TRANSACTIONS OF THE SECTIONS. 



Mathematics. 



On the Mercator's Projection of a Surface of Revolution. 

 By Prof. Catlet, F.B.S. 



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 in- 

 cluded 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 di-aw the parallels at such intervals as to divide the siu-face into infinitesimal 

 squares ; the meridians and parallels are then in the projection represented by two 

 systems of equidistant parallel lines dividmg the plane into squares. And if the 

 angidar 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 revolu- 

 tion : 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'^) constnict the posi- 

 tions of the 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 orthographic projections of the parallels ; the figure thus exhibits the ortho- 

 graphic projection (on the plane of a meridian) of the hyperboloid divided into 

 squares as above. The other tigiu-e, which is the Mercator's projection, is simply 

 two systems of equidistant parallel lines dividing the paper into squai-es. I remark 

 that in the first figure the projections of the right lines on the surface are the tan- 

 gents to the bounding hyperbola ; 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 projection of the right line on the surface ; viz. this is a ser- 



Sentine curve included between the right lines which represent two opposite meri- 

 ians 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 

 m a reversed position. 



_ A Mercator's projection mi^ht be made of a skew hyperboloid not of revolution ; 

 viz. the ciuT^es of curvature might be drawn so as to divide the suiface into squares, 

 and the cui-ves 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. 



On some Curves of the Fifth Class. By Professor W. K. Cliffokd. 



On a Surfaxe of Zero Curvature and Finite Extent. 

 By Professor W. K. Clifford, 



