Application of the Fluwional Ratio, &c. 93 
8. Small pieces of sodium projected upon a bath of mercury, were 
not found to exhibit the phenomena indicated by Serullas; that is, 
they were not thrown off with explosions accompanied with light and 
caloric. The effects are, however, curious. 'The amalgamation of 
the sodium takes place slowly, without any rotary motion; althoug 
sometimes, when breathed upon, a motion of short duration is indu- 
ced. When several pieces are put upon the bath at the same time, 
they show no disposition to come together, but rather the contrary. 
But when one piece is pushed towards another, there appears to be, 
within a certain distance, an attractive force exerted, which is imme- 
diately succeeded by a repulsive one of some comparative energy. 
Many pieces being accumulated in a small space, they become vio- 
lently agitated, as if alternately attracting and repelling each other, 
until they finally separate. 
University of Maryland, August Ist, 1833. 
Arr. XII.—On the application of the Flucional Ratio to particu- . 
lar cases; and the coincidence of the several orders of Flucxions, 
with the binomial theorem; by Exizur Wrieut, Esq. 
Concluded from Vol. xxiv. p. 312. 
By comparing fluxions with trigonometry in regard to properties, 
features, and results, we shall find a striking analogy to exist between 
them ; in trigonometry the triangles must be similar, for between an 
equilateral and a scalene triangle the relation g- 
of proportion does not exist; so in fluxions no 
proportion exists between two fluents and their 
fluxions, when those fluents are dissimilar ; for 
instance, in the two consecutive fluents ABD, 
AOE (Fig. 6.) the correspondent lines AB, AO, 
are not parallel, and by the definition not simi- 
lar ; consequently the fluents ABD, AOE, are 
not proportional to their fluxions. : 
In trigonometry, if the three sides of any triangle whatever are si- 
milar to the three sides of another triangle, the like sides are propor- 
tional ; in fluxions, if any fluxional quantity whatever is similarly con- 
stituted as another given fluxional quantity, the relation of proportion 
between the fluxions and their respective fluents subsists. 
