THEORY OF POLYCONIC PROJECTIONS. 157 



Of all the right angles which are formed by the tangents 

 at the point those of the lines OP and OQ and their pro- 

 longations are the only ones one side of which remains 

 parallel to the tangent plane after the rotation which was 

 described above; these are the only ones then which are 

 projected into right angles. We can now state an addition 

 to the proposition which has just been proved, and we can 

 express the whole in the following form: At every point of 

 the surface which we wish to represent there are two per- 

 pendicular tangents, and, if the angles are not preserved, 

 there are only two, such that those which correspond to 

 them upon the other surface also intersect at right angles. 

 So that, upon each of the two surfaces, there exists a sys- 

 tem of orthogonal trajectories, and, if the method of rep- 

 resentation does not preserve the angles, there exists 

 only one of them the projections of which upon the other 

 surface are also orthogonal. 



We shall denote, by first and second principal tangents, 

 the two perpendicular tangents the angle between wmch is 

 not altered by the projection. We shall continue to denote, 

 respectively, by c and d the ratio of lengths in the direc- 

 tions of these tangents, and we shall suppose that c is 

 greater than d. 



If the infinitesimal curve drawn around the point is a 

 circumference of which is the center, the representation 

 of this curve will be an ellipse the axes of which will fall 

 upon the principal tangents, and these will have the values 

 2c and 2a, the radius of the circle being taken as unity. 

 This eUipse constitutes at each point a sort of indicatrix 

 of the system of projection. 



In place of projecting orthogonally the circumference, 

 the locus of the points M in figure 43, which gives the 

 ellipse the locus of the points N, then increasing this in the 

 ratio of c to unity, which gives the locus of the points Jf', 

 we can perform the two operations in the inverse order. 

 We should then in figure 44 obtain the point M' of the 

 elliptic indicatrix which corresponds to a given point M 

 of the circle by prolonging the radius OM until it meets at 

 R the circumference described upon the major axis as 

 diameter, and then by dropping a perpendicular from R 

 upon OA, the semimajor axis, and, finally, by reducing this 

 perpendicular RS, starting from its foot 8 in the ratio of d 

 to c. The point If' thus determined will be the required 

 point. 



In figure 44 let us draw Oi/', and let us call, respectively, 

 u and u' the angles AOM and AOM' which correspond 

 upon the two surfaces. Inasmuch as the second is the 



