DR WALLACE ON A FUNCTIONAL EQUATION, 



680 



Let ABH be that curve, which is now the figure of the chain ; draw a hