cubic passing, respectively, through the six points Aj E;^ Gj, 
 
 and the six points Ag Bg Cg Dg Eg Gg. Exclusive of the five points 
 Ag Bg Cg Dg Eg aud of anothcr point Sg, which may be termed the 
 satellite of this group of five out of the seven points Ag Bg Cg Dg Eg Eg Gg, 
 the above two cubic loci in the second plane intersect in three points 
 2g, which, with their associated points 2)]^? satisfy the relation (1). 
 
 The excluded point Sg is that which corresponds to every point Sj on 
 the conic (Ai^Bj^CjDj^Ej) in such a manner that 
 
 Si (Ai Bi Cl Di El) = Sg (Ag Bg Cg Dg Eg).(4). 
 
 It is, obviously, the fourth intersection of two conics (S 2 ) (S^) circum¬ 
 scribed to the triangle AgBgCg, one of which passes through Dg, and 
 makes (S^) (Ag Bg Cg Dg) = Ei (Ai Bi Ci Di), 
 
 and the other passes through Eg, and makes 
 
 (S 2 ) (Ag Bg Cg Eg) = Dl (A^ Bi Cl Ei). 
 
 This point Sg is excluded for the following reason : Each of the cubic 
 loci, in the first plane, determined by (2) and (3), cuts the conic 
 (AiBiCiDiEi) once again; but under the assumption in art. 27 these two 
 points of intersection will not be coincident, although by (2) and (3) each 
 will have Sg for its correspondent. Hence, although Sg is common to 
 the two cubics in the second plane, it cannot be regarded as one of the 
 points Sg with which a point 2i is so associated as to satisfy the rela¬ 
 tion (1). It is easy to see, however, that the satellite Sg, defined by (4), 
 is the only one of the four intersections of the two cubics, in the second 
 plane, which fails to lead to a solution of (1). 
 
 We conclude, therefore, that the system under consideration contains 
 three exceptional correlations ivith singular points ; accordingly, we have 
 the values X = 0, tt = 3, /x = 1, v = 2. 
 
 ( 0061 ) 
 
 A=0 
 
 7r = 6 
 
 38. We proceed to the second system (0061) of Group VI., the symbol 
 
 for which is 
 
 /Ai B; Cl Dl El El gi\ 
 
 \A2 B2 C2 D2 E2 F2 gj' 
 
 Eor the reasons stated in art. 34, this system can contain no exceptional 
 correlations with singular lines. 
 
 If Si, S 2 be the associated singular points of any exceptional eorrela- 
 tion, the condition 
 
 Si (Ai Bi Cl Dl El El) — 22 (A2 B2 C2 D2 E2 E2) 
 
 ( 1 ) 
 
 must be satisfied. Hence, as we have seen in art. 37, Si and Sj must 
 be corresponding points on two cubics ; one of which passes through 
 Ai Bi Cl Dl El El, as w^ell as through the satellite of each of the six groups 
 of five selected from these six points, and the other passes through 
 
19 
 
 Ag B 2 C 2 D 2 E 2 F 2 , as well as through the six satellites similarly con¬ 
 nected therewith. 
 
 Besides satisfying the relation (1), however, it is to be remembered 
 that if be not situated on ^ 1 , then S 2 must lie on ; and further, that 
 if S 2 be not on g^^ then 2^ must be on ^ 1 , whence we conclude that in 
 the system there are six, and only six, exceptioyial correlations with singular 
 points, and that in three of these 2i is on gi, whilst in the other three 
 •S 2 is on g^. 
 
 We have thus the following system of values : 
 
 X = 0, TT = 6, fx =2i, V = 4. 
 
 39. We pass now to the third system (0052) of the Group VI., the 
 
 symbol for which is • 
 
 VA2 B2 C2 D2 E2 /2 ^2/ 
 
 Here again there are no exceptional correlations with singular lines. 
 
 One condition to be satisfied by the singular points of every excep¬ 
 tional correlation which the system may contain is 
 
 2 i (Aj Bi Cl Di El) =: ^2 (A2 B2 C2 D2 E2).( 1 ), 
 
 in virtue of which to each point 2i will correspond, in general, one, and 
 only one, point ^ 2 , and vice versa* Hence we infer that there can he hid 
 one position for Si exterior to hoth the lines f and gi, for in such a case its 
 associate would necessarily coincide with (/ 2 y 2 ) (art. 17. a.). That there 
 will be one such position of S^ however, follows from the assumption 
 in art. 27. 
 
 In like manner, there ivill he one, and only one, exceptional correlation 
 in ivhich Sj will he coincident with (/il7i), Cind its associate exterior hoth to 
 /g and r/g. 
 
 In all other cases, 2^ will lie on and therefore Sg on g^ ; or else Sj 
 will be on g-^, and 2^ on f^. To decide how often the first of these 
 cases will present itself, it will clearly suffice to determine the order of 
 the locus of the points Sg which, by the relation (1), correspond to the 
 several points 2^ of the right line/j. 
 
