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 SOME INVARIANTS AND CO VARIANTS 
 OF TERNARY COLLINEATiONS 
 
 HENRY BAYARD PHILLIPS 
 
 A DISSERTATION 
 
 Submitted to the Board of University Studies of the Johns Hopkins 
 
 University in Conformity with the Requirements 
 
 FOR the Degree of Doctor of Philosophy 
 
 Press of 
 
 The new Era Printing Company 
 
 Lancaster, Pa. 
 
 U)OT 
 
SOME INVARIANTS AND COVARIANTS 
 OF TERNARY COLLINEATIONS 
 
 BY 
 
 HENRY BAYARD PHILLIPS 
 
 A DISSERTATION 
 
 Submitted to the Board of University Studies of the Johns Hopkins 
 
 University in Conformity with the Requirements 
 
 FOR the Degree of Doctor of Philosophy 
 
 Prem of 
 
 TNC NtW EDA PRINTINQ OOMPAHY 
 UNOASTER, Pa. 
 
 1907 
 
SOME INVARIANTS AND COYARIANTS OF TERNARY 
 COLLINEATIONS. 
 
 Introduction. 
 
 1. The analytical basis of the present paper is the form of Grassmann's 
 Liickenausdruck which Gibbs called a dyadic. This, as the sequel shows, is 
 merely a general bilinear function from which the variables are omitted. It 
 may then represent a collineation or correlation and may be manipulated prac- 
 tically like the ordinary symbolical bilinear form. 
 
 Starting with this as a basis, the object is in the next place to give an inter- 
 pretation by means of the invariant theory of various double products suggested 
 by Gibbs and incidentally to obtain some of the properties of the invariants 
 and covariants involved. The field of operation is plane projective geometry 
 and the products are formed according to the combinatory multiplication of 
 Grassmann. 
 
 Finally, in the third part, there is considered a skew symmetric function 
 of any number of collineations which is called an alternant. It is a combinant, 
 linear in the coefficients of each collineation, and presenting in some ways for 
 functions of two sets of variables properties analogous to those of the expres- 
 sions resulting from the combinatory multiplication of linear manifolds. 
 
 Part I. Notation. 
 I. The open product or dyadic. 
 
 2. In a space of two dimensions a sum of mixed products of similar construc- 
 tion, each containing a single factor x, may be written in the form 
 
 A^xB^ + A^xB^ + A^xB^, 
 
 where the dot is used to show that the order of multiplication is from left to 
 right. A^ , B^ and x are geometric quantities, points or lines of the plane, and 
 all products are formed according to the combinatory multiplication. This 
 may be considered as resulting from the operation of x on the expression 
 
 A,{yB,^A,{).B, + A,{)-B,, 
 
 the operation consisting in placing the variable x in the parentheses. This last 
 expression is an example of what Grassman called an open product.* 
 
 *"Au8dehnung8lehre" (1878), p. 265. 
 
 165252 
 
4 SOME INVARIANTS AND (X) VARIANTS OF TERNARY COLLINEATIONS. 
 
 Gibbs wrote the open product in the form 
 
 and from the nature of its construction called it a dyadic.* The variable is 
 supposed to operate on the dyadic from the outside and so give as result 
 
 xA^ • B^ + xA^ ■ B^ + xA^ • B^ 
 or 
 
 A^- B^x + A^-B^x + A^- B^x 
 
 according as x is used as prefactor or postfactor. 
 
 In the present paper the notation of Gibbs will be used and combinatory 
 products will be represented either by placing the letters in parentheses or by 
 placing a bar over them. It is found convenient to use the parentheses when 
 the product reduces to a scalar, or number, and in all other cases to use the bar. 
 Unless otherwise expressly stated the variable will enter the dyadic as post- 
 factor, i, e., the dyadic will operate on the variable. From analogy with the 
 ordinary symbolism for a row product we shall write 
 
 AB = ^,^j + A^B^ + ^3 J53. 
 
 It is to be observed that A^ and B^ in this expression have a definite size or 
 intensity. If they are only projectively given the dyadic will have the form 
 
 AB = \A^B^ + \A^B^ + \A^B,, 
 
 where the X's are numbers determined when definite intensities are given to 
 A^ and B^. 
 
 3. As an operator the dyadic gives a linear transformation of quantities con- 
 tragredient to B^. For, x being such a quantity, since (B^x) is a number, 
 
 A(Bx) = X,{B,x)A, + \{B,x)A, + \{B,x)A, 
 
 which as a function of A^ is a simple manifold involving x linearly. 
 
 There are two cases of present interest. When A^ and B^ are contragredient 
 we have a collineation ; when cogredient, a correlation. 
 
 A dyadic of the form 
 
 aa = \a^a^ + \a^a^ + X3a3a3, 
 
 where the a's are points and the a's lines, represents a point collineation. f In 
 
 *GlBB8'8 "Vector Analysis" (Wilson), chap. V. 
 t In the notation of Clebsch this is of course 
 
 (af)(cur) = 2A^(a<f)(aia;)=0, 
 
 when X is given and f variable. If f were given and x variable the dyadic would be written aa. 
 The dyadic is thus regarded not as giving a connex, but as setting up a definite transformation. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINS ATIONS. 
 
 particular to the point a^a^ corresponds the point 
 
 To the vertices of the triangle a then correspond the points a.. Since the 
 triangle a merely presents a set of points to be operated upon it is obvious 
 that this may be chosen at random, the collineation then determining a as its 
 correspondent. While aa as an operator gives the collineation of points it 
 involves internally the collineation of triads. 
 Similarly the dyadic 
 
 represents a correlation in which the lines a. correspond to the points of the 
 triangle yS. A like interpretation may be given for the dual cases aa and ab. 
 
 II. Tetrad and counter-tetrad. 
 
 4. We have seen that the dyadic in trinomial form involves the correspond- 
 ence of triads. Since, however, a collineation or correlation in the plane is 
 determined by four pairs of corresponding elements, it is of greater interest to 
 have the dyadic involve a correspondence of sets of four. And as a dyadic 
 always operates on a contragredient quantity this end can only be attained by 
 the use of a self dual scheme of four-point and four-line. 
 
 With a 4-point we associate the 4-line obtained by taking the polar of each 
 point with respect to the triangle of the other three. It is well known then 
 that conversely the 4-point is obtained by taking the polar of each line with 
 respect to the triangle of the other three. These mutually related systems have 
 been called tetrad and counter-tetrad.*^ 
 
 Supposing the points a. to satisfy only one linear relation (which is the only 
 case of interest) their intensities may be chosen such that 
 
 (1) a^ + a2 + a^+ a^ = 0. 
 
 Operating on this identity with the products a^a. we find that the triple products 
 (a^a.aj^) are in absolute value all equal. If then we write 
 
 we obtain 
 
 (2) (a.a,.aj=±4 {i<j</c), 
 
 where the sign is positive or negative according as a.a.af^ is complementary to 
 an odd or an even term in the sequence a^a^a^a^. 
 
 * F. MOELEY, Trans. Amer. Math. Society, vol. 4, p. 291. 
 
6 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 Making use of these formulas the counter-tetrad a^ may be written in the 
 canonical form 
 
 4a^ = — G^] 0^2 — ^2^4 — ^4^1> 
 
 4a^ = «! Og + ajttg + 03(1^. 
 From these equations by addition we obtain 
 
 (4) aj + a2 + a3+a^=0. 
 
