676 



SCIENCE 



[N. S. Vol. XXXI. No. 800 



of order n — 1. We will represent it by 

 A„_i. The solution of the system of equations 

 for any unknown, say M^ wiU be a fraction 

 with A„_i for the denominator. The numera- 

 tor of the fraction will be a determinant of 

 order n — 1 with the same elements as An-i ex- 

 cept that each element in the r — 1th column 

 is 1. By expanding An-i it is easy to see that 

 the general formula 



A„ — 4A„.i — ^n-, 



holds. Since Ai ^ 4 and Aj may be defined as 

 1, any A may be computed. 



For computing the determinant in the 

 numerator we let D„ represent a determinant 

 of the nth order which has the same elements 

 as A„ except that each element of the first 

 column is 1. Expanding D„, it is found that 



Z)„ is to be defined as 0. Now expanding the 

 numerator of the fraction representing Mr in 

 terms of minors of the upper r — 2 rows, we 

 find 



Ar-jD„_rrt — Dr-.A„.r 



Mr = 



An-i 



and multiplying this result by — wl'/^ we 

 have the general expression given at the be- 

 ginning of this article. In computing a table 

 from this formula it is of course not neces- 

 sary to compute all the Ms, for the bending 

 moments at supports equidistant from the 

 ends are equal, that is, 



Arthur E. Crathoene 

 Univeesity of Illinois 



SOCIETIES AND ACADEMIES 



THE BOTANICAL SOCIETY OF WASHINGTON 



The sixtieth regular meeting of the society was 

 held at the Ebbitt House, February 19, 1910, at 

 eight o'clock p.m.; President Wm. A. Taylor pre- 

 sided. The following papers were read: 

 Sprout Leaves of Western Willows: C. R. Ball, 



U. S. Bureau of Plant Industry. 



A knowledge of the range of variation in the 

 leaves of willows is important because a large 

 proportion of the herbarium material must be 

 determined from foliage specimens only. This is 

 due to the precocious flowering of many species 

 and the quick disappearance of the staminate 

 aments from all, thus leaving fully half the plants 

 in this dioecious genus with only the leaves as 

 determining features. The pistillate aments also 

 are gone from plants of the dioecious species be- 

 fore most collectors reach the field. The leaves of 

 the so-called water sprouts are interesting because 

 of their wide departure from the normal, especially 

 in size and to some extent in form also. 



A series of collections shows that the proportion 

 of breadth to length found in the normal leaves is 

 maintained in sprout leaves from the same indi- 

 vidual in several species of the sections Pentandrse, 

 Longifolise and Cordatse from the western United 

 States. A variation of form was found in a speci- 

 men of S. scouleriana (section Capreae) from 

 Arizona, in which the normal leaves are obovate, 

 but those of this sprout were broadly ovate. The 

 paper was illustrated by numerous specimens. 

 Bull-horn Acaoias in Botanical Literature, ivith 



a Description of two new Species: W. E. Saf- 



FOBD, U. S. Bureau of Plant Industry. 



There has been much confusion as to the iden- 

 tity of certain acacias of Mexico and Central 

 America having large inflated horn-like stipular 

 thorns, which are usually inhabited by ants. 

 Linnseus placed all which had been described pre- 

 viously to the publication of his " Species Plan- 

 tarum," under a single species Mimosa cornigera. 

 Schlechtendal and Chamisso recognized the fact 

 that the supposed synonyms cited by Linnseus 

 included more than one species. These authors 

 described two species found in the collections of 

 Schiede from the state of Vera Cruz, Mexico, 

 which they named A. spadicigera and A. sphwro- 

 cephala. They leave it in doubt whether either 

 of these species is the Arior cornigera, figured and 

 described by Hernandez (ed. Rom., p. 86, 1656), 

 which in all probability is identical with the first 

 plant cited by Linnseus, under his description of 

 Mimosa cornigera. 



In the National Herbarium are specimens of a 

 buU-horn acacia from the type region of Hernan- 

 dez's plant, collected by Dr. Edward Palmer. 

 There are also at least two others quite distinct 

 from any species hitherto described, one of them 

 from Guatemala, with the inflorescence in spher- 

 ical heads and with very long slender dehiscent 

 pods; the other from the state of Chiapas, south- 

 ern Mexico, with spadix-like inflorescence and 