 To do this we will first enquire into the number of points in which 
 any line a^, passing through Ag, is intersected by the required locus. 
 Now we have already proved that the locus of the point 2^ which 
 satisfies the condition 
 
 2i (Aj Bj^ Cj E;^) = 22 (a^ ^2 ^2 ^ 2 ) 
 
 is a cubic passing through B^^ Cj Dj E^ and having a double point at Aj 
 (art. 34). Hence we may say that, exclusive of the point Ag, which 
 has a perfectly arbitrary position on a^, there are three positions of 
 on ^ 2 , to which correspond three positions of 2^ on /j, such that the 
 relation (I) is satisfied. 
 
 * The only exceptions to this arise when 2i (or 22 ) coincides with one of the five 
 points in its plane, or with the satellite S) (or S 2 ) of these five (art. 37). 
 
 C 2 
 
 (0052) 
 
 A = 0 
 
 7r = 12 
 
20 
 
 It is obvious, however, that this relation will also be satisfied when 
 Hg coincides with A 2 , and with either of the intersections ofand 
 the conic (Sj, through Ej, determined by the anharmonic relation 
 
 (S.)(BiCiDiEi) = A2(BjC,D2E,). 
 
 Hence, and from symmetry, we conclude that if 2 ^ describe any right 
 line fi, the locus of the point ^ 3 , associated therewith by the relation ( 1 ), is 
 a quintic which has a double point at each of the five points A 2 B 2 C 2 l) 2 E 2 . 
 
 This quintic, ivhich has moreover a sixth doubte point (at the satellite 
 S 2 of the above five points) corresponding to the intersections of f and 
 the conic (AiBiCiDjEi), cuts in five points S 2 , to each of which cor¬ 
 responds, by ( 1 ), a point 2 i situated on/i; whence we infer that there 
 are five excepjtional correlations in the system under consideration, of which 
 one singular point lies inf^, and its associate in g^. In like manner we 
 conclude that there are five exceptional correlations of which one singular 
 point lies in pi, and its associate in / 2 . 
 
 Altogether, therefore, the system includes twelve exceptional correlations 
 with singular points, but none with singular lines. Accordingly we 
 have the following numerical values : 
 
 X 0, IT ■= 12, ju = 4, V = 8. 
 
 (0043) 40. AVe have now arrived at the last of the fundamental systems 
 
 which need investigation. Its signature is (0043) and its symbol 
 
 / Aj Bi Cl Dj 61 fi yA 
 
 \A 2 B2 C2 B2 62 A ^ 2 / 
 
 A =6 For reasons before stated (art. 36), every singular line cti must pass 
 through two of the four points AiBjCiDi, and its associate o ’2 must 
 pass through the conjugates of the remaining two; moreover, it is 
 easily seen that the seven given conditions can be satisfied, and that in 
 one way only, by an exceptional correlation whose singular lines have 
 any one of the six pos.sible positions above indicated. Hence in the 
 system there are six correlations ivith singular lines. 
 
 „_i 2 The associated singular points 2 i, ^2 of every exceptional correlation 
 in the system must, in the first place, satisfy the anharmonic relation 
 
 2 i(AiBiCiDi) = 22 (A2B2 021)2). 
 
 The position of one of the two points being known, therefore, we 
 have at once a conic on which the other must be situated. 
 
 Now the point Sj must lie in one, ^t least, of the three lines eifgi, 
 otherwise the poles of these lines would be coincident in 22 (art. 17. a.), 
 which is obviously inconsistent with the condition of their being situ¬ 
 ated, respectively, on e,, / 2 , and ^ 2 - Again, if 2 i lie on one only of the 
 three lines e^ f g^, then its associate will necessarily coincide with the 
 intei section of the conjugates of the other two, and 2 i itself will be 
 
21 
 
 coincident witli one of the two points in which the first line is inter¬ 
 sected by a conic determined by (1). Lastly, if Sj coincide with an 
 intersection of any two of the lines ei fy gy, then will necessarily 
 coincide with one of the two points in which the conjugate of the third 
 line is intersected by another conic, likewise determined by (1). In 
 every possible case, therefore, one of the associated points Si, must 
 coincide with the intersection of two of three given lines, and the other 
 must have one of two known positions on the conjugate of the third. 
 
 Taking all possible combinations into consideration, and remember¬ 
 ing that each one of them leads to two exceptional correlations satis¬ 
 fying the seven conditions, we conclude that the system tinder consi¬ 
 deration contains twelve exceptional correlations with singular points. We 
 have consequently the following system of values : 
 
 X = 6, TT = 12, jj, = S, V = 10. 
 
 41. The class, order, and singularities of each of the thirteen funda¬ 
 mental systems, arranged in the left-hand column in art. 24, having 
 now been determined, those of the remaining thirteen systems may, as 
 stated in art. 25, be deduced by the Principle of Duality. The 
 results may be thus tabulated: 
 
 Group. 
 
