646 



DR WALLACE ON A FUNCTIONAL EQUATION. 



therefore 



dy? + dy^ 



and 

 and 



dz 



dx' 



— I 6" +e " +2} , 



If- - 



y 



i a 



Pig. 6. 



dx 

 adz=:y d x. 



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

 ness, a rod is suspended, whose weight may be expressed hj ydoc, and here we 

 have found that the rod is equivalent in weight to adz, 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 curve 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 affirmed 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 amplitude 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 4>' the angle which a straight line 

 drawn at Y 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 a' for CB, 



BC the parameters of the curves. Because tan = —^ and tan(i'=-^ there- 

 to a; ' ^ dx '' 



fore tan : 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 (p -. tan (p' = a: a' ; 



and cot (p : cot (p' — ay. a ; 



and ycotcp: y' cot (f)'=y a' :y' a. 



Now, ya'=y'a; therefore y' cot =y cot (p', but y cot (p and y cot (p' express 

 the segments of the axis between the ordinates y, y\ and the hues touching the 

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



