We have briefly reviewed the most important projections which are derived from the sphere by direct geometrical construction, and we pass to that more important branch of the subject which deals with projections which are not subject to this limitation. _Conical Projections._ Conical projections are those in which the parallels are represented by concentric circles and the meridians by equally spaced radii. There is no necessary connexion between a conical projection and any touching or secant cone. Projections for instance which are derived by geometrical construction from secant cones are very poor projections, exhibiting large errors, and they will not be discussed. The name conical is given to the group embraced by the above definition, because, as is obvious, a projection so drawn can be bent round to form a cone. The simplest and, at the same time, one of the most useful forms of conical projection is the following: [Illustration: FIG. 15.] _Conical Projection with Rectified Meridians and Two Standard Parallels._--In some books this has been, most unfortunately, termed the "secant conical," on account of the fact that there are two parallels of the correct length. The use of this term in the past has caused much confusion. Two selected parallels are represented by concentric circular arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by their true lengths; and the other parallels are circular arcs through points so determined and are concentric with the chosen parallels. Thus in fig. 15 two parallels Gn and G´n´ are represented by their true lengths on the sphere; all the distances along the meridian PGG´, pnn´ are the true spherical lengths rectified. Let [gamma] be the co-latitude of Gn; [gamma]´ that of Gn´; [omega] be the true difference of longitude of PGG´ and pnn´; h[omega] be the angle at O; and OP = z, where Pp is the representation of the pole. Then the true length of parallel Gn on the sphere is [omega] sin [gamma], and this is equal to the length on the projection, i.e. [omega] sin [gamma] = h[omega](z + [gamma]); similarly [omega] sin [gamma]´= h[omega](z + [gamma]´). The radius of the sphere is assumed to be unity, and z and [gamma] are expressed in circular measure. Hence h = sin [gamma]/(z + [gamma]) = sin [gamma]´(z + [gamma]´); from this h and z are easily found. In the above description it has been assumed that the two errorless parallels have been _selected_. But it is usually desirable to impose some condition which itself will fix the errorless parallels. There are many conditions, any one of which may be imposed. In fig. 15 let Cm and C´m´ represent the extreme parallels of the map, and let the co-latitudes of these parallels be c and c´, then any one of the following conditions may be fulfilled:--