 Signature. 
 
 (a/3y^) 
 
 Characteristics. 
 
 Singularities. 
 
 T i (3010) 
 
 . L (0301) 
 
 TT i (2110) 
 
 . I ( 1201 ) 
 
 f (2030) 
 
 ttt J 
 
 . 1 ( 0212 ) 
 
 ^ (0203) 
 
 r(1130) 
 
 TV d 
 
 . I (1112) 
 
 1(1108) 
 
 
 
 
 (1050) 
 
 (1041) 
 
 (1032) 
 
 (0123) 
 
 (0114) 
 
 (0105) 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 2 
 
 1 
 
 2 
 
 2 
 
 2 
 
 1 
 
 2 
 
 4 
 
 5 
 4 
 2 
 
 2 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 2 
 1 
 
 2 
 
 2 
 
 2 
 
 1 
 
 2 
 
 4 
 
 5 
 
 4 
 
 o 
 
 TT 
 
 3 
 
 0 
 
 1 
 
 1 
 
 3 
 
 4 
 1 
 0 
 
 3 
 
 2 
 
 2 
 
 0 
 
 3 
 
 6 
 
 6 
 
 3 
 
 0 
 
 0 
 
 0 
 
 3 
 
 1 
 
 1 
 
 0 
 
 1 
 
 4 
 
 3 
 
 0 
 
 2 
 
 2 
 
 3 
 
 0 
 
 0 
 
 3 
 
 6 
 
 6 
 
 3 
 
22 
 
 Group. 
 
 VI. 
 
 Signature. 
 
 Characteristics. 
 
 Sing 
 
 ularities 
 
 (ciflyB) 
 
 
 V 
 
 TT 
 
 A 
 
 ^(0070) 
 
 I 
 
 2 
 
 3 
 
 0 
 
 (0001) 
 
 2 
 
 4 
 
 0 
 
 0 
 
 (0052) 
 
 4 
 
 8 
 
 12 
 
 0 
 
 (0043) 
 
 8 
 
 10 
 
 12 
 
 6 
 
 (0034) 
 
 10 
 
 8 
 
 0 
 
 12 
 
 (0025) 
 
 8 
 
 4 
 
 0 
 
 12 
 
 (0010) 
 
 4 
 
 2 
 
 0 
 
 0 
 
 .(0007) 
 
 2 
 
 1 
 
 0 
 
 3 * 
 
 Number of Correlations satisfying eight Elementary Conditions. 
 
 42. The number of correlations wliicb satisfy any eight elementary 
 conditions may be readily determined from the preceding Table. In 
 fact, if we indicate this number by the symbol where 
 
 /3, 7, ^ have the same signification as before (art. 23), but now satisfy 
 the equation 2a + 2/3-f = 8, 
 
 we shall clearly have [a/lyr] egual to the class of the system of 
 correlations (a/3y —1^), as ivell as to the order of the system of cor¬ 
 relations (afoy d-l)- But the systems (o/Sy^) and (/3ay^) being 
 identical (art. 23), and the class and order of the system (a/3y^) being 
 the same, respectively, as the order and class of the system (/3a^y) (art. 
 25), it follows that [a/3y^] = [payB] = [a/3ay] = [/3acV] , 
 in other words, that y and B as well as a and (3 are interchangeable in 
 the symbol [apy^]. The following Table, therefore, gives the number 
 
 * The meanings of the symbols being suitably modified, the results contained in 
 this Table are at once applicable to the case of two homographic planes. For by 
 the method employed in art. If), we have, corresponding to every system of corre¬ 
 lations (a^yd) in two planes 111 , 112 , ^ system of homoyraphic relations' (a^yS) 
 in the planes rij, n', which satisfy seven conditions of the following type : a points in 
 the first plane correspond to given points in the second; 13 lines in the first plane 
 correspond to given lines in the second ; y points in the first plane each correspond 
 to a point on a given line in the second; and 8 lines in the first plane each correspond 
 to a line through a given point in the second. Every such system of homographic 
 relations coiitains exceptional ones. In tt of these there is a singular point in the 
 first plane and a singular line in the second, whilst in A others there is a singular 
 line in the first plane and a singular point in the second (art. 14). The points which 
 correspond, in the several homographic relations of a system, to a given point in the 
 first plane, lie on a curve of the order p in the second ; those which correspond to a 
 point in the second plane, however, lie on a curve of the order v in the first. Simi¬ 
 larly, the lines which correspond to a given line in the first })lane envelope a curve 
 of the class v in the second, whilst those which correspond to a line in the second 
 plane envelope a curve of the class y in the first. 
 
