THEORY OF POLYCONIC PROJECTIONS. 155 



TISSOT'S INDICATRK. 



To represent one surface upon another we imagine that 

 each surface is decomposed by two systems of hnes into 

 infinitesimal parallelograms, and to each line of the first 

 smiace we make correspond one of the lines of the second; 

 then the intersection of two lines of the different sys- 

 tems upon the one surface and the intersection of the 

 two corresponding lines upon the other determine two cor- 

 responding points; finally, the totality of the points of the 

 second whicn correspond to the points of a given figure of 

 the first forms the representation or the projection of this 

 figure. We obtain the different methods of representation 

 by varying the two series of lines which form the graticide 

 upon one of the surfaces. 



If two surfaces are not applicable to. each other, it is 

 impossible to choose a method of projection such that there 

 is similarity between every figure traced upon the first and 

 the corresponding figure upon the second. On the other 

 hand, whatever Qie two surfaces may be, there exists an 

 infinity of systems of projection preserving the angles, and, 

 as a consequence, such that each figure infinitely small and 

 its representation are similar to each other. There is also an 

 infinity of others preserving the areas. However, these 

 two classes of projections are exceptions. A method of 

 projection being taken by chance, it will generally happen 

 that the angles will be changed, except, possibly, at par- 

 ticular points, and that the corresponding areas will not 

 have a constant ratio to each other. The lengths will thus 

 be altered. 



Let us consider two ciirves which correspond to each 

 other on the two surfaces. In figure 41 let and -3f be two 

 points of the one, 0' and M the corresponding points of 

 the other, and let The the tangent at to the firet curve. 

 If the point if approaches the point indefinitely, the point 

 M! will approach indefinitely the point 0', and the ratio of 

 the length of the arc O'M! to that of the arc OM will tend 

 toward a certain limit; this limit is what we caU the ratio of 

 lengths at the point upon the curve Oif or in the direction 

 2; In a system of proj ection preserving the angles the ratio 

 thus defined has the same value for all o&rections at a given 

 point; but it varies with the position of this point, unless 

 the two surfaces are applicable to each other. When the 

 representation does not preserve the angles except at par- 

 tictdar points, the ratio of lengths at all other pomts 

 changes with the direction. 



