TRANSACTIONS OF THE SECTIONS, 3 
the three axes are unequal. Dr. Booth mentioned as the results of the method he 
developed, that every umbilical surface of the second order has four directrix planes 
parallel to the circular sections of the surface ; that these directrix planes—when the 
surface is an ellipsoid—pass two by two through the directrices of the principal sec- 
tion, in which lie the greatest and mean axes; that every such surface has four foci 
situate two by two on the umbilical diameters; that when the surface is an oblate 
spheroid, the four directrix planes are reduced to two parallel to the equator; and 
he concluded by showing that every graphic property of the sphere may be repro- 
duced in an analogous form in umbilical surfaces of the second order having three 
unequal axes. 
On the Mutual Relations of Inverse Curves and Inverse Curved Surfaces. 
By the Rev. J. Bootu, LL.D., F.R.S. 
Inverse curves and inverse curved surfaces were defined as curves and surfaces, the 
product of whose coincident radii vectores is constant ; he showed that the tangents 
to the two curves are equally inclined to the common vectors; that the sum of the 
vectors drawn from the common pole to the two inverse points, each divided by its 
corresponding cord of curvature drawn through the pole, is constant and equal to 
unity; that if a=) y being the radius vector of the primitive curve, the arc of the 
r 
derived curve is the same function of wu that the arc of the original curve is of r. He 
showed, also, that the element of the arc of the inverse curve represents the velocity 
of a planet in its urbit through the corresponding element of its orbit, assumed as 
the primary curve. He moreover proved that the circle is inverse to a circle, but 
when the pole is on the circumference the inverse curve is a right line; that when 
the focus of a parabola is the pole, the inverse curve is a cardioid ; but that when the 
vertex is the pole, the inverse curve is the cissoid; that the spiral of Archimedes is 
inverse to the hyperbolic spiral, while the logarithmic spiral is inverse to itself. He 
drew attention also to certain curves which are inverse to themselves, pointed out the 
facilities which this inverse method affords of passing from curves of the second order 
to corresponding properties of curves of the higher orders, and concluded by stating 
that the theory would be fully developed in the forthcoming Numbers of the 
© Quarterly Journal of Pure and Applied Mathematics.’ 
On the Notion of Distance in Analytical Geometry. 
By A. Cayrey, FR. 
The author remarks that the principles of modern geometry show that any metrical 
proposition whatever is really based upon a purely descriptive proposition, and that 
these principles contain in fact a theory of distance; but that such theory has not 
been disengaged from its applications and stated in a distinct and explicit form. The 
paper contains an account of the theory in question, viz. it is shown that in any 
system of geometry of two dimensions, the notion of distance can be arrived at from 
descriptive principles by means of a conic termed the Absolute, and which in ordi- 
nary plane geometry degenerates into a pair of points. 
On Dr. Whewell’s Views respecting the Nature and Value of Mathematical 
Definitions. By J. Pore Hennessy, of the Inner Temple. 
Having briefly referred to the several controversies which had taken place on this 
subject, the author remarked that the question at issue was equally interesting to two 
great divisions of the scientific world—the mathematical and the metaphysical. 
That question is, Are the deductions of mathematical reasoning absolute truths, or 
are they merely hypothetical? Against the form of the reasoning there could be no 
objection. It was that of the ordinary Sorites, and might, with a little labour, be 
split up into separate syllogisms. This had, in fact, been done, with some of the 
books of Euclid, by two foreign mathematicians. The reasoning process then being 
perfectly sound, the absolute truth of the conclusion must depend on the absolute 
: 1* 