 Multiplying the equations (3) by a. and making use of (2) it is easily seen that 
 
 (5) (a,a,) = 3, {a,a.)=-l (i+j). 
 We have here sixteen equations. From the identity 
 
 i J 
 
 it is seen however that seven of these equations are superfluous, their effect 
 being to make a^ and a^ subject to conditions (1) and (4). When one tetrad is 
 given there are then nine conditions to be satisfied by the other. And since a 
 tetrad subject to the conditions (1) or (4) in addition to its eight geometrical 
 constants involves an undetermined intensity it follows that there is a single 
 solution. The equations (5) may therefore be taken as canonically defining a 
 tetrad and counter-tetrad. Their symmetry in a^ and a. indicates the mutuality 
 previously mentioned. 
 
 From the equations (3) by direct multiplication we obtain 
 
 (6) ^.= a^-a^, 
 
 where i,jj h, /is a positive permutation of the numbers ], 2, 3, 4. Multiply- 
 ing by flj^ and making use of (5) 
 
 (a,a.aj==±4 {i<3<k), 
 
 the sign being positive or negative according as (^if^jOi,^ is an odd or even minor 
 of a^a^a^a^. From the symmetry of the entire system in a^ and a. we may 
 finally write 
 
 (8) ^^.= a,-a,, 
 
 where the rule of subscripts is the same as before. 
 
 5. The application of the preceding to the study of dyadics in the plane is 
 now simple. Consider the collineation 
 
 (9) E(«2«3«4)(^2/33/9J«,A. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 7 
 
 Taking a^ and /8. subject to the conditions (2) and (7) the products 
 (a^a.a^)(y8^yS./3j) all become equal and (9) becomes 
 
 Operating on this with 6^, a point of the counter-tetrad of /8, and making use 
 
 of (5) we get 
 
 Sttj — ttj — ttg — a^ = 4a^ . 
 
 The points h. pass by (9) into the points a^ . The collineation therefore trans- 
 forms the associated system 6, /3 into the system a, a and so the dyadic in this 
 form involves a correspondence of tetrads. 
 In the same way we see that 
 
 (10) Z(«2«3«J(^2^3^4)«l^I 
 
 is a correlation which transforms the counter-tetrad of /S into a and so carries 
 the system b , yS into the system a, a. 
 
 Part II. Multiple Products. 
 I. Multiple products are complete invariants. 
 
 6. With two dyadics AA', BB' is connected a form ABA' B' which Gibbs 
 called the double product of the two dyadics.* It is formed by multiplying 
 the dyadics distributively, each pair of terms combining to form a product in 
 which the prefactor is product of prefactors and postfactor product of post- 
 factors. Gibbs showed that this double multiplication is distributive with 
 respect to a resolution of either dyadic or is invariantive as is readily seen upon 
 expansion. 
 
 So with a system of dyadics are a series of multiple products given by the 
 various ways in which prefactors and postfactors may be independently com- 
 bined. From their construction it is evident that such forms retain their 
 significance when the prefactors and postfactors are transformed separately and 
 therefore belong to the class of functions that Pasch called complete invariants.f 
 If the dyadics appear as transformations operating on the elements of a certain 
 field, since a transformation of postfactors amounts to a transformation of that 
 field, it follows that the geometric interpretation of a multiple product must 
 involve an arbitrary initial field. If, for example, a system of collineations and 
 correlations in the plane operate upon four points, the multiple products will 
 give results independent of the initial 4-point, i. e., invariants and covariants 
 
 * Loo. cit., p. 306. 
 
 The function here considered is a double product only in the sense that it is formed by a cer- 
 tain double process. It is neither the scalar nor the vector but the combinatorial double product. 
 All of these have certain properties in common which characterize double multiplication. 
 
 t " Vollkommene Invariante," Math. Ann., Bd. 52, p. 128. 
 
 
8 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 of the resulting tetrads. Illustrations of this property will appear in the dis- 
 cussions that follow. 
 
 II. Apolarity of coUineations. 
 
 7. Consider two contragredient coUineations aa and /86 , the first an operator 
 on points, the second an operator on lines. They have a double product invar- 
 iant (ay8)(a6). When this vanishes the coUineations will be called apolar. 
 
 To see the meaning of this write 
 
 aa = \a,a^ + \a^a2 + \a^(^3, 
 
 and take A6 = Aa as reference triangle. That is, place 
 
 (6,aJ=l, (6,aJ = 0. 
 We then have 
 
 (afi){ab) = \/^,(a,A) + \/*,(«2^2) + \f'si<^z^s) - . 
 
 This obviously vanishes when for each value of the subscript a^ is on )8^ . Two 
 triangles so related that each point of the one lies on the corresponding line of 
 the other will be called incident. Now Aa and AyS are the correspondents of 
 the reference triangle with respect to aa and yS6 . Hence apolarity is the con- 
 dition under which two coUineations can send a triangle into a pair of inci- 
 dent triangles.* 
 
 A coUineation apolar to the identical collineation sends certain triangles into 
 incident or inscribed triangles. Such a collineation has been called normal. 
 
 Write the given collineation 
 
 aa = XjOjaj + \a^a^ + ^^s^s 
 
 and the identical collineation 
 
 a^a^-\-a^a^+a^a^, 
 
 where Aa = Aa is reference triangle. The apolarity condition is then 
 
 Hence the condition for normal collineation is the vanishing of what Gibbs 
 called the scalav^ of the dyadic. 
 
 8. Another interpretation for apolarity is obtained by using the system of 
 counter-tetrads explained in Art. 4. The invariant (a/S)(a6) gives the con- 
 dition that the collineation 
 
 a(a6)yS 
 
 ♦ M. Pasoh, Math. Annalen, Bd. 23, p. 431. 
 fLoo. cii, p. 275. 
 
SOME INVARIANTS AND COVAEIANTS OF TERNARY COLLINEATIONS. 9 
 
 be normal. Hence if aa and ^b operating on a certain tetrad and counter- 
 tetrad give a 4-point a^ and a 4-line yS^, the coUineation which transforms 6* 
 (the counter-tetrad of /3J into a. will be normal. According to (9) the coUinea- 
 tion which carries 6^ into a^ is 
 
 Therefore the condition that the coUineations aa and Bb be apolar is 
 
 (11) Z(«2«3«4)(^2^3/34)(«lA) = 0, 
 
 where a^ and yS^ correspond respectively through aa and y86 to a tetrad of points 
 and its counter-tetrad of lines. 
 
 It is to be observed that (11) is linear in each of the quantities a^ and ^^. 
 Hence if all of those quantities except a point a^ are given it will lie on a defi- 
 nite line, and if all except a line yS^ are given it will pass through a definite 
 point. Therefore a 4-point and 4-line subject to the condition (11) determine 
 a tetrad of lines through a^ and a tetrad of points on /S^ , consisting, in fact, of 
 the evectants of (11) with respect to a^ and ^^. 
 
 If we write a^ and /3,. in such form that the equations (2) and (7) hold, the 
 apolarity condition (11) may be written 
 
 Placing {a^a^a^) = A , we get for the evectant with respect to a^ 
 
 47l == 4^1 - (a2^2)«3^4 + {(^Z^i)^2^i - («4^4)«2«3- 
 
 Placing a^ as counter-tetrad of a^ this may be written in the form 
 
 47, = 4 { /3, - i i{aMa, + {a,^,)a^ + {^M^z + («4^4)«4] } , 
 
 as is readily seen upon multiplying the right hand members of both equations 
 by each of the points a^ and using the identity 
 
 (a^^J + {aM + (a3^3) + {aM = 0, 
 
 to which (11) reduces. Since the expression in brackets is symmetrical with 
 respect to the numbers 1, 2, 3, 4 we may finally write 
 
 (12) % = ^,-V 
 
 where 
 
 V = J [(aiA)«l + («2^2)«2 + («3/53)S + («4^4)«4] ' 
 
 From (12) and (8) we have 
 where 6 is the counter-tetrad of ^. Hence if we order to the lines 7^ the 
 
10 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 points 6^ it follows that each pair of lines intersect on the join of the remain- 
 ing pair of points. Such a 4-point and 4-line may be called chiastic.* 
 
 From the symmetry of (11) in a^ and y8^ we are now able to write the evect- 
 ant with respect to yS^. in the form 
 
 (13) c. = a.-y 
 
 where 
 
 Therefore the four-point c. and the four-line a. are chiastic. 
 
