496 



Mr. C. B. Clarke. 



[Mar. 12, 



are in this case no longer true, but are approximately true only if the 

 height be very small compared with the span. The correction to the 

 curvature as calculated from the usual formula is found to be a 

 constant. 



The paper concludes with an account and a short discussion of the 

 work of Lame and Clapeyron, de Saint- Venant, Boussinesq, and, more 

 recently, of M. Mathieu,* M. Eibiere,f and Mr. J. H. Michell^ 

 which bears upon the subject of rectangular beams. Although, in 

 certain cases, some of the results overlap, the attempt has been made 

 in the paper to co-ordinate them, and to present them in a more com- 

 plete form, and to develop further the two-dimensional theory, so as 

 to obtain solutions to various interesting questions relating to the 

 effects of isolated loads. 



"Antarctic Origin of the Tribe Schcenea?." By C. B. Clarke, 

 F.B.S. Eeceived March 12 — Eead April 24, 1902. 



[Plate 14.] 



The map annexed to this paper is designed to illustrate the geogra- 

 phic distribution of all the species of the Schcenese — a sub-ordo or 

 tribus of the Cyperacese. 



The result suggests a flow in geologic time of the sub-order from the 

 South Pole up the three great southern prolongations of land, viz., 

 Oceania, South Africa, Temperate South America ; the number of 

 species dying away rapidly as we recede from the South Pole. 



I explain how the map is made. I take the outline map of the 

 World divided into twenty-three geographic sub-areas, and my MSS. 

 of the sub-order Schoenese which show the distribution of every 

 species with reference to these twenty- three sub-areas. 



The first species is Carpha alpim, E. Br., which I see in the MS. has 

 been collected in the sub-areas 12, 13, 14, 23. I put a spot of black 

 paint in each of these four sub-areas, and proceed to the next species. 

 I have treated 262 species in black dots, two in rings, two in crosses. 

 The black dots do not signify anything as to the abundance of a 

 species ; nor in Australia and the Cape do they indicate more than 

 that the species has been collected in that sub-area. But the outlying 

 scattered spots in Central Africa, Japan, Jamaica, &c, are placed as 

 accurately as the scale of the map would admit. 



* ' Theorie de l'Elasticite,' Paris, 1890 ; also ' Comptes Eendus,' vol. 90, pp. 1272 

 —74. 



f ' Sur Divers Cas de la Flexion des Prismes Rectangles,' Bordeaux, .1889 ; also 

 ' Cornptes Rendus,' vol. 126, pp. 402—404 and 1190—92. 

 X ' Quart. Journ. Math.,' vol. 32. 