 It is worth observing that if each point in the plane 111 he connected with every 
 point in the plane n' which corresponds thereto, in the several homographic relations 
 of a system whose characteristics are y and r, we shall have, in sjiace, a complex of 
 the degree \x3-v — 'K\ir [art. 20 (2)J, of which n, and n' are singular planes. We 
 have consequently twenty-six distinct comi)lexe8 associated with the several funda¬ 
 mental systems of homographic relations satisfying seven conditions. 
 
23 
 
 of solutions in all possible cases where the eight given conditions are of 
 the elementary kind described in art. 22 : 
 
 [4000] = 1 ; 
 
 [3100] = 1 ; 
 
 [ 2200 ] = 0 ; 
 
 [3020] = 1, 
 
 [3011] = 2; 
 
 [ 2120 ] = 1 , 
 
 [ 2111 ] = 1 ; 
 
 [2040] = 1, 
 
 [2031] = 2, 
 
 [2022] = 3; 
 
 [1140] = 1, 
 
 [1131] =- 2, 
 
 [ 1122 ] = 2 ; 
 
 [1060] = 1, 
 
 [1051] = 2, 
 
 [1042] = 4, 
 
 [1033] = 5; 
 
 [0080] = 1, 
 
 [0071] = 2, 
 
 [0062] = 4, 
 
 [0053] = 8, 
 
 [0044] = 10.* 
 
 43. The only results in the above Table which cannot be obtained in 
 the manner described in art. 42 are the first three. Of these, however 
 the first is well known, and has already been alluded to in art. 3. The 
 second differs in form only from the first; for whenever the poles of 
 three lines are given, as in the first case, the polars of three points (the 
 intersections of the given lines) are always known, and vice versa. With 
 respect to the third result, it will be at once seen, on writing the 
 eight conditions in full thus : 
 
 Pi) Qi) '^1) ^1 
 
 1^2) 2'2) ^2) ^2 
 
 that they cannot possibly be satisfied by any correlation unless the an- 
 harmonic ratios (ri5i)(PiQiri5j) and K 2^2 (P 2 5 ' 2 P 2 S 2 ) are equal to one 
 another. The assumption of any such equality, however, would be in¬ 
 consistent with that already made in art. 27, in virtue of which the 
 positions of the given points and lines are perfectly arbitrary. 
 
 * This Table obviously gives, also, the number of ways in which two planes may 
 be rendered homographic (or put in perspective with each other), so as to satisfy 
 eight conditions of the kind described in the note to art. 41. 
 
24 
 
 Connexes determined hy Fundamental Systems of Correlations. 
 
 44. In art. 18 allusion was made to the series (doubly infinite) of 
 curves of the class fi, and to the series of curves of the order v. which 
 every system of correlations, satisfying seven conditions, determines in 
 each of the two planes ; and in art. 22 it was observed that upon a 
 more intimate knowledge of the properties of these curves must 
 depend the solution of the general problem of correlation. Although 
 unprepared, at present, to treat this wide subject exhaustively, I pro¬ 
 pose, before terminating the present paper, to indicate briefly the more 
 salient features of a few of the simplest series of curves of the above 
 kind. 
 
 45. Before commencing to do so, I may observe that, according to the 
 terminology employed by Clebsch in a very suggestive posthumous paper 
 recently published in the “ Mathematische Annalen” (vol. 6, p. 203), 
 each point of one of our two planes, and its polar in any correlation of a 
 system, constitute an element o f a connex of the class y and order r;* whilst 
 each point in the other plane, and any one of its polars constitute an ele¬ 
 ment of the conjugate connex. As illustrations, therefore, of the Theory 
 originated by the eminent geometer, whose loss to science is so uni¬ 
 versally deplored, as well as on account of their application to the 
 general problem of correlation, the well-defined conjugate connexes to 
 which the several fundamental systems of correlations lead are well 
 worthy of full investigation.f 
 
 46. Each curve of the class y may be termed the representative of the 
 point by whose polars it is enveloped, and each curve of the order v 
 the representative of the line of whose poles it is the locus. It is ob¬ 
 vious that every representative of a point touches all the singular lines in 
 its planeand every representative of a line passes through all the sin¬ 
 gular points in its plane. 
 
 C miex 47. Each of the systems (3010), (2030), (1130), (1050), and 
 (0070) leads to a pair of conjugate connexes of the first class and 
 second order. In each of these five systems, a point in either plane is 
 represented by a point in the other plane, (in other words, every point 
 of the plane has its conjugate, exactly in the same manner as the given 
 points have,) whilst every line is represented by a conic passing 
 through the three fixed singular points of the three exceptional cor¬ 
 relations which each system includes. 
 
 * A connex (/xi/), of the class ix and order v, is defined by a given relation 
 
 (I. m O'* (o', y, zf — 0 between the coordinates x, y, z oi a. jDoint, and the coordi¬ 
 nates 7], f of an associated right line. 
 
 t I niay here mention that this investigation was in a tolerably advanced state 
 eighteen months ago, when my studies were interrupted by other duties. Uncertainty 
 as to when these studies may be resumed, has at length induced me, contrary to my 
 original intention, to publish the present paper, before completing the whole enquiry. 
 
