TRANSACTIONS OF THE SECTIONS. 3 
supposed variable density is so small as to be insensible. Since there is a certain 
elastic force of vapour in contact with its fluid corresponding to every temperature, 
may we not assume that the density of this indefinitely thin envelope may vary 
from the density of the fluid inside to the density of the vapour outside ? 
On the Differential Equations of Dynamics. By Professor Boots, F.R.S. 
Referring to the reduction, by Hamilton and Jacobi, of the solution of the dyna- 
mical equations to that of a single non-linear partial differential equation of the first 
order, and to that, by Jacobi, of the latter to the solution of certain systems of 
linear partial differential equations of the first order,—the author showed, Ist, how, 
from an integral of one equation of any such system, a common integral of all the 
equations of the system could, when a certain condition dependent upon the pro- 
perties of symmetrical gauche determinants is satisfied, be deduced by the solution 
of a single ordinary differential equation of the first order susceptible of being made 
integrable by means of a factor; 2ndly, how the common integral could be found 
when this condition was not satisfied. 
On an Instrument for describing Geometrical Curves ; invented by H. Jounston, 
described and exhibited by the Rev. Dr. Boorn, F.R.S. 
This instrument supplies a want which has been felt by architects and sculptors. 
By its help, geometrical spirals of various orders may be described with as much 
manual facility as a circle may be drawn on paper by a common compass. 
On a Certain Curve of the Fourth Order. By A. Caxuny, F.R.S. 
The curve in question is the locus of the centres of the conics which pass 
through three given points and touch a given line; if the equations of the sides of 
the triangle formed by the three points are z=0, y=0, z=0, these coordinates being 
such that 2+-y+2=0 is the equation of the line infinity, and if ar+fy+yz=0be 
the equation of the given Jine, then (as is known) the equation of the curve is 
Vax (y+2—2) + V By GF2—y) + Veet y—)=0. 
The special object of the communication was to exhibit the form of the curve in 
the case where the line cuts the triangle, and to point out the correspondence of the 
positions of the centre upon the curve, and the point of contact on the given line. 
On the Representation of a Curve in Space by means of a Cone and Monoid 
Surface. By A. Carter, £.R.S,. 
The author gave a short account of his researches recently published in the 
‘Comptes Rendus.’ The difficulty as to the representation of a curve in space 
is, that such a curve is not in general the complete intersection of two surfaces ; 
any two surfaces passing through the curve intersect not only in the curve itself, 
but in a certain companion curve, which cannot be got rid of; this companion curve 
is in the proposed mode of representation reduced to the simplest form, viz. that 
of a system of lines passing through one and the same point. The two surfaces 
employed for the representation of a curve of the mth order are, a cone of the nth order 
haying for its vertex an arbitrary point (say the point r=0, y=0, s=0), and a monoid 
surface with the same vertex, viz. a surface the equation whereof is of the form 
Qw—P=0, P and Q being homogeneous functions of (x, y, 2) of the degrees p and 
p—lrespectively (where p is at most=x—1). The monoid surface contains upon 
it p (p—1) lines given by the equations (P=0, Q=0); and the cone passing through 
n( p—1) of these lines (if, as above supposed, p >> n—1, this implies that some of 
these lines are multiple lines of the cone), the monoid surface will besides intersect 
the cone in a curve of the th order, 
On the Curvature of the Margins of Leaves with reference to thei Growth. 
By W. Esson, M.A. 
Leaves have a right and left margin on each side of their axis, These margins 
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