379] 
BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE. 
551 
5. On Carves of the Third Order. Report, 1861, p. 2. 
A curve of the third order or cubic curve is a section of a cubic cone and such 
cone is intersected by a concentric sphere in a spherical cubic. It is an obvious 
consequence of a theorem of Sir Isaac Newton’s that there are live principal kinds 
of cubic cones, or what is the same thing five principal kinds of spherical cubics— 
but the nature of these five kinds of spherical cubics was first distinctly explained by 
Mobius. They may be designated the simplex, the complex, the crunodal, the acnodcil 
and the cuspidal: where crunode, acnode, denote respectively the two species of double 
points (nodes), viz. the double point with two real branches, and the conjugate or 
isolated point. The foregoing results are known: the special object of the paper is 
to establish a subdivision of the simplex kind of spherical cubics. The simplex kind 
is a continuous reentering curve cutting a great circle, to fix the ideas say the 
equator, in three pairs of opposite points, which are the three real inflexions of the 
curve. The three great circles which are the tangents at the inflexions and the 
equator divide the entire surface of the sphere into fourteen regions whereof eight 
are trilateral and the remaining six are quadrilateral. The curve may be entirely in 
six out of the eight trilateral regions, and it is in this case said to be simplex 
trilateral; or it may lie entirely in the six quadrilateral regions, and it is in this 
case said to be simplex quadrilateral; and there is an intermediate form, the simplex 
neutral; viz. in this case the three great circles tangents at the inflexions meet in 
a pair of opposite points and there are in all only twelve regions all of them 
trilateral; the curve lies entirely in six of these regions. 
6. On a Certain Curve of the Fourth Order. Report, 1862, p. 3. 
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 x = 0, y = 0, z — 0, these coordinates being 
such that x + y -f- z = 0 is the equation of the line infinity, and if ax + (3y + = 0 be 
the equation of the given line, then (as is known) the equation of the curve is 
Vax (y + z - x) + Vfiy (z + x - y) + \/>yz (x + y — z) = 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.