 The geometric interpretation of apolarity then leads to the following state- 
 ment of a theorem of Pasch : f 
 
 Two apolar collineations transform any four-point and associated four-line 
 into a four-point a^ and a four-line jS^ such that there is a four-line through a»- 
 chiastic to the counter-tetrad of ^. and a four-point on yS^ chiastic to the counter- 
 tetrad of a^ . 
 
 From (13) it is observed that the tetrads c. and a. are perspective, y being 
 the center of perspective. Since counter-tetrads are chiastic this is a special 
 case of a theorem of Pasch which states that any pair of four-points chiastic 
 to the same four-line are perspective. 
 
 9. It has already been observed that the apolarity of the collineations aa 
 and y86 amounts to the vanishing of the linear invariant (scalar) of 
 
 a{ab)^. 
 For brevity we shall write 
 
 s^ = aa, Sj = bfi, 
 
 a^ = aa, o-^ = /86. 
 
 And generally we shall designate a collineation by s and its inverse by a. The 
 collineation written above is then Sj Sj . As a convenient abbreviation we shall 
 denote the linear invariant of any collineation by the symbol of that colline- 
 ation placed in parentheses. Thus the apolarity of aa and /86 is given sym- 
 bolically by 
 
 From the definition it follows immediately that 
 
 («i«2) = («2«i) = (o-iO-J = (o-2<r,). 
 
 Similarly we shall sometimes find it convenient to denote the linear invariant 
 
 *Cf. Sir Robert Ball, "Theory of Screws," p. 306. 
 t Math. Annalen, Bd. 26, p. 311. 
 
 X «i and 8] in this case will be called harmonic, retaining the word ai>olar to express the rela- 
 tion of «i and Of , or «, and Oi . 
 
SOME INVARIANTS AND COVAEIANTS OF TERNARY COLLINEATIONS. 11 
 
 of the product of any number of collineations Sj, s, •••, s^ by (s^s^- • - s^). 
 Writing the collineations in the form aa, b^ ,cy, etc., it is immediately seen that 
 
 (14) («1 ---VlS.) =(«.«! •••«r-l)- 
 
 That is, the linear invariant of the product of any number of collineations is 
 not affected by a cyclic permutation of those collineations. 
 
 10. Suppose we have three collineations each of which transforms a 3-line 
 a. into a 3-line /8^ . If a. and 6^ are the points of the triangles a and fi the 
 collineations may be written 
 
 ^2 = /*1 A«X + f^2^2^2 + /*3^3«8J 
 0-3 = I/,/3,a^ -I- I'2^2«2 + ^3^3 «3> 
 
 where \, fi^, and y^are all different from zero. Furthermore let 
 
 be a non-singular coUineation apolar to cr^, a^, and 0-3. Taking a as reference 
 triangle the conditions required are 
 
 PlM^^^i) + PM^2<^2) + Pz\i^^<^s) = 0, 
 PlM'l(fil^l) + P2P'2{^2^2) + PzP'A^Z^) = 0, 
 Px "X ( A^l) + P2 ^2(^2^2) + Pz ""ii^A) = 0. 
 
 These equations can only coexist when either 
 
 = 
 
 or 
 
 Pl(^lOl) = P2i^2'^2) = PlC/^sCs) = 0. 
 
 The first of these expresses that the collineations are linearly related ; the 
 second that the 3-line /3^ and the 3-point c^ are incident. Hence if a non-singu- 
 lar coUineation is apolar to three linearly independent collineations having a 
 common triangle pair, it transforms the first of those triangles into one incident 
 to the second. In particular, if a. and ^. are identical, their triangle is trans- 
 formed by s into an inscribed triangle and is consequently a Pasch triangle* 
 of s. 
 
 Suppose a second set of collineations T^ , T^ and T^ which transform a 3-point 
 
 *F. MOELEY, loc. cit., p. 295. 
 
 \ 
 
 \ 
 
 \ 
 
 P'l 
 
 P'2 
 
 f^s 
 
 ^1 
 
 ^2 
 
 ^3 
 
12 SOME INVARIANTS AND CO VARIANTS OP TERNARY COLLINEATIONS. 
 
 d. into a 3-point c. . The same is then true of any coUineation of the net 
 
 Suppose one of these coUineations should transform a^ into a 3-point incident 
 to ^.. By the last paragraph the conditions required are 
 
 These equations may be satisfied if 
 
 {S,T,) {S,T,) (S,T,) 
 
 (15) {S,T,) (S,T,) {S,T,) =0. 
 (S,T,) (S,T,) {S,T,) 
 
 From the symmetry of this condition in S and T we conclude that if there is a 
 coUineation which transforms d. into c. and a^ into a triad incident to /8^., then 
 there is a coUineation that transforms 6,. into a^ and c^ into a triad incident to B^ . 
 If in the above theorem we take a^ equal to b^ and c^ equal to d^ we have the 
 theorem of Hun that the relation of Pasch triangle and fixed triangle in a nor- 
 mal coUineation is mutual. 
 
 III. The intermediate.* 
 
 11. Take in the next place two cogredient point coUineations aa and 6/3. 
 They have a double product aba^.-f This is a covariant line coUineation 
 which has been called the intermediate of aa and 6/S . 
 
 The geometric interpretation is easily seen. Let r/ (Fig. 1) be the correspon- 
 dent of any line f two of whose points are x and y - Then by a well known 
 identity 
 
 (16) r}='^b{a^^) = ab{{ax)ifiy)-{ay){^x)}=¥^"-Wi' 
 
 in which x' and x", y' and y" are the correspondents of a and y with respect to 
 aa and 6y8. Therefore ?; is a line through the join of x'y" and x"y' j or, as we 
 may say, through the cross-join of the correspondents of x and y. Since x and 
 y are any points on ^, ij is the locus of cross-joins of pairs of points, on |. This 
 from the known construction of a polarity amounts to saying that x'x" envelopes 
 a conic tangent to x'y' and x"y" at their junction with 17. 
 
 *A. B. Coble, Trans. Amer. Math. Soc., vol. 4, p. 70. Prof. Morley has used the word 
 Clebschian to represent a form of this kind (IVans., vol. 4, p. 471). 
 
 t In the symholic notation of Clebsch this would of coarse be written 
 
 (ate)(ay3f)=0, 
 where f is given and x variable. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 13 
 
 In case of three points jc , y , s of | the preceding construction involves Pas- 
 cal's theorem for the hexagon inscribed in a two-line. 
 
 In case the two coUineations are the same, rj is the join of x and y' and the 
 intermediate reduces to the reciprocal or line form. Hence the line equation 
 
 X 
 
 V 
 
 z 
 
 Fig. 1. 
 
 of a given coUineation is gotten by taking the double product of the dyadic 
 with itself. 
 