25 
 
 We have here, in fact, the well-known case of the Quadric Cor¬ 
 respondence of two planes, presented in its most general form; the 
 three pairs of singular points being identical with the three pairs of 
 Principal Points. The five systems above enumerated correspond, 
 obviously, to the five different ways of determining a Quadric Cor¬ 
 respondence by means of given Corresponding Points, Principal Points, 
 and Principal Lines.* 
 
 48. The systems (0301), (0203), (1103), (0105), and (0007) all 
 lead to connexes of the second class and first order. Every right 
 line, in each of these systems, is represented by a right line ; in other 
 words, every line has its conjugate, in exactly the same sense as the 
 given lines have. Every point, however, is represented by a conic 
 touching the singular lines of the three exceptional correlations which 
 each system includes. In short, we have here a correspondence estab¬ 
 lished between two planes which is, simply, the correlative of the ordi¬ 
 nary Quadric Correspondence.f 
 
 49. Of the remaining sixteen fundamental systems of correlations, by 
 far the simplest are the two whose signatures are (2110) and (1201), 
 both of which lead to one and the same kind of connex of the first class 
 and first order. 
 
 In illustration of this I will consider briefly the first of the two sys¬ 
 tems. Its complete definition is given by the symbol 
 
 /Pi Qi ’"i ^i\ 
 
 % P'2 ^2/ 
 
 and, as shown in art. 29, it contains two exceptional correlations; of 
 which one has the singular lines = P^ Qj and = Rg Ag, whilst 
 the other has the singular points ^ 2 = (^ 2 ^ 2 )’ 5 latter point 
 
 being so situated, on r-^, that 
 
 ^1 (Pi Qi -^i) — ^2 (P 2 % P2 '^2)- 
 
 Since the characteristics are /^ = I and r = l, it is obvious, first, that 
 the polars of any point (say M^) in any two correlations of the system 
 {e.g., in the two exceptional ones) determine, by their intersection, the 
 point M 2 , through which all the other polars of pass ; and, secondly, 
 that the poles of any right line in the two exceptional correlations 
 determine the line upon which all the other poles of are situated. 
 Now if be neither coincident with 2^ nor situated on its polar in 
 the exceptional correlation with singular lines will coincide with the 
 
 * See Rpye on Gcometrische Verwandschaften zweiten Grades, in the Zeitschrift fur 
 Mathematik und Fhysik, vol. xi., 1866. 
 
 t The fact that we have no distinctive name for this correspondence is doubtless 
 due to the circumstance that no terms correlative to the very convenient ones quadric, 
 cubic, &c. are yet in general use. A special case of the correspondence in question 
 was briefly described in my paper “ On the Quadric Inversion of Plane Curves,” 
 published in the Proceedings of the Royal Society, vol. xiv., 1865, and in the 
 “Annali di Matematica pura ed ap])licata,” tom. vii., Roma, 1865. It was also 
 distinctly referred to in the paper by Rcye above cited. 
 
 Connex 
 
 ( 21 ) 
 
 Connex 
 
 ( 11 ) 
 
26 
 
 singular line o-g (17. a.), and its polar Wg, in the exceptional correlation 
 with singular points, will pass through S 2 (17. b.), and be determined by 
 the relation Sj (P^ Qj = Sg (jpg Rg ??^g). 
 
 Consequently, every such point is rejpreserited, in the connex, by a 
 point Mg = on the singular line Cg. If were coincident with 2j, 
 
 however, mg would be perfectly indeterminate (art. 16.), and its repre¬ 
 sentative Mg would, as a consequence, be an indeterminate point on erg. 
 Lastly, if M;^ were situated on then, although mg would be deter¬ 
 mined by (1), as before, the polar of M^ in the exceptional correlation 
 with singular lines would be an indeterminate line through 
 (art. 16.) ; and since the latter in one of its possible positions would be 
 coincident with mg, the representative of M;^ must be regarded as an 
 indeierminate point in m^* 
 
 50. The peculiar point hy q)oint representation to which we are led by 
 the system of correlations under consideration is such, therefore, that 
 1) the singular point of either plane is represented by an indetermi¬ 
 nate point in the singular line of the other plane ; 2) a point in the 
 singular line of either plane is represented by an indeterminate 
 point in a determinate line through the singular point of the other 
 plane; 3) every other point in the first plane is represented by a per¬ 
 fectly determinate point in the singular line of the second plane. From 
 similar considerations we infer that the line hy line representation, to 
 w'hich the present system of correlations leads, may be briefly described 
 thus : 1) a singular line is represented by an indeterminate line through 
 a singular point ; 2) a line through a singular point, by an indetermi¬ 
 nate line through a determinate point on a singular line ; 3) every other 
 line, in either plane, is represented by a perfectly determinate line 
 through a singular point. 
 
