is the probability that the error x = a m of the measurement m will lie between a' and x"; here a is the true value. Under the G. L. there is, indeed, "no most probable value" for the unknown, the probability of each of the infinite number of possible values being zero. Definition: The probability of a function F of the measurements will be said to be greater than that of another function f if the probability that F will differ from the true value a by less than a is greater than the probability that ƒ will differ from a by less than a, for all positive values of a less than some a'. Then, under the G. L., the measurements having the same "measure of precision" h, the probability of the root-meansquare, √(m2 + m2), of two measurements m1, må is greater than that of the arithmetic mean M = (m1+m2) if ha > 2. But in the case of three measurements the probability of the arithmetic mean is the greater. = Let M be the arithmetic mean of any number of measurements, with the same h. There exist values of the constant b, a little less than unity, such that the probability of bM is greater than that of M. But if b > 1, the probability of bM is less than that of M. Under the G. L., the probability of the geometric mean, and of the median and of M+ c,-this c being a constant, not zero, is less than the probability of M. ... ... • + Pn = = If the measures of precision are h1, h2, ..., hn, respectively, the probability of the weighted mean W P11 + P2m2 + +Pnmn (where p1 + P2+ 1) is greatest when the weights are given their usual values p1 = h/2h; but values of b exist making the probability of bW greater than that of W. 5. In his first paper Dr. Blumberg sketches the main results in his dissertation, entitled "Ueber algebraische Eigenschaften von linearen homogenen Differentialausdrücken" (Göttingen, 1912). The dissertation deals with algebraic properties (such as those connected with reducibility, irreducibility, etc.) of expressions of the form where the functions a。(x), a1(x), . . . belong to a fixed domain of rationality, in the sense of the Picard-Vessiot group theory of linear homogeneous differential equations. The chief point to be emphasized is that the proofs make no use of integrals, but are given with the aid of a very small number of properties of linear homogeneous expressions, so that the results are of a much more general character than those hitherto obtained. 6. A rational plane curve of order n, an R", possesses 2n2 tangents through every point of the plane; also, there are 2n 2 osculant (n-1)-ics of the R" through every point of the plane. Dr. Rowe's paper consists in proving that any projective property imposed upon the parameters of the 2n - 2 osculant (n - 1)-ics of the R" through a point holds for the tangents through the point as a pencil of lines. 7. In a recent paper (Proceedings of the London Mathematical Society, November, 1911) Burnside made a determination of all finite collineation groups in n variables with rational coefficients which contain the symmetric group on those variables. In addition to the already known groups, those of order (n+1)! (for any n) and 27.34.5 (for n = 6), he found two additional primitive groups, of order 29.34.5.7 (for n=7) and 213.35.52.7 (for n = 8). Dr. Mitchell solves a more general problem, i. e., the determination of all primitive groups in n(>4) variables which contain homologies, the results for n≤4 being already known. A transformation in the symmetric group which interchanges two of the variables and leaves the rest unaltered is evidently an homology of period 2. In addition to the groups mentioned, there are two others, of order 26.34.5 (for n = 5) and 28.36.5-7 (for n = 6). Neither of these can be represented with rational coefficients. The former is isomorphic with the known simple group of that order, and the latter contains a self-conjugate subgroup of index 2, which is (1, 1) isomorphic with a known simple group. 8. Professor Ranum shows that in the Lobachevskian plane there are nine polygons whose trigonometry is equivalent to that of the triangle and three special polygons whose trigonometry is equivalent to that of the right triangle. Among the former is the rectangular hexagon (whose angles are all right angles). Among the latter is the rectangular pentagon, which bears somewhat the same relation to the right triangle in the Lobachevskian plane that Gauss's "Pentagramma Mirificum" bears to the right triangle in the Riemannian plane. The hyperbolic cosine of any side of the rectangular pentagon is equal to the product of the hyperbolic sines of the two opposite sides and to the product of the hyperbolic cotangents of the two adjacent sides. 10. A matrix (pq) is unlimited if the transformation Eqpqq does not transform every system of variables {x} of finite norm into a system of finite norm. Linear differential and differential-integral equations, and other types of linear equations, correspond directly to linear equations in infinitely many unknowns for which the matrix of the coefficients is unlimited. In Mrs. Pell's paper conditions are given for the existence of solutions of systems of homogeneous and nonhomogeneous linear equations with an unlimited matrix of coefficients. 