 12. An identity for which we shall find frequent use is obtained by develop- 
 ing (a6as) (otySf ) according to the ordinary rule for multiplication of determi- 
 nants. We thus obtain 
 
 {aa) (a/3) (a|) 
 
 {ahx){a^^)= (ba) (b/S) (6f) 
 
 {xa) (x^) (jc|) 
 
14 SOME INVARIANTS AND CO VARIANTS OF TERNARY COLLINEATIONS. 
 
 =(ah^)(x^)+{a^){b^)(xa) + (a|)(6a)(x;S) - {aa){b^){xfi) - {b^){a^)(ax). 
 
 Writing 
 
 s^ = aa, §2 = 6)8 , 
 
 and using the notation of Art. 9 for the linear invariants, we have 
 
 (17) «l«2=(«l«2)+<^l0"2 + °"2°"l-(«l)0'2-(«2)<^l> 
 
 where the bar over s^s^ is used to signify the operation of forming the inter- 
 mediate. 
 
 13. The intermediate belongs to a type of correspondence that occurs in any 
 number of dimensions. And though we are at present concerned with the col- 
 lineation in the plane it may be worth the trouble to indicate the general theory. 
 
 It is very easy to extend the construction already given for the plane inter- 
 mediate to the case of higher dimensions. Using the equations (16) we have 
 for the intermediate ab ayS where aa and bS are point coUineations in space that 
 to any line | corresponds a linear complex with reference to which xy" and x"y' 
 are polar lines, x and x", y and y", being correspondents with respect to aa. and 
 6y3 of two points x and y on^. 
 
 In the same way for the intermediate ahca^fy we have 
 
 (18) abG{a^y7r) = abc{(au){^yyz) + {ay){^yzx) -\- (az){^y'xy)} 
 
 where x, y, z are three points of the plane ir . If then we take any triangle on 
 the plane tt, transform its points by the collineation aa, transform the opposite 
 lines by the collineation bc^y , and join the corresponding elements, we get a 
 set of three planes intersecting on the correspondent of tt with respect to the 
 collineation abca^y. If bc^y is the identical line collineation, or quadratic 
 complex to which every line belongs, the preceding construction is readily 
 translated into a descriptive property of two complete four-points in planes 
 in space. 
 
 Proceeding in this way we may construct intermediates in any number of 
 dimensions. Another method was however presented by Kraus * and Muth.f 
 Consider in the first place the plane intermediate written in the form 
 
 {abx){afi^) = 0. 
 
 This expresses the apolarity of the coUineations axa^ and bx^^. For on form- 
 ing the double product we get 
 
 x{abx){afi^)^ = 0. 
 
 ♦"Dissertation" (Giessen), 1886. 
 ■fMath. AnncUen, Bd. 33. 
 
SOME INVARIANTS AND COVARIANTS OP TERNARY COLLINEATIONS. 15 
 
 Now axa^ and bx^^ may be considered as binary projectivities which give for 
 points on | lines joining x to the correspondents with respect to aa and 6y8. 
 Then since two binary apolar projectivities give rise to an involution it follows 
 that if we transform the points of a line | by the collineations aa and 6/3, the 
 correspondent of | with respect to a6a/3 is the locus of points from which the 
 transforms appear in involution. 
 
 Considering x and | as lines in space it follows from the argument of the 
 last paragraph that the collineation ab a^ in three dimensions gives for a line | 
 the complex consisting of lines which joined to the correspondents of points on 
 f give pairs of planes belonging to an involution. 
 
 In order to interpret the triple intermediate abooL^y we need a new invariant. 
 Three plane collineations aa, 6/3, and cy have in fact a triple product invariant 
 
 {abc)(a^y). 
 
 When this vanishes the collineations have been called harmonic* Its vanish- 
 ing simply expresses that the intermediate of any two of the collineations is 
 apolar to the third. Since we are able to construct the intermediate and to inter- 
 pret the condition of apolarity this harmonic relation may be supposed known. 
 The intermediate of three collineations aa, 6/3, and cy of space may be writ- 
 ten in the form 
 
 (abcx)( a^yir ) = . 
 
 This expresses that the three projectivities ax air, bx^ir , and cxyrr are har- 
 monic. For on forming the triple product we have 
 
 x(abcx) {a^y7r)'7r = 0. 
 
 But axaTT^ bx^ir , and cxyjr are ternary correspondences which give for points 
 on TT lines joining x to their correspondents with respect to aa, b^, and 07. 
 Therefore if we construct with respect to aa, 6/3, and ey the correspondents of 
 points belonging to a plane tt, the correspondent of tt with respect to abcafiy 
 is the locus of points from which those plane systems appear harmonic. 
 
 So by a process of continuous induction we may build up intermediates of 
 any degree of complexity. An intermediate of R collineations is reducible to 
 a harmonic invariant of R collineations \n R — 1 dimensions. This may be 
 interpreted as the apolarity of the intermediate oi R —1 collineations and 
 the remaining one. Thus given the knowledge of apolarity the intermediate 
 of jR collineations is reducible to that of i2 — 1 . 
 
 14. A collineation is singular when there is an element whose correspondent 
 is indeterminate. Thus the intermediate aba^ in the plane is singular when a 
 line ^ can be found such that 
 
 (19) ^(a/8|) = 0. 
 
 * J. Keaus, Math. Annalen, Bd. 29, p. 234. 
 
16 SOME INVAMANTS AND COVARIANTS OF TERNARY CX>LLINEATIONS. 
 
 From the construction of the intermediate it is evident that | must in this case 
 pass by da and b^ into the same line 77. Now (19) is the condition for the 
 correlation 
 
 to be symmetrical, i. e., to be a polarity. But a(a^^)b sets up a binary corre- 
 lation on rj consisting of pairs of points given by aa and 6/S for points of ^. 
 Therefore since every line of the plane cuts | , it follows that in case of singular 
 intermediate every line of the plane passes by aa and 6/S into a pair of lines 
 apolar to a definite pair of points, i. e., the double points of the binary polarity 
 on the correspondent of f , 
 
 IV. Apolarity of coUineation and correlation.* 
 
 15. A coUineation aa and a contragredient correlation be may be apolar, i. e., 
 may satisfy the condition a6(ca) = 0. The meaning of this is easily seen. 
 Write them 
 
 aa = \a^a^ + X^a^a^ + \«3«3> 
 
 be = fi^b^G^ + fM^b^c^ + t^A^zy 
 
 and take Ac = Aa as reference triangle. The condition of apolarity is then 
 
 ab{ca)= \ti^aA + X^fi^a^ + X^fM^aA = 0. 
 
 That equation expresses that the triangles a^ and b^ are perspective. Therefore 
 since they are the correspondents through aa and bo of the reference triangle, 
 it follows that a coUineation and an apolar correlation transform any triangle 
 into a pair of perspective triangles. 
 
 Conversely if a^ and b. are perspective and aa is given, values ^Ji.^ may be 
 found such that (20) holds. Taking those values as the coefficients in 6c we 
 have a correlation apolar to aa. Therefore if a coUineation a/i transform the 
 points of a triangle a into a triad perspective to b^, there is a correlation apolar 
 to a/x which has a^ and b^ as corresponding pairs. In particular a coUineation 
 and correlation are apolar if they transform respectively the points and lines of 
 a triangle into the same triad of points. 
 
 The condition that the intermediate ah ayS of two collineations should be apo- 
 lar to a correlation cd is 
 
 {abc)a^d^ (abc) { a{^d) - fi{ad) } 
 
 = bc{^d)aa^ac{ad)b^=0. 
 
