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NOTICES OF COMMUNICATIONS TO THE BRITISH ASSOCIATION.
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successive parallels; I complete the figure by drawing the hyperbolas which are the
orthographic projections of the meridians, and the right lines which are the ortho
graphic projections of the parallels; the figure thus exhibits the orthographic projection
(on the plane of a meridian) of the hyperboloid divided into squares as above. The
other figure, which is the Mercator’s projection, is simply two systems of equidistant
parallel lines dividing the paper into squares. I remark that in the first figure the
projections of the right lines on the surface are the tangents to the bounding hyper
bola ; in particular, the projection of one of these lines is an asymptote of the
hyperbola. This I exhibit in the figure, and by means of it trace the Mercator’s pro
jection of the right line on the surface; viz. this is a serpentine curve included
between the right lines which represent two opposite meridians and having these lines
for asymptotes. It is sufficient to show one of these curves, since obviously for any
other line of the surface belonging to the same system the Mercator’s projection is
at once obtained by merely displacing the curve parallel to itself, and for any line of
the other system the projection is a like curve in a reversed position.
A Mercator’s projection might be made of a skew hyperboloid not of revolution;
viz. the curves of curvature might be drawn so as to divide the surface into squares,
and the curves of curvature be then represented by equidistant parallel lines as above;
and the construction would be only a little more difficult. The projection presented
itself to me as a convenient one for the representation of the geodesic lines on the
surface, and for exhibiting them in relation to the right lines of the surface; but I
have not yet worked this out. In conclusion, it may be remarked that a surface in
general cannot be divided into squares by its curves of curvature, but that it may
be in an infinity of ways divided into squares by two systems of curves on the
surface, and any such system of curves gives rise to a Mercator’s projection of the
surface.