60 



On Draining. 



torily performed my task within straiter limits ; and now, with an 

 invocation of all the indulgence it may need from yourself and 

 others my possible readers, 1 commend it to your joint and 

 separate notice, for the extraction of whatever good it may really 

 contain. 



Believe me yours very truly, 



Wharncliffe. 



Worthy, Sheffield, April, 1851. 



Note. 



Let A B or C D be supposed to be two of the parallel deep- 

 drains placed at any distance (say 20 yards) asunder. Then if 

 the smaller drains are extended from C D (at whatever angle) to 

 reach to within one-fifth, or 4 yards, of the opposite deep-drain, 

 A B ; their farther ends will be at the constant distance of 16 

 yards from the line C D. Let the line F E, at right angles to 

 C D, be taken to represent that distance ; and draw from the 

 point F the lines F H, F and F K, at any angles, say re- 

 spectively of 45°, 60°, and 70°. Then the inner, or acute 



Fig. 2. 



angles at H, I, and K will be the complements of these, or 

 respectively 45^, 30^, and 20^ ; and it is obvious enough that, as 

 these latter angles diminish, the length of the intermediate 

 drains must increase. Now, if we draw the circular arc E G 

 with the radius F E, it will be seen that the lines F H, F I, 

 and F K, become the secants to the angles at the point F ; or, 

 which is the same thing, the co-secants to their complements, 

 the angles at the points H, I, and K ; and therefore so long as 

 the distance F E remains constant, the length of the several 

 branch drains so drawn will vary as the cosecant of the angle at 

 their junction with the main drain C D, corresponding to the 

 radius F E. 



This, then, being the law of variation for their lengths, it next 

 becomes necessary to determine that for their number ^ assuming 