 Now bc{^d) = 0, and ac ( ad) = , are the conditions of apolarity of 6y9 and aa 
 with cd. Hence if two collineations are apolar to a correlation, so is their inter- 
 
 * F. AscHiKBi called snoh correspondences harmonic. Compare his article, " SuUe omografle 
 binarie e ternarie," Bend, del R. Isiituto Lonibardo, (2 J yoI. 24, p. 289. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 17 
 
 mediate. By an entirely analogous process it follows that if two correlations 
 are apolar to the same coUineation their intermediate is also. 
 
 The intermediate of a coUineation or correlation with itself is the reciprocal 
 or adjoined form. Hence if a coUineation and correlation are apolar the same 
 relation subsists when either or both are replaced by their adjoined forms. 
 
 If, for example, two coUineations transform the points of a triangle a into 
 perspective triads we have seen that there is a correlation apolar to both coUine- 
 ations which transforms the lines a^ into either of those triads. The preceding 
 paragraph then expresses that, if two coUineations transform a. into a pair of 
 perspective triangles, the intermediate gives a triangle perspective to both.* 
 
 16. A correlation apolar to the identical coUineation transforms any triangle 
 into a perspective one and is therefore a polarity. The condition that 6c be a 
 polarity is then 
 
 (21) 6c=0. 
 
 Suppose a coUineation aa. is apolar to a polarity 6c . We then have 
 
 a6(ac) = 0, 
 
 (22) 6^=0. 
 
 Let f be a fixed line of aa given by the root \ of the characteristic equation, 
 i. e., such that 
 
 (|a)a = X^ 
 
 Multiplying by | we get from (22) 
 
 (ac){(|6)a-(^a)6} = (ac)(^6)a - X(|c)6 = 0, 
 (|c)6-(|6)c = 0. 
 Combining these equations we obtain 
 
 (|c)(6a)a = X(|c)6. 
 
 For a fixed line of a coUineation, an apolar polarity gives a fixed 'point corre- 
 sponding to the same root of the characteristic equation. If the characteristic 
 equation has three distinct roots the fixed triangle is then self-conjugate or 
 polar with respect to any apolar polarity. 
 
 From (1 7) we have for the adjoined form of a coUineation s , 
 
 (23) ss = {is) + 2a^ - 2{s)<7 , 
 
 where o- is the inverse of s. Since a polarity is symmetrical, if it is apolar to 
 5, it is apolar to o-. In that case we have also seen that it is apolar to »s and 
 *Cf. MuTH, Math Annalen, Bd. 40, p. 98. 
 
18 SOME INVARIANTS AND CO VARIANTS OF TERNARY COLLINEATIONS. 
 
 identity. Therefore from (23) it follows that if a polarity is apolar to s it is 
 apolar to <7* . All collineations that are co variants of a are however expressible 
 linearly in terms of o-", o- and o^.* Consequently, a 'polarity apolar to a col- 
 lineation is apolar to all of its covariants. 
 
 If a collineation s transforms the points of a triangle a into a perspective 
 triad, there is a polarity which transforms the lines of a into the same triad. 
 That polarity is obviously apolar to s and to all of its covariants. Further 
 there is a polarity which leaves a fixed and is apolar to s. Therefore we have 
 Muth's theorem that if a collineation s transforms a triangle into a perspective 
 one, then all of the covariants of s transform it into triangles perspective to 
 each other and to the original triangle.f 
 
 17. For a correlation to be apolar to a collineation requires the identical 
 vanishing of a line and therefore subjects the coefficients of either to three linear 
 conditions. There is not then in general a correlation apolar to each of three 
 given collineations. The condition for such is the vanishing of the determinant 
 of the nine equations expressing the three apolarities. This invariant, the 
 explicit form of which does not concern us, has been called A.J It is a com- 
 binant symmetrical in the coefficients of the three collineations, and of the third 
 degree in each. We will now consider the peculiarities of a system of three 
 collineations for which A vanishes. 
 
 In the net 
 
 (24) S = Pl«l + P2«2+P3«5 
 
 there are a single infinity of singular collineations. The singular points lie on 
 a cubic that we may call C; the singular lines envelop a cubic that we may 
 call r . It is well known that the adjoined form of a singular collineation con- 
 sists of the product of singular line and singular point. And we have seen 
 that a correlation apolar tj a collineation is apolar to its adjunct. Therefore, 
 if a correlation is apolar to the collineations s^ , s^ and s^ , the singular lines and 
 points of (24) are correspondents in that correlation and consequently the cubics 
 r and C are reciprocal through it. 
 
 With a point of C is associated in two ways a line of F . In the first place 
 the point a appears as singular point in a definite collineation of (24) which has 
 a singular line /3. Secondly, it is transformed by those collineations into the 
 points of a definite line 7 . 
 
 To say that a correlation is apolar to each of three collineations amounts to 
 saying that those collineations operating on the inverse of that correlation give 
 polarities. Such a transformation of the collineations does not afifect the cubic 
 r which is therefore the Cayleyan of the three polarities. We saw above how- 
 
 * Clebsch, Vorlesungen iiber Geometric, Bd. 1, p. 991. 
 
 fMuTH, loo. cit., p. 97, 
 
 % ROSANES, Orelle, Bd. 95, p. 254. 
 
SOME INVARIANTS AND CO VARIANTS OF TERNARY COLLINEATIONS. 19 
 
 ever that the line yS passes by the inverse of the apolar correlation into the 
 point a . Therefore /S passes by the three polarities into points of 7 and conse- 
 quently j8 and 7 are corresponding lines of T . 
 
 Conversely, suppose with each point of O are associated a pair of correspond- 
 ing lines of V . Two coUineations^ of (24) whose common polar triangles are 
 not singular may be written 
 
 The lines yS^ are singular in the collineations whose singular points are a^. 
 The points ttj , a^, a^ pass by the net of collineations respectively into points of 
 
 Fm. 2. 
 
20 SOME INVARIANTS AND COVARIANTS OF TERNARY CX)LLINEATIONS. 
 
 7ij 72> 73 where 7. is incident to 6.. By supposition /Sj7j, P^'y^, and ^8373 are 
 three pairs of corresponding lines with respect to the curve. If we should 
 start with h^ we could construct a complete four-point, all of whose lines touch 
 the curve. It would contain those three pairs of lines as diagonal pairs and 
 therefore 7i, 73, 73 pass through a point (Fig. 2). The triad a^ then passes by 
 two of the coUineations into the triad h^ and by the third into one perspective 
 to 6,.. According to Art. 15, the three coUineations therefore have a common 
 apolar correlation. 
 
 The vanishing of the invariant A is the necessary and sufficient condition that 
 icith any point, of C should he associated (in the above way) a pair of correspond- 
 ing lines of V , and, dually y with any line of T should be associated a pair of 
 corresponding points of C. 
 
 The similarity of a set of coUineations for which A vanishes to a net of 
 polarities is noticeable. It is due to the fact that a set of polarities are apolar 
 to the identical coUineation, or, what amounts to the same thing, that a net of 
 coUineations with vanishing A may be transformed into a net of polarities in 
 such a way as to have either the initial or the resultant field invariant. 
 
 Part III. The Alternant. 
 
