On Draining. 



61 



that the direct distance between any two of them, measured at 

 right angles to their parallel directions, is a fixed quantity. 



Fig. 3. 



For this purpose let A B and C D again represent a pair of 

 the deep drains bounding an interval of 20 yards in width, and 

 let the auxiliary drains have their outlets along the line of C D, 

 and be carried to within a certain distance, say 4 yards as before, 

 of A B. Then the direct distance of each of those ends from 

 C D will be, as before, 16 yards, a constant quantity. Now^ 

 suppose that from any point, E, a line be drawn at an angle of 

 45® to the point L, at 16 yards from C D, and that, from E also, 

 a line, E F, be drawn at right angles to E L, and corresponding 

 to a given distance, say as before of 8 yards. Then a third line 

 J K, drawn through F, and parallel to E L, will represent the 

 next auxiliary drain laid at an interval of 8 yards, and at an 

 angle of 45° with the main drain. 



Now, it is obvious that the distance E K between the junc- 

 tions on this last line must be larger than E F the direct measure 

 of the interval between the small drains, in all cases except 

 where the primary and secondary lines are at right angles 

 to each other ; and that if we similarly draw other pairs of lines 

 at smaller angles, as one at 40°, at 30*^, or at 20°, these distances 

 will continually increase, as E K', E K E K'", as the angle 

 diminishes, in comparison with the perpendicular representing the 

 true measure of the interval between these parallel lines : and 

 as the number of the points of junction must diminish in pro- 

 portion as the said distances increase^ the length of these seg- 

 ments of the line C D will determine the number of the se- 

 condary drains in a given length of the primary. Now, if from 

 the point E we describe the circle H F G, with the lines E F, 

 E F', &c., which are assumed all equal, as radii, the segments 

 E K, E K', &c., become the secants to the several angles F E K, 

 F' E K', &c., and therefore cosecants to their complements, the 

 angles F K E, F' K' E, &c., which are the angles of junction, 

 as before. But the number of these minor drains must vary 

 inversely as the distances between their origins along the main 



