The November Meeting in Raleigh

s of the papers presented follow below. Abstracts whose numbers are followed by the letter "t" were presented by title. Paper number 51 was read by Professor Rutledge. Professor J. W. Ellis was introduced by Dr. Goodner, Mr. McAuley by Professor Jones, Dr. Jaeger by Professor Cairns, Dr. Gordon and Mr. Capel by Professor Wallace, Mr. Pellicciaro and Dr. Shields by Professor Whyburn. ALGEBRA AND THEORY OF NUMBERS 37. A. T. Brauer: On the distribution of Jacobian symbols. Let p be a prime and ^ = 0 or ±1 (i"=l, 2). N. Aladow (Mat. Sbornik vol. 18 (1896) pp. 61-75) determined the number of integers a in a complete system of residues (mod p) for which the Legendre symbols (a/p) =* e\ and (a+1/p) = e% for each given ei and e%. Other proofs for these results were given by E. Jacobsthal {Anwendung einer Formel aus der Theorie der quadratischen Reste, Dissertation, Berlin (w. Fr. Kâstner, Göttingen, 1906)) and by O. Perron (Math. Zeit. vol. 56 (1952) pp. 122-130). In this paper the corresponding problem for Jacobian symbols for odd squarefree moduli is solved. These results will be used in another paper for the construction of Hadamard determinants of certain orders. (Received November 28, 1952.) 38£. Leonard Carlitz: A note on modular invariants. Consider a system of forms/i, • • • , fr, with coefficients €E.GF(q), which are subjected to the transformation of the full linear group over GF(q) ; thus the number of classes is some finite number k. Let the invariants of the system take on values in an arbitrary field $. We show that if $ contains at least k distinct numbers, then there exists an invariant / such that every invariant can be exhibited as a polynomial in J of degree ̂ £ — 1 and with coefficients in *. More generally if $=*GF(qi) and q'f <kSqv then there exist s invariants Ji, • • • , J» such that every invariant is a polynomial in the / ' s . Moreover the number 5 cannot be diminished. (Received October 6, 1952.) 39t. Leonard Carlitz: A reciprocity formula f or weighted quadratic partitions. This note is concerned with a reciprocity theorem for the sum S(at X, Q) « Lo(5)-a«(2Mi+ • • * +2Xr£r), where a, X», ZiÇzGF(q) and Q is a quadratic form with coefficients in GF(q). Some extensions and a rational analogue are also discussed. (Received October 6, 1952.) 40. Leonard Carlitz: Invariant theory of systems of equations in a