 51. It will be observed that the representation just described is not, 
 even in an exceptional sense, homographic ; since one of the character¬ 
 istic properties of homographic correspondence is not fulfilled by it,—I 
 mean that property, in virtue of which to a point in a line always cor¬ 
 responds a point in the corresponding line. The representation above 
 indicated, however, as well as that termed homographic, is to be 
 reo-arded as one of the forms to which a connex of the first class 
 
 o 
 
 and first order may lead. 
 
 52. It is also worth observing that in the connex now under con¬ 
 sideration the locus of the points which represent those situated in a 
 given right line of either plane, breaks up into the representative of 
 that line in the other plane, and the singular line of the latter plane ; 
 
 * It is obvious that, in all the correlations of the system, except that which has 
 sinpfular lines, the polars of a point on a singular line in either plane are coincident 
 with a line which passes through the singular point of the other plane. 
 
27 
 
 and further, that the envelope of the lines which represent those passing 
 through a given point of either plane, breaks up into the representative 
 of that point in the other plane, and the singular point of the latter 
 plane. 
 
 We have here, in fact, the simplest case of a general theorem which 
 holds for all the connexes determined by the fundamental systems of 
 art. 41, and which may be thus enumerated: 
 
 The curve, of the order v, which represents any straight line, is always 
 a constituent part of the envelope of the curves, of the class /u, ivhich repre¬ 
 sent the several points oj that line; and, vice versa, the curve, of the class 
 fx w^iich represents any point, is always a constituent part of the enve¬ 
 lope of the curves, of the order y, ivhich represent the several right lines 
 passing through that point.^ In proof of this, it will be sufficient to 
 show that if and (Xg be pole and polar in any correlation of a system, 
 then, in that same col relation, the tangent at to the representative 
 of and the point of contact of with the representative of A^^, 
 will also be pole and polar.f Now in the correlation immediately fol¬ 
 lowing the frst, let a'g be the polar of A;,^, and A'^^ the pole of a^, in 
 which case, of course, (ag af) will be the pole of Then, ulti¬ 
 
 mately, as the latter correlation approaches to identity with the first, 
 
 ((Xg df) becomes the point of contact of ^2 with the representative of 
 A;^, and AjA'^^ becomes the tangent at A^ to the representative of a^. 
 
 53. The connexes, of the second class and second order, to which the Conne 
 
 . . . . (22J 
 
 systems (1121) and (1112) give rise, are essentially the same in cha¬ 
 racter. We may confine our attention, therefore, to the first system, 
 which has been denoted in art. 33 by the symbol 
 
 /Pi Py Ai Bi CjX 
 \P3 ^2 ^2 ^2 ^2/ 
 
 and found to include four exceptional correlations. One of these has 
 the singular lines a\ = Pj A^, ^ P 2 Ba; another the singular lines 
 
 <Ti ^ Pi Bi and al ^ P 2 A 2 ; a third has the singular points S'l = (_PiCi) 
 and 1 ^ 2 , the latter being so situated on p 2 that 
 
 (pi ^1) (pi P1 Ai Bi) = ^2 (P2 P2 A2 B2) ; 
 and the fourth has the singular points Sj and ^^'^(paCa), of which 
 the former is so situated on pi that 
 
 {Pi Pi Ai Bi) = (pa C2) (P2 P2 A2 Ba). 
 
 * III studying the properties of the various connexes, this theorem is of the greatest 
 utility. It was to it, as a fruitful source of theorems on Envelopes, that I refeiTed in 
 a Note published in Educational Times iox December, 1871 (Reprint, vol. xvi., 
 p. 5)9). The theorems demonstrated in that Note were special cases of more general 
 ones, dcducihle from the connex treated of in art. 63. 
 
 t It is in consc(iuence of this that the two connexes, to which every system of 
 correlations leads, are conjugate in the niamier defined by Clehsch, at p. 208 of the 
 paper referred to in art. 46. 
 
28 
 
 The conjugate connexes to which we are led by the present system are 
 such that every point, in either plane, is represented by a conic which 
 touches the two singular lines of the other plane, and every right line 
 in the first plane is represented by a conic which passes through the 
 two singular points of the second. 
 
 54. It should be observed, however, that the conics which re-present 
 the several points of each singular line, degenerate to point-pairs. 
 
 Thus, if be a point on the singular line (t[, its polar, in the excep¬ 
 tional correlation whose singular lines are (7[ and is an indeterminate 
 line through the point M,, on which is determined by the relation 
 
 ( t \ (Pi Pi Cl Ml) = 0-^P 2 P 2 ^2 M 2 ) .(1). 
 
