July 29, 189S.J 



SCIENCE. 



12c 



— 34(j3V* + 5V') — 292(5V +/r') 

 + p'q^ + p'?V 



+ 530(/5/ +pqV) + bQ{p'qr'^ + pq^r) 



— 4(5=' + _pV') - 12{p'qS^ + pYr) 



in which^, q, r in terms of the homogeneous 

 variables are 

 p = Z^ +Z.^ + Z^,q== Z^Z^ + Z^Z, + Z,Z^, 



It is to be noted that P itself is not ex- 

 pressible in terms of ^, q, r, but 



P = z,z,z, {z, - z;) iz, - z,) {z, - Z3). 



As a remarkable coincidence it was 

 found that the three invariants of the com- 

 plete form-system of the binary quintic 

 form, when written in terms of a fundamen- 

 tal system of two cross- ratios of the roots, 

 are precisely these forms A, P^, C, when 

 similarly expressed in terms of the cross- 

 ratios. It is shown that A, P'' and C are 

 the complete form- system of our quadratic 

 group, Gj2„, by a series of theorems of which 

 the most important are the following : (1) 

 An invariant under a quadratic operator 

 must be a fraction whose numerator and 

 denominator throw off a common factor in 

 the Za. (2) The numerator and denom- 

 inator of an invariant fraction must be 

 absolute or relative invariants under the 

 linear subgroup and hence rational integral 

 functions of the known invariants in its 

 complete form-system. (3) There can be 

 no invariant fraction whose numerator and 

 denominator are of odd degree or of un- 

 equal degree. (4) The most general in- 

 variant form suitable for numerator or 

 denominator of an invariant fraction under 

 Gj2o is of degree 6?i and throws off the 

 factor r'" (r = Z^Z^Z^) under the quadratic 

 generator; Z\:Z\:Z, = Z,Z,: Z,Z,: Z,Z,. (5) 

 The most general invariant form under 

 Gi2o(a) is of the form J?6„=P^'^-Bc(„- 2^),where 

 /i = or a positive integer and R^i^,, _ 2^) con- 

 tains no factor of P; and (6) has at each of 

 the four critical points a multiple point of 

 order 2{n + ii). (6) If a and /5 are two 



such ternary forms having the binary forms 

 a and b, of degi-ee /I and /i, respectively, as 

 tangential quantics at one of the critical 

 points, then a,? has ab as its tangential 

 quantic at the same multiple point, and 

 a + /3 has a, b or a -i- b according as ^ is less 

 than, greater than or equal to ti. (7) Na 

 ternary form can have a binary tangential 

 quantic at any critical point of odd degree 

 in either or both of the cubic invariants be- 

 longing to the dihedron subgroup which 

 leaves the critical point fixed. (8) Two 

 reduced ternary forms of the same degree 

 which have the same tangential quantic at 

 any critical point can differ only in such 

 terms as involve P^ as a factor. 



By means of these theorems it is then 

 shown, by a process of successive reduction, 

 that the most general invariant form under 

 G121, is expressible as a rational integral 

 function of ^, P', C, and thus a system of 

 fundamental forms is established in terms 

 of which all invariant fractions under the 

 quadratic group can be expressed. The 

 above forms are absolute invariants. The 

 only relative invariant fractions are those 

 expressible in terms of A, P and C, which 

 are invariant except for change of sign. 



THE CONFERENCE OF SCIENCE TEACHERS 

 IN THE TRANS-MISSISSIPPI EDUCATIONAL 

 CONVENTION. 



A FEW months ago the undersigned was 

 requested by the program committee to 

 arrange a series of conferences of science 

 teachers in connection with the Trans-Miss- 

 issippi Educational Convention, to be held 

 in Omaha, June 2Sth, 29th and 30th. As 

 a result there were held seven conferences, 

 namely, in Chemistry, Physics, Astronomy, 

 Botany, Zoology, Geography and Geology, 

 occupying the afternoon sessions of the 29th 

 and 30th. The following abstracts of the 

 principal papers will give some idea of thee e 

 meetings. The attendance was not large, 



