555]
BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE.
567
a I 2 d — 3abc + 26 3 ; the general value of u is u = any function whatever of the last-
mentioned three functions. We have to find the rational non-integral functions of
these functions which are rational and integral functions of the coefficients; such a
function is
^ \{a 2 d — 3abc + 2b 3 ) 2 + 4 (ac — b 2 ) 3 },
= ci 2 d 2 + 4ac 3 + 4 b s d — 3 b 2 c 2 — imbed,
and the original three solutions, together with the last-mentioned function a 2 d 2 + &c.,
constitute the complete system of the seminvariants of the cubic function; viz. every
other seminvariant is a rational and integral function of these. And so, in the general
case, the problem is to complete the series of the n solutions a, ac — b 2 , a 2 d — 3abc + 2b 3 ,
a 3 e — 4ni 2 bd + 6ab 2 c — 3b*, &c. by adding thereto the solutions which, being rational but
non-integral functions of these, are rational and integral functions of the coefficients;
and thus to arrive at a series of solutions such that every other solution is a rational
and integral function of these.
4. Note on certain Families of Surfaces. Report, 1871, pp. 19, 20.
See the paper numbered 538, of which this Note is a duplicate.
5. On the Mercator’s Projection of a Surface of Revolution. Report, 1873, p. 9.
The theory of Mercator’s projection is obviously applicable to any surface of re
volution; the meridians and parallels are represented by two systems of parallel lines
at right angles to each other, in such wise that for the infinitesimal rectangles
included between two consecutive arcs of meridian and arcs of parallel the rectangle in
the projection is similar to that on the surface. Or, what is the same thing, drawing-
on the surface the meridians at equal infinitesimal intervals of angular distance, we
may draw the parallels at such intervals as to divide the surface into infinitesimal
squares ; the meridians and parallels are then in the projection represented by two
systems of equidistant parallel lines dividing the plane into squares. And if the
angular distance between two consecutive meridians instead of being infinitesimal is
taken moderately small (5° or even 10°), then it is easy on the surface or in piano,
using only the curve which is the meridian of the surface, to lay down graphically
the series of parallels which are in the projection represented by equidistant parallel
lines. The method is, of course, an approximate one, by reason that the angular distance
between the two consecutive meridians is finite instead of infinitesimal.
I have in this way constructed the projection of a skew hyperboloid of revolution:
viz. in one figure I show the hyperbola, which is the meridian section, and by means
of it (taking the interval of the meridians to be = 10°) construct the positions of the