 Hence we at once infer that the conic which represents M^ breaks up 
 into the point M'l, and another point M 2 . The latter point must be 
 situated on 0 - 2 , because this line is the polar of Mj|^ in the exceptional 
 correlation whose singular lines are a’i and (17. c.), and it must 
 also satisfy the relation 
 
 O’! (Pi Pi ^1 Ml) = 0-2 (_P 2 P2 ^2 M2) .( 2 ), 
 
 because the polars of M;^, in the correlation whose singular points are 
 2i and S 2 , as well as its polar in the correlation whose singular points 
 are S'/ and S'/, obviously pass through the point M 2 thus determined. 
 This point Mo, it will be observed, is conjugate to Mi in precisely the 
 same sense that A 2 is conjugate to Ai; in fact, we may say that every 
 point on a singular line of either plane has a conjugate point situated on 
 the non-associated singular line of the other plane. 
 
 The relations (1) and (2) also show that there is a (1, 1) correspond¬ 
 ence between the points M 2 and M '2 of a nature such that when one of 
 tliem coincides with P 2 , or falls upon po, the other does the same. 
 Hence, by a well known theorem, we conclude that M 2 M '2 always 
 passes through a fixed point S/ on p.^. In like manner, it can be shown 
 that every point Hi on the singular line o-/ is represented by a point- 
 pair, H 2 (on a'j) and N/ (on o-/), whose double tangent N 2 N/ passes 
 through a fixed point S/ on po,. It can be shown, moreover, that the 
 six points 2/, S/, M 2 , M/, ^ 2 , N/ always lie on a conic; and by the 
 theorem of art. 52 we at once infer that this is the conic which repre¬ 
 sents the arbitrary line Mi Ni. 
 
 55. In a precisely similar manner, the following properties may be 
 established :— 
 
 Every line through a singular point of either plane has a conjugate line 
 passing through the non-associated singular point of the other plane. 
 
 Every right line passing through the singular point is repre¬ 
 sented by a line-pair mo^ni'i, of which nC passes through 2/ and through 
 
29 
 
 22 , and the double point of this line-pair lies on a fixed line s\ 
 
 passing through P 2 . 
 
 ^very line n^ passing through 2i is represented by a line-pair n.^n'^^ of 
 which nl passes through 22 , and n^ through 'L' 2 ; and the double point 
 (>^ 2 ^^ 2 ) of this line-pair lies on a fixed line s'^ passing through P 2 . 
 
 ' Every four such lines m-i-, W 2 , n-i-> nf together with the singular lines 
 0 - 2 , 0-2 constitute six tangents of the conic which represents the arbitrary 
 point (^ 1 ^ 1 ). 
 
 56. The degenerate conics of the first plane which represent points 
 and lines in the second, have precisely similar properties, that is to 
 say,— 
 
 The points of the point-pairs lie on the singular lines <r'i, tr'i', whilst 
 their double tangents envelope a point-pair S'/, whose own double tan¬ 
 gent is coincident loith the line p^. The lines of the line-pairs pass 
 through the singular points 2^, 2'/, whilst their double points describe a 
 line-pair s\s'[^ whose own double point is coincident with Pj. 
 
 57. A degeneration such as that above described occurs in every 
 fundamental system of correlations. We may in fact say, generally : 
 
 The curve of the class p which represents any point through which pass 
 K singular lines^ degenerates to k points, situated on the associated sin¬ 
 gular lines, and to a curve of the class p — K, which touches the remaining 
 \ — K singular lines. 
 
 The curve of the order v which represents any line in which lie k sin¬ 
 gular points, degenerates to k lines through the associated singular points, 
 and to a curve of the order v — k ivhich passes through the remaining tt — k 
 singular points. 
 
 58. Next in point of simplicity are the connexes determined by the 
 systems of correlations of which (2021) and (2012) are the sig¬ 
 natures. The first of these connexes is of the second class and third 
 order, the second of the third class and second order. 
 
 The properties of the latter being at once deducible from those of the 
 former by the Principle of Duality, we may confine our attention to the 
 first of the above systems, the symbol for which is 
 
 /Pi Qi Ai cA 
 \ P2 T A2 B2 c-i) 
 
 In art. 31 the existence was proved of a pair of singular lines P^ Qj 
 and A 2 B 2 , and four pairs of singular points, viz.. Pi and {q^c^, Qj and 
 (P 2 C 2 ), 2'i and (^ 222 )? ^^d 2'/ and (^> 2 ^ 2 )) where 2'i, 2'/ are the intersec¬ 
 tions, with Cl, of the conic (Si) which passes through PiQiAiBi so as to 
 make the anharmonic ratio of these four points on it equal to 
 
 {PtT) (P 2 qi A2B2). 
 
 59. The mode in which the conics and cubics representing special 
 
 Connex 
 
 ( 23 ; 
 
30 
 
 points and lines degenerate, is very instructive. Instead of attempting 
 an exhaustive discussion of the question, however, I must limit myself 
 here to those cases of degeneration which most elucidate the general 
 character of the representative curves. 
 
