xxviii Transactions of the Royal Society of South Africa. 
of the coast and from Natal as far inland as Pietermaritzburg. The Middle 
and High Yeld of the Transvaal are only represented by a single specimen, 
a species of Bimeriella collected at Bandolier Kop. 
All that is known of the South African Perisporiales up to the present is 
comprised in diagnoses and descriptions of fungi collected by Professor 
MacOwan and Dr. J. Medley Wood, and in a few descriptions of fungi 
more recently collected and published in the Annales Mycologici and 
elsewhere. 
All the earlier work was done in the G-rahamstown District and the 
Coast Pegion of Natal, so that a large part of the Union was left totally 
unexplored so far as this group was concerned. 
" The Arrangement of Successive Convergents in Order of Accuracy." 
By Alexander Brown. 
One of the most important uses of Simple Continued Fractions is for 
the solution of the problem to find the fraction whose denominator does not 
exceed a given integer, which shall most closely approximate to a given 
number commensurable or incommensurable. A practically complete 
solution was provided by Lagrange in 1769 in his paper " Sur la Eesolution 
des Equations Numeriques " in the Memoires de I'Academie royale des 
Sciences et Belles -Lettres de Berlin. 
His results give the fraction nearest in defect and the fraction nearest 
in excess satisfying the conditions. He does not, however, consider the 
question of deciding which of these two fractions is nearest in absolute 
value to the given number. The author gives a proof of the rule and a 
method of arranging the convergents in one set so as to show the nearest in 
defect, the nearest in excess, and the nearest in absolute value satisfying the 
stated condition. 
" The Use of a Standard Parabola for Drawing Diagrams of Bending 
Moment and of Shear in a Beam Uniformly Loaded." By Alexander 
Brown. 
The important stresses in a uniform continuous beam are the Shear and 
the Bending Moment ; they are best shown in the form of graphs where 
length along the beam is taken as abscissa and the required function as 
ordinate. 
Ordinary Meeting. 
October 20, 1915. 
The President, Dr. L. Peringuey, was in the Chair. 
The Minutes of the Ordinary Meeting held on September 15th, 1915, were 
confirmed. 
