646 



DR WALLACE ON A FUNCTIONAL EQUATION. 



therefore 



and 



=}' 



and 



adz=y d z. 



Fis. 0. 



36. By hypothesis (art. 26), from every point of a chain of unifonn thick- 

 ness, a rod is suspended, whose weight may be expressed by y d x, and here we 

 have found that the rod is equivalent in weight to a d z, which may represent an 

 element of the chain ; hence, it follows that, whether the chain be loaded, accord- 

 ing to the hypothesis, with rods, or be composed of some perfectly flexible ma- 

 terial, like gossamer, of uniform thickness, and not loaded, the cm-ve it forms will 

 be the very same, that is, it will be a catenary. So that the properties which 

 have been proved to belong to the equilibrated curve, 



in its general form, may be all affinned to be true 

 of the simple catenary, that is, a curve formed by a 

 chain or cord of uniform thickness, hanging in a ver- 

 tical plane from two fixed points. 



37. Let APBH be a common catenary, and 

 A'P'B'H' a curve of equilibration, such as it has 

 been defined in art. 25, which have a common hori- 

 zontal axis EF, and their vertical axes is the same 

 straight line ; let PQ, P'Q be ordinates which have 

 the same amphtude CQ ; let <p denote the angle 

 which a straight line PK, touching the catenary 

 ABH at the top of the ordinate, makes with the 

 axis EF, and (p' the angle which a straight line 

 drawn at P' the top of the other ordinate, touching 

 the curve A'B'H', makes with the same axis : Put x 

 for CQ, the common amplitude, y for the ordinate 

 PQ, and / for the ordinate P'Q, and a and «'for CB, 



BC the pai-ameters of the curves. Because tan(^ = --^, and tan(^'=-^ , there- 



dz 



dx 



fore tan (p -. tan (p'=dy : dy' : Now, y having to y' a constant ratio, viz. that of a to a\ 



we have y ■.y' = dy: dy' — a : a' ; 

 therefore tan ^ : tan (/>' = «: a' ; 



and cot </< : cot (^' = «' : a ; 



and y cot (p : y' cot (p' =y a' : y a . 



Now, ya:=y'a; therefore y' cot<p =y cot(p', but ycot^ and ycot(p' express 

 the segments of the axis between the ordinates y, y', and the lines touching the 

 curves. On the whole, then, we have these two propositions. 



