The profile of a valley at the stage of its development corresponding to the above section is represented diagrammatically by the curve A B in Fig. 19. The sketch (Pl. XIX, Fig. 1) represents a bird's-eye view of a valley in the same stage of development. [Illustration: Fig. 20.--Diagrammatic cross-section of a valley at a stage corresponding with that shown in Plate XIX, Fig. 2.] At a stage of development later than that represented by the V-shaped cross-section, the corresponding section is U-shaped, as shown in Fig. 20. The same form is sketched in Plate XIX, Fig. 2. This represents a stage of development where detritus descending the slopes is not all carried away by the stream, and where the valley is being widened faster than it is deepened. Its slopes are therefore becoming gentler. The profile of the valley at this stage would be much the same as that in the preceding, except that the gradient in the lower portion would be lower. Still later the cross section of the valley assumes the shape shown in Fig. 21, and in perspective the form sketched in Plate XX, Fig. 1. This transformation is effected partly by erosion, and partly by deposition in the valley. When a stream has cut its valley as low as conditions allow, it becomes sluggish. A sluggish stream is easily turned from side to side, and, directed against its banks, it may undercut them, causing them to recede at the point of undercutting. In its meanderings, it undercuts at various points at various times, and the aggregate result is the widening of the valley. By this process alone the stream would develop a flat grade. At the same time all the drainage which comes in at the sides tends to carry the walls of the valley farther from its axis. [Illustration: Fig. 21.--Diagrammatic cross-section of a valley at a stage later than that shown in Fig. 20.] A sluggish stream is also generally a depositing stream. Its deposits tend to aggrade (build up) the flat which its meanderings develop. When a valley bottom is built up, it becomes wider at the same time, for the valley is, as a rule, wider at any given level than at any lower one. Thus the U-shaped valley is finally converted into a valley with a flat bottom, the flat being due in large part to erosion, and in smaller part to deposition. Under exceptional circumstances the relative importance of these two factors may be reversed. It will be seen that the cross-section of a valley affords a clue to its age. A valley without a flat is young, and increasing age is indicated by increasing width. Valleys illustrating all stages of development are to be found in the Devil's lake region. The valley of Honey creek southwest of Devil's lake may be taken as an illustration of a valley at an intermediate stage of development, while examples of old valleys are found in the flat country about Camp Douglas and Necedah. [Illustration: WISCONSIN GEOL. AND NAT. HIST. SURVEY. BULLETIN NO. V., PL. XIX. FIG. 1.