11. In this paper Dr. Robinson obtains a complete set of invariants of two tetrahedra, the one taken in points, the other in planes. The tetrahedra are written (ag)(bg) (c§)(d§) = 0 and (ax) (ẞx)(yx)(dx) = 0. An invariant is defined as a rational integral function of the coefficients homogeneous in the Greek and Roman letters and unaltered by any permutation of the letters and the subscripts. The group of permutations is of order 576. A rather full discussion is given of this group and the results obtained are compared with those obtained by Feder in a paper in the Mathematische Annalen, volume 47. (1) 12. The Klein quartic (Mathematische Annalen, 1879) and the Ciani quartic (Palermo Rendiconti, 1899) (for a special value of λ) are both invariant under 21 harmonic homologies. Ciani states that (2) must therefore be trans formable into (1) by some opportune transformation. In Professor Sharpe's paper the requisite transformation is determined and is applied to prove the following theorems. There are 63 conics that pass through one vertex of each of the 8 inflectional triangles, 21 of which pass through each vertex. There are 168 conics that pass through 2 vertices of each of 4 of the 8 inflectional triangles, 56 of which pass through each vertex. There are 28 conics that pass through the 6 vertices of each pair of inflectional triangles and through the points of contact of an associated bitangent. 13. Noether (Mathematische Annalen, 1889), reduced the double planes that can be rationally mapped on simple planes to three types according as the curve of branch points is (1) a Can with a (2n-2)-fold point, (2) a non-singular quartic, (3) a sextic having 3 branches touching the same line at a common point. A complete analytical treatment of the second type was given by De Paoli (Atti Lincei, 1878), and of the first type by Boyd (American Journal, 1912). In Professor Sharpe's second paper precise analytical expressions are determined for the remaining type of (2, 1) correspondence by means of the Grassmann and Wiman (Mathematische Annalen, 1895) depictions of a cubic surface on a simple and double plane. 14. Professor Sharpe and Dr. Morgan consider a quartic surface whose section by any plane through a line AB is the line and a cubic through A and B. If the line joining A to any point P on the surface meets the surface in Q and if BQ meets the surface in R, the transformation used converts P into R. In the most general case the conditions for periodicity give 17 relations amongst the 26 coefficients of the equation of the surface. When certain terms are absent from the equation the conditions for periodicity are simple. 16. In his second paper Dr. Blumberg deals with sets of postulates for rational, real, and complex numbers. These sets grew out of conversations with Professor Zermelo and it is the intention of the latter and Dr. Blumberg to publish their results in detail in the near future. Starting with a set (F) of postulates defining the most general field (Körper), where the commutative law of multiplication need not hold,-such a field will be henceforth simply called a "non-commutative field"we obtain by the addition of a postulate R a set of postulates defining the most general non-commutative field that contains as a sub-field the field of rational numbers. By the further addition of one postulate C, we obtain a set of postulates defining the most general non-commutative field that contains as a sub-field the field of real numbers. Finally, the set of postulates (F), R, C and a new postulate I define the most general non-commutative field containing as a sub-field the field of complex numbers. To obtain sets of postulates defining the rational, real, and complex numbers a postulate P is added respectively to the second, third, and fourth set of postulates mentioned above, as shown below: For the rational numbers: (F) R, P. There are no postulates of order. In the proofs the concept of finite number is not presupposed. 17. Professor Veblen's paper gave a proof of the theorem that an (n − 1)-dimensional polyhedron decomposes an n-dimensional space into two regions. The proof is of an essentially combinatorial character and involves few geometrical ideas beyond the principle that an n-dimensional convex region is decomposed into two n-dimensional convex regions by an (n - 1)-space containing one of its points. 18. Triad systems in thirteen letters were given by Kirkman, Reiss, Netto, and De Vries. But the complete determination of these systems, of which there are only two distinct types, was first effected by De Pasquale* and Brunel,† both of whom based their examination on the configurations of S. Kantor. Professor Cole determines the possible systems by direct construction without the use of any special apparatus. In the discussion of the paper, Professor Veblen pointed out that the two systems might be obtained as an application of finite geometry. 19. In a triple system every pair of elements (or dyad) * "Sui sistemi ternari di 13 elementi," Rendiconti R. Istituto Lombardo, ser. 2, vol. 32 (1899). "Sur les deux systèmes de triades de treize éléments," Journal de Math., ser. 5, vol. 7 (1901). |