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Two basic assumptions are employed in this treatment of the statistics of stream lengths: (1) All topologically distinct networks with a given number of sources are equally likely (after Shreve). (2) Lengths of interior links in a given network are independent random variables drawn from the same population. The mathematical development leads to an approximate expression for L• that contains no adjustable parameters and that depends only on the stream numbers and the mean link length. This expression gives somewhat better agreement with data on actual stream systems than does Horton's law of stream lengths; other advantages over Horton's law are cited. Quantitatively, our procedure appears to account for about 65% of the variance in mean stream length data for third-and fourth-order basins. If exact values of the link numbers are introduced into the calculation, the unexplained variance is reduced to about 15%. s. s•AgTThe Horton analysis m•y be regarded as an attempt to give a quantitative description of drainage basin composition in terms of five parameters' Rs, RL, RA, •, and •, or some equivalent set [Horton, 1945, p. 295]. Once these five numbers are known, approximate values for other geomorphic parameters, such as total stream length, total basin area, drainage density, and stream frequency, can easily be obtained. Leopold and Miller [1956] have shown how the analysis can be extended to predict width, depth, discharge, sediment load, and other hydraulic variables. Although the Horton and Strahler ordering techniques have been much used by hydrologists and geomorphologists, they have also attracted a certain amount of criticism. In

SUMMARYThe Horton method of analyzing drainage network composition is reviewed, with the conclusion that it has not been very effective either in improving understanding or in developing useful methods of characterizing drainage basins. New methods which are based on the link rather than the Horton (or Strahler) stream are described. A number of detailed examples of the application of these new methods to the topologic and geometric properties of networks are provided. The results are compared with the predictions of the random model.Data used in the analysis were obtained from 1 :24,OOO U.S.G.S. topographic maps of eastern Kentucky. Thirty drainage basins were selected and their channel networks were outlined first by the contour-crenulation (CC) method and then by another, more objective, method (SC) in which stream sources were identified by a quantitative slope criterion. The CC and SC samples comprise about 8,700 and 1,700 links, respectively.The three most important results of the analysis are: (1) the channel networks are slightly but significantly more elongated than predicted by the random model, (2) there are fewer second magnitude links than predicted, and (3) the length distribution for interior links depends upon the kind of link (interior or exterior) joined downstream. These features are found in both CC and SC networks.

The Néel theory of ferrimagnetism is a molecular-field theory of magnetic ordering in systems which contain nonequivalent substructures of magnetic ions. The two principal results of the theory are (a) the curve of reciprocal susceptibility vs temperature above the Curie temperature is an hyperbola, and (b) the curve of spontaneous magnetization vs temperature below the Curie temperature may have any one of a number of unusual forms. The exact form depends on the distribution of magnetic ions between the substructures and on the relative values of the molecular field coefficients. The agreement between the predictions of the Néel theory and some experimental observations on ferrites is discussed briefly.