 Every point C 2 on the singular line A 2 B 2 is represented, as we have 
 already stated in art. 57, by a point-pair Ci, Ci. The point C'l is so 
 situated on the associated singular line PiQi as to satisfy the condition 
 
 PibJi (Pi Qi Cl Cl) = A 2 B 2 (^ 222 ^ 202 ) .(1) ; 
 
 the polar of C 2 , in the exceptional correlation whose singular lines are 
 PiQi and A 2 B 2 being an indeterminate line through Cl. The point Ci, 
 through which the polar of C 2 in every other correlation of the system 
 passes, is situated on the conic (Si) and fulfils the condition 
 
 (Si)(Pi Qi A;^ B^ Cj) = (p2 g2)(P2 9^2 ^2 ^2 ^ 2 ) . (2)> 
 
 as is at once obvious on considering the polars of C 2 in the two corre¬ 
 lations whose singular points are 2l and and 21' and (i ?2 9 ' 2 )- 
 
 60. Between the points of the conic (S,), and those of the line A 2 B 2 , 
 therefore, a (1, 1) correspondence is established, such that any two 
 corresponding points C]^ C 2 thereof are conjugate to each other in the 
 same sense as A;|^A 2 and Bj^Bg are by hypothesis. The system of cor¬ 
 relations, in fact, would suffer no change if C^Cg were substituted in 
 place of either of the latter pairs of conjugate points. 
 
 61. The conic, in the second plane, which represents any arbitrary 
 point, sayAj, of the conic (Si), is a line-pair tohose double point coincides 
 tuith the corresponding point Ag on AgBg. This follows from, the cir¬ 
 cumstance that, in the system now under consideration, there are two 
 correlations for which the polar of A^ coincides with an arbitrary line 
 Og passing through Ag; in other words, there are two correlations 
 which satisfy the eight conditions 
 
 Qi5 h 
 
 Pit (lit (^1t Bg, Cg 
 
 In proof of this we need merely refer to the Table in art. 42, where it 
 will be seen that [3011] = 2. 
 
 62. From the above we infer that every line in the second plane is 
 represented by a cubic x\ which has a double point Cj on the conic (Si). 
 For if C| be the point on (Si) which corresponds, by art. 60, to the point 
 Cg in which AgBg is intersected by ^g, then, as we have just seen, there 
 are two correlations of the system for which the pole of is coincident 
 with Cj. The cubic Xi obviously passes also through the singular points 
 Pj, Qj, 21, 21' as well as through the point Cl on PjQi, which, by (1) of 
 art. 59, corresponds to Cg on AgBg. 
 
 63. Lastly, since two of the singular points of the second plane coin- 
 
31 
 
 cide in obvious that every line a;^ in the first plane is repre¬ 
 
 sented hy a cubicwhich has a double point at (^ 2 ^ 2)5 and passes, 
 likewise, through the singular points (^ 2 ^ 2 )’ ( 2 ' 2 ^ 2 )’ through 
 
 the point Q^,on A 2 B 2 , which corresponds, by (1) of art. 59, to the point 
 Cl in which oc^ intersects P^Q^, and through the points Ti^and^^i 011 A 2 B 2 , 
 which correspond, by (2) of art. 59, to the points and in which 
 intersects the conic (Sj). 
 
 64. The remark in art. 61 is susceptible of generalisation in the form 
 of the following useful Theorems, with the enunciation of which I will 
 conclude the present paper : 
 
 Theorem (i.) :— In a system of correlations (a/3y3), the curve, of the 
 class [a/3(y+ 1)^] (art. 42), which represents either of two conjugate points 
 A 2 , breaks up into the other, together with a point on each of the singular 
 lines associated with those which pass through the former. The multi¬ 
 plicity of Ag on the representative of A^ is [(a + l)/3 (y —1) ^], and 
 that of on the representative of is [a (/3 + 1) (y —1) ^]. The num¬ 
 ber of singular lines which pass through Aj is 
 
 [a/3(y + l)^]-[(a + l)/3 (y-l) 
 and the number of those which pass through A 2 is 
 
 [a/3 (y+1) ^] —[a (/3 + 1) (y-1) ^]. 
 
 Theorem (ii.) :— In a system of correlations whose signature is (a/3y^) 
 the curve, of the order [a/3y(^ + 1)] (art. 42), which represents either of two 
 conjugate lines a^, a^, breaks up into the other, together with a line through 
 each of the singular points associated with those situated on the former. 
 The multiplicity of a,^ on the representative of is [a (/3 + 1) y (^ —1)], 
 and that of a^ on the representative of a^ is [(a-j-l)/3y — 1)]. The 
 
 number of singular points situated on a^ is 
 
 [«/3y(^ + l)]-[a (/3 + 1) y(^!-l)], 
 and the number of those situated on a^ is 
 
 [a/3y(J + l)]-[(a + l)/3y(a-l)]. 
 
 May 2nd, 1874. 
 

 •, ' A 
 
 s