Decembeb 14, 1906.] 



SCIENCE. 



769 



1906, Vice-president Hayford in the chair. 



R. A. Harris presented a paper entitled 

 * Elementary Notions relating to Integrals in 

 General Function Theory' which treated of 

 integration around a pole of a function, and 

 of Cauchy's theorem. 



It was pointed out that any element Zdz, 

 where Z = 1/z — c and dz is an elementary 

 arc of a small circle about c, is simply a 

 vector in the ^/-direction, 2Tr/n in length, n 

 denoting the number of parts into which the 

 circumference is divided. Consequently, 



'27ri 



/ 



Zdz: 



-Xn^= 27ri. 



This established, it readily follows that 

 fzdz = 2mf{c) where Z= ^, 



f having no zero in the neighborhood of c. 



Cauchy's theorem was demonstrated for cir- 

 cular paths, for paths enclosing slender areas, 

 and for any path. 



In the first case the truth is obvious if Z 

 denote a power-series. For, the argument of 

 Z^dz varies v + 1 times as fast as does the 

 argument of dz^ and as dz takes all directions 

 uniformly on account of the path being cir- 

 cular, so does Z^dz. (v refers to any par- 

 ticular term of the power-series.) The aggre- 

 gate of the dz^s being zero, so must be the 

 aggregate of the Z ^dz's or of the Zdz's. 



For slender strips the theorem is obvious 

 because the variation in the value of the func- 

 tion for an infinitesimal z-rectangle or square 

 can be ignored in such products as Xdx, etc., 

 if only 00^ of these rectangles go to make up 

 the given strip. 



Finally, by taking into account the varia- 

 tion of the function when z describes an in- 

 finitesimal square, so that instead of X, say, 

 we have for the X-coordinates of the middles 

 of the sides of the transformed square 



X— I ^dy, X+ J ^dx, etc. 



dx 



With these values for X and similar ones for 

 Y, the value of Zdz for the four sides of the 

 elementary square is readily obtained. This 

 sum will vanish, including infinitesimals of 

 the second order, if the conditions for a mono- 



genic function are satisfied. A given area 

 comprises oo^ elementary squares, and so the 

 neglected infinitesimals of the third order are 

 of no consequence in the result. 



Mr. A. Press presented ' Studies in Soil 

 Capillarity.' 



There seems to have been considerable mis- 

 conception of what constitutes the capillary 

 height of soils. A mere presence of moisture 

 can not be an index of how much water will 

 rise in a soil by virtue of capillary action. If 

 one will drive home the analogy of capillary 

 movement in fine bore tubes the ' capillary 

 height ' can be defined as that height to which 

 moisture will rise in a soil where there is com- 

 plete filling of the pore-space of the soil. This 

 maximum height will be definite just in pro- 

 portion as the soil-particles are homogeneous 

 in size. However, if an equilibrium-condition 

 is established in a soil, the indefiniteness of 

 the capillary height is not so marked. This is 

 proved by the experiments of Professor King 

 when ten-foot cylinders of soil were employed, 

 and after being first surcharged with moisture 

 d]?aining was continued for a period of about 

 two years. Moisture above the capillary 

 height can only be adsorption moisture and 

 the pore-space need not necessarily be entirely 

 filled. There are two types of adhesion-water : 

 the one dry, the other wet; the first is hygro- 

 scopic water and is explained on Laplace's 

 theory of molecular (surface) attraction; the 

 second is of the same nature but differs in 

 degree, because in the one the water-vapor has 

 first to be condensed to water before strict 

 adhesion can take place, and in the other no 

 extra amount of energy is called for, because 

 by wetting the soil (above the capillary height) 

 the bonds of attraction of soil for moisture 

 are more easily satisfied. Mulching, to be 

 beneficial, depends upon whether the capillary 

 height is exceeded or not. A so-called ' natural 

 mulch ' would be where the capillary height is 

 below the surface of the soil. 



Dr. Parks's experiments, in the Philosoph- 

 ical Magazine, show that the attraction of 

 powders for moisture practically follows the 

 law: 



