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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.