 I. Introduction. 
 
 16. In contrast with the symmetrical forms just considered are a series of 
 combinants that we will call alternants. The alternant of n coUineations s^ is 
 defined by the equation 
 
 [«i •••«„] = 
 
 where the determinant is supposed to be developed in the order of its columns, 
 i. e., in each term of the development the first letter is taken from the first row, 
 the second from the second row, etc. This determinant is readily seen to follow 
 the ordinary rules so far as its rows are concerned. If, for example, a linear 
 relation exists between the coUineations Sj, •••, s^, the alternant is zero. 
 Using the ordinary rule of signs, the determinant may be developed as the sum 
 of products by their minors of determinants of rth order in the first r rows. 
 The alternant cannot, however, in general be developed in terms of its columns. 
 Obviously, there will be a marked difference according as the order of the 
 alternant is odd or even. If the order n is even, for every term of the form 
 s^-" s.Sj^ will be a term — «j«. • • • s^ where the intervening letters in both are 
 
 h 
 
 «2 • 
 
 • «» 
 
 «1 
 
 «2 • 
 
 •• K 
 
 «1 
 
 «2 • 
 
 •' «„ 
 

 SOME INVAEIANT8 AND CO VARIANTS OF TERNAEY COLLINEATIONS. 21 
 
 the same. The linear invariant is therefore 
 
 r{(Si---ssJ-(s4«i •••«•)}. 
 
 The alternant of an even number of collineations is normal. 
 Again if n is even the alternant may be written in the form 
 
 (26) T,{Ps,Q - Qs,P) 
 
 where P and Q are products not containing s. . This is harmonic with s. * since 
 
 The alternant of an even number of collineations is harmonic with each of them. 
 
 Finally from (26) placing s^ to be identity we see that the alternant of an even 
 number of collineations vanishes when one of those collineations is identity. 
 
 In case of an odd alternant, since the members of a cyclic group are all of 
 the same sign, we have 
 
 (27) ([«i---«J) = '^(«x[«3---«„]). 
 
 If the alternant of an odd number of collineations is normal, each collineaMon is 
 harmonic vnth the alternant of the remaining n — 1 . 
 Write the alternant in the form 
 
 where 8^ is the minor of s,. in the first row of the alternant. If n is odd and 
 one of the collineations «j is identity, since the first minors are even, all those 
 containing identity vanish and the alternant takes the form 
 
 [«1 •••««] =«o(«2 •••««] = [«2---«n]- 
 
 The alternant of an odd number of collineations containing identity is equal to the 
 alternant of the remmning n — 1 . 
 
 II. The binary alternant. 
 
 17. The alternant [sjSg] of two collineations is normal and harmonic with 
 each of the collineations. A binary normal coUineation, or polarity, is apolar 
 to a coUineation s only when the fixed points of s form a pair in that polarity. 
 Therefore [SjSg] is the polarity determined by the common harmonic pair of 
 the fixed points of s^ and s^ .f 
 
 Consider in the next place the triple alternant 
 
 s. 
 
 [«1«2«3] 
 
 1 "2 "3 
 1 ^2 ^3 
 1 *2 ^3 
 
 IJ^^ 
 
 * See Art. 9, footnote. 
 
 t Study, " Benare Formen." 
 
22 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 It is normal when s^ is harmonic with [SgSg] . Therefore the vanishing of the 
 linear invariant of [SiSj^s] expresses the condition that the fixed points of 
 Sj, Sj, and s^ should lie in an involution. 
 The condition that a collineation 
 
 (28) ^ PiSl + p2^2 + Psh 
 
 be harmonic with [SiSj^sl ^^ 
 
 = Pl{(«l«2«3)-(«?«3«2)} +P2{(«2«3«l)-(«2«1»3)}+P3{(«3»l«2)-(«3«2«1)}- 
 
 Making use of the identity «^ = *s — A , this becomes 
 
 = { PlM + PiM + PsM} {(«1«2«3) - («3«2«l)} • 
 
 The alternant is therefore harmonic with all of the normal coUineations of the 
 pencil pi8^ 4- ^2*2 + Psh' ^^ fixed points of [s^s^s^'] give the coUineations of 
 (28) whose dyadics are squares. 
 
 It is readily seen that the collineation 
 
 s= [s,s,s,] - 'i {{s^s.s^) - (s^s.s,)} 
 
 is harmonic with s^,s^, and s^. When s is identity [s^Sg^s] ^^ identity and con- 
 versely. TTie condition that s^, s^, and s^ should be polarities is that the alter- 
 nant l^s^s^s^'j should coincide vnth the identical collineation. 
 
 When [SjSjSj] vanishes, the invariant ( «j«2 S3 ) — (SjSgS^) also vanishes and 
 hence s is zero. The three coUineations s^, s^, s^ have no definite apolar col- 
 lineation and therefore satisfy a linear identity. The vanishing of [SjSgSj] is 
 the condition for the three coUineations to satisfy a linear identity. 
 
 From any four linearly independent binary coUineations any other may be 
 linearly derived. In particular to such a set of four belongs the identical 
 collineation. Since an alternant of even order containing identity is zero, it 
 follows that the binary alternant of fourth order vanishes identically. There- 
 fore with the alternant (s^s^^s^) just considered the discussion in the binary 
 domain comes to an end. 
 
 III. The alternant of two ternary collineoiions. 
 18. We shall usually write the coUineations in the form 
 
 Sj = aa , 
 
 The alternant is then 
 
 [^1*2] = *i*2~*2*i = {ch)aP — {^a)ha. 
 
 The CO variants of a collineation « are linear functions of s^, s, and «*, where 
 «j is the identical collineation. Since each of these is commutative with « it 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 23 
 
 follows that the alternant of a collineation and any of its covariants vanishes 
 identically. 
 
 The invariant relations of the alternant and covariants of s^ and s^ may be 
 summed up in two general theorems. 
 
 (i). The alternant is apolar to all the covariants of s^or s^. 
 
 For let cy be any covariant of s^. The condition of apolarity with the 
 
 alternant is 
 
 = (a6)(a7)(/3c) - {Pa){hri){ac) 
 
 = 6)S' [(07)00 — (ac)7a] , 
 
 where the dot is used to represent the process of forming the double product, 
 or bilinear invariant. Since the expression in brackets is the alternant of aa. 
 and a covariant 07, the function vanishes as was required. 
 
 (ii). Hie alternant is apolar to the intermediate of one of the coUineations and 
 any covariant of the other. 
 
 For let 07 again be a covariant of aa . The intermediate of this with b^ is 
 
 60)87. 
 
 The condition of apolarity with the alternant is 
 
 = {ab'){abc){0'^y) - {^a){b'hG){a^y), 
 
 where b'^ is a new symbol for 6yS . Interchanging b^ and 6'y8' and adding, 
 the last expression becomes 
 
 ^ {{fi'/3y)(b'ba- ca) - {b'bc){fi'^a ya)} = l^jSb'b ■ [yaca-ayac] . 
 The expression in brackets expands into 
 
 [aa)yG — (^ac)ya — (aa)yc + (^ay)ac = (ay)ac — [aG)ya, 
 which is zero since it is the alternant of aa and a covariant. 
 
 19. Since all covariants of s are expressible linearly in terms of 5g, s,s^, the 
 alternant is found to be apolar to the eight coUineations 
 
 (29) 
 
 ''OJ <^1> «■?? <^2> O'L «lSj> «1«2^ «1«2> 
 
 where, as in Art. 9, s^ s^ is the intermediate of s^ and s^ and o- is the same connex 
 as s but considered reciprocally. If the eight coUineations are linearly inde- 
 pendent, the eight apolarity conditions are sufficient uniquely to determine the 
 alternant. It is our purpose in the next place to see whether or when such is 
 the case. 
 
 By the formula (17) we have 
 
 h4 = {^l) + <^>l-^<^>l-(^l)<rl-(sl)<rl. 
 
24 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 Since the alternant [s^Sj] is apolar to <r^, tr^ and al, we see that the condition 
 of apolarity of [SiSj] and s^sl is the vanishing of the linear invariant of the 
 collineation 
 
 Hence by direct expansion of this last named invariant we obtain 
 
 ,,^, [«I«J •«!«i = («1«2«1«2) + («1«2«D - («2«?«2) - («2«lS2«?) 
 
 {6{J) 
 
 = («1«2«1«2)-(«2«1«2«?)> 
 
 since by Art. 9 (SiSgSj) and {s^sls]) are equal, both passing by a cyclic permu- 
 tation into (s'Sj). 
 Again we have 
 
 Since [ s' *i ] is apolar to o-^ , c^ and a^ , by the same argument as before, the 
 apolarity condition of [ Sj «i ] and si s^ is found to be the linear invariant of the 
 collineation 
 
 {sls,-s,sl)(sls, + s,sl). 
 
 Expanding and making use of the cyclic permutation we then obtain 
 
 .„,, [«2«l] • M2 = («2«1«2) + («2«l«2«l) - («1«2«1«2) " i^A^l) 
 
 [61) 
 
 = («l«2«f«2)-(«2«l«2«l)- 
 
 In like manner, making use of the identity 
 
 M2 = («1«2) + °"l<^2 + <^2°"l -(«l)°"2 - («2)<^U 
 
 we obtain the apolarity condition of [ s^ Sg ] and s^ s.^ as the linear invariant of 
 
 («?«2-«2«l)(«l«2+«2«l) 
 
 under the form 
 
 (32) [S^SI] •M2=(«l«2«l«2)-(«2«l«2«?)- 
 
 Since [s^Sj] is a normal collineation the adjoined form is given by (17) as 
 
 [«i«2] [«i«2] = ( [«i«2] [«i«2] ) + ^[o-xo-j'. 
 The discriminant of [SjSj] is then 
 
 ^l = K«l«2] • [«1«2] [«1«2] = *[«1«2] • [<^l'^2]' = J( [«1»2]') 
 = i {(«1«2«1«2«1«2) - («l«2«l«2«l) - («1«2«N2) " ('2«f «2«1»2) 
 
 (33) 
 
 - (S2«,«2»i*j«i) + («2«i«2«l«2) + («2»l«2«l) + ( «I «2 «1 «a«l )} 
 = («l«2«?«2)-(«2«l«2«f). 
 
SOME INVAEIANT8 AND CO VARIANTS OP TERNARY COLLINEATIONS. 25 
 
 in which the notation A^^ is used to show that it is the discriminant of an alter- 
 nant involving Sj and s^ each to the first degree. 
 
 A comparison of (30), (31), and (32) with (33) shows that the various invari- 
 ants there considered are equal to each other and to the discriminant of [s^Sj]. 
 
 Now suppose that A^^ does not vanish and that there exists a linear relation 
 of the form 
 
 (34) Po^o + ^i<^i + P20"? + P3<^2 + pA + P5M2 + P6^2 + P7M2 + Ps^l = 0- 
 
 Since according to (29), [SjSg] is apolar to the first eight collineations in that 
 sequence but by (30) and (33) is not apolar to the last, it follows that Pg is zero. 
 Likewise, operating in turn with [SjSg], [«2*i] and [Si^z]? ^® ^^^ *^** 
 Py = /Og = /jg = . Hence the relation (34) must be of the form 
 
 (35) Po^o ■^Pi'^i + P2<^1 + PsS + Py2 = 
 Replacing a^ by s^ and forming the alternant with s, we obtain 
 
 (36) Pd^.s,] +/>,[«,«!] =0. 
 Forming the bilinear invariant with s[ $1 , this gives 
 
 Hence p^= 0. Likewise on operating with si s^, it is seen that p^ = 0. Simi- 
 larly p, = p^ =/>„= . 
 
 Hence if the discriminant Aj^ is different from zero no linear relation of the 
 type (34) can exist. If however the discriminant is zero, since the nine colline- 
 ations are apolar to [ «i ^2 ] > ^^^J must satisfy a linear relation. Therefore, the 
 vanishing of the discriminant of the alternant is the necessary and sufficient con- 
 dition for the existence of a linear relation of the type (34). 
 
 Since in the usual case the discriminant of the alternant is not zero, it follows 
 that in general the apolarity conditions of (29) are independent and give an in- 
 variant determination of the alternant. When the discriminant is zero, all the 
 collineations of the net 
 
 (37) ^[^1^2] +\[«1«2]+\[*1«2] +\[«^2] 
 
 satisfy those conditions and the determination is not unique. 
 
 20. When the discriminant vanishes there is always a collineation of (37) 
 that vanishes identically, i. e., the four alternants satisfy a linear relation. For 
 since all the collineations of (37) are apolar to all those in (34) it follows that 
 the two sets must contain four linear relations. If one of these belongs to (37) 
 the point at issue is settled. If not there must be four equations of the type 
 (34). Either one of those is of the form (35) and the collineation in question 
 is (36), or it is possible to solve for one of the intermediates and so obtain 
 
26 SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 
 
 an equation 
 
 Developing s^s^ by (17), inverting, and forming the alternant with s^, gives 
 
 [5j«J -(«l)[«l«2] =\[«1«2] +^4[Sl«2'] 
 
 which is the relation desired. 
 
 Suppose conversely that the four alternants satisfy a linear relation 
 
 In this equation there must be at least one coefficient, for instance X^ , that is 
 different from zero. Operating on the equation with s^sl we see that A^j must 
 then vanish. Therefore, the vanishing of the discriminant is the necessary and 
 sufficient condition for a linear relation between the four alternants. 
 
 21. The symmetry resulting when Aj^ vanishes suggests that it is an invar- 
 iant common to the four alternants. In order to prove that such is the case, 
 take in the first instance the alternant [ «f Sj ] . According to (33), the discrim- 
 inant has the form 
 
 ^21 = («lVt«2)--(«2«l«2«J)- 
 
 From the characteristic equation for s^ we have 
 
 4 = \ + \^i + \4' 
 
 Substituting this value for s* we have 
 
 (38)^ . A,,=.\{(sls,s,sl)-{s,slsls,)]^-\A,,. 
 
 Making use of 
 
 «2 = /*0 + /*l«2 + A*2«2 
 
 and following out the same argument we obtain the discriminants of [^i^j] 
 
 and [SjSj] ^° *^^ form 
 
 (39) |^. = -''.An. 
 
 If \j is zero the characteristic equation for sj is 
 
 {sir = \ + \sl. 
 
 The equation is quadratic and hence the collineation is a perspectivity. There- 
 fore \ and fij are respectively the invariants whose vanishing expresses that sj 
 and si are perspectivities. From (38) and (39) we see then that tlie altemarU 
 of a perspectivity and any collineaiion is singular. 
 
 If Xj and ^l^ are not zero Ajj is a combinant of the two systems 
 \Sq 4- \j8j + X.2*i ^^^ f^o^o + A''i^2 + ^2*2- -^^ ""^^^^ ^^®" express a property of 
 the fixed triangles of those systems. What that property is we shall see later. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 27 
 
 22. The alternant [s^Sj] ^^ ^ combinant of the net 
 
 (40) \s^ + fis^+vs^. 
 
 In forming it any two independent collineations may then be chosen. Let s be 
 a singular collineation, x its singular point, and | its singular line. The alter- 
 nant may be written 
 
 where s^ is some other collineation of the net. Since sx is zero the correspondent 
 of X with respect to the alternant is 
 
 Now s transforms every point (and in particular Sj x) into a point on f . There- 
 fore the alternant transforms the singular point of any collineation of (40) into 
 a point of the associated singular line. For varying X, the singular points and 
 lines obtained are the fixed points and associated fixed lines in the collineation 
 /xSj -|- i/Sg. Therefore the fixed triangles of all the collineations of the pencil 
 fis^ + vs^ are JPasch triangles of the alternant. 
 
 The significance of the apolarity relations satisfied by the alternant is here 
 suggested. In fact two collineations (one in points, the other in lines) that send 
 a triangle into a pair of incident triangle are apolar. Now if s is any colline- 
 ation in (40) the triangle that s leaves fixed is sent by the alternant into an 
 inscribed triangle. Therefore the alternant is apolar to s and to all of its 
 covariants. 
 
 Two collineations have in general a common pair of polar triangles. In 
 terms of these they may be written 
 
 s^ = /^i^iOCj + fi^\a^ + fi^h^s- 
 Their adjoined forms and intermediate are respectively 
 «i«i = 2 { W^^a^ + W^i^i + W^iCii } > 
 «2«2 = 2 { /^i^a/^a^a + ^2^3A«i + /*3/*i^2«2 } f 
 
 «1«2 = {\f^2 + \/*l)^3«3 + (\/*3 + \f^2)^l^l + {\f^l + \f^3)^2^2' 
 
 Hence s^^s^, s^s^, and s^s^ have a common pair of polar triangles, i. e., the com- 
 mon pair of Sj and s^ considered contragrediently. Since [s^Sg] is apolar to 
 each of those collineations, according to Art. 10, it transforms the triad a^ into 
 one incident to /3^. Taking any two collineations and the associated interme- 
 diate belonging to (29), we obtain a pair of polar triangles such that the points 
 of the first pass by the alternant into a triad incident to the second. These 
 relations are therefore the geometric equivalent of the eight apolarity conditions. 
 
28 SOME INVARIANTS AND COVAEIANTS OP TERNARY <X)LLINEATIONS. 
 
 23. In Arts. 19-21 the entire theory seemed to hinge on the vanishing or 
 non-vanishing of the discriminant of the alternant. It is our purpose in the 
 next place to consider the geometrical interpretation of that invariant. 
 
 For that purpose suppose a polarity (? to be apolar to s^ and 8^ . Designating 
 the collineations respectively by aa and bjS, the conditions required are 
 
 (41) ^(ac) = 6^(^c) = 0. 
 
 There are six equations in all, three of them linear in the coefficients of aa, 
 three linear in the coefficients of b^ , and all linear in the coefficients of c^. 
 Therefore if we eliminate the coefficients of c^ from these equations we get an 
 invariant of the third degree in the coefficients of aa and b^ whose vanishing 
 is the necessary and sufficient condition for the equations (41). 
 
 Now we have seen that a polarity apolar to a collineation is apolar to all of 
 its covariants, and that a correlation apolar to two collineations is apolar to 
 their intermediate. Therefore c^ is apolar to the following nine collineations, 
 
 (42) a^, <7j, 0-2, 0-2, 0-2, SjSj, sfsj, s^s], s^sl. 
 
 Since the nine collineations are apolar to the same polarity they must satisfy 
 a linear relation, in fact, must satisfy three linear relations. Therefore accord- 
 ing to Art. 19, the discriminant of the alternant is zero. But the discriminant 
 is of the third degree in the coefficients of aa and 6/3. Therefore, the vanish- 
 ing of the discriminant of the alternant is the necessary and sufficient conditions 
 for two collineations to have a common apolar polarity. 
 
 If a polarity is apolar to a collineation there are two cases to be considered 
 according as the polarity is singular or is not. 
 
 If the polarity is singular it either consists of the square of a point or its 
 adjoined form consists of the square of a line. In both cases if the polarity is 
 apolar to a collineation its fixed triangle contains the double element. Hence 
 if a singular polarity is apolar to each of two collineations their fixed triangles 
 have an element in common. In that case all the collineations of (42) have a 
 fixed point or line in common. 
 
 In general, however, if a polarity is apolar to a collineation, the fixed triangle 
 (or, a fixed triangle) of the collineation is self-conjugate with respect to the 
 polarity. If two triangles are self-conjugate with respect to the same polarity 
 they lie on a conic. And, conversely, if two fixed triangles lie on a conic, they 
 are self-conjugate with respect to a polarity, which is consequntly apolar to their 
 associated collineations. Therefore we see that the vanishing of the discriminant 
 of the alternant is the necessary and sufficient condition for any two collineations 
 formed linearly from those of (42) to have fixed triangles lying on a conic. 
 
 24. As we have seen the conditions for a polarity c^ apolar to aa and 6/8 are 
 
 (ac)ac = (/Sc)6c = 0. 
 
SOME INVARIANTS AND COVARIANTS OF TERNARY COLLINEATIONS. 29 
 
 Multiplying these equations by 6)8 and aa and writing in the variables to avoid 
 confusion we get 
 
 (ac){(/3c)(a|)-(^a)(c|)}(6^)^0, 
 
 (/9c) { («c)(6|) - («6)(c|) } (ar;) ^ 0. 
 
 Placing I and 77 equal and subtracting, these equations give 
 
 (43) {(a6)(^c)(a|)-(^a)(ac)(6|)}(c|) = 0. 
 
 Comparing this with the general form 
 
 we see that in the present case the product of the alternant with c^ is a correla- 
 tion having no particular coincidence conic, i. e., a nullsystem or line. 
 
 Suppose for the moment we designate the alternant by dh . The equation 
 
 (43) is then 
 
 (44) (c^l)(Sc)(c|) = 0, 
 
 where | is any line whatever. Let 7; be a fixed line of the alternant. We 
 
 then have 
 
 {d7))h = \r}, 
 
 which substituted in the preceding equation gives 
 
 (d7;)(8c)(c7/) = X(77c)2 = 0. 
 
 Hence the fixed lines of the alternant touch the conic of c^. 
 
 If again we follow out the argument of this article with 7^, the adjoined 
 form of c^, we obtain as correlative to (44) 
 
 (45) (7rf)(cca)(a;7) = 
 
 where x is any point whatever. Taking cc as a fixed point of the alternant, it 
 follows as before that the fixed lines of the alternant lie on 7^. 
 
 Now, we have found that when c^ is apolar to s, and s^, it is apolar to a set 
 of nine co variants. Therefore, all the alternants that can he formed of those 
 Govariants are singular, all their fixed lines are tangent to c^ and aU their fixed 
 points lie on 7^ . 
 
 It was shown by Study that the fixed points of the alternant of two binary 
 coUineations consist of the common harmonic pair of the fixed points of those 
 colli n cations. The fixed triangles of two ternary coUineations are not in gen- 
 eral polar with respect to the same conic. If such however is the case, we 
 have just seen that the fixed lines of the alternant are tangent to and the fixed 
 points lie on that conic. 
 
30 SOME INVARIANTS AND <X>VARIANTS OF TERNARY COLLINEATIONS. 
 
 VITA. 
 
 Henry Bayard Phillips was born at Yadkin College, N. C, September 27, 
 1881. He received his preliminary and part of his collegiate education at that 
 place. In 1898 he entered Erskine College, Due West, S. C, and graduated, 
 B. S., in June, 1900. After teaching for one year at Linwood, N. C, he 
 entered the Johns Hopkins University, attending courses in mathematics, 
 physics, and philosophy. There he was successively scholar and fellow and 
 graduated, Ph.D., in June, 1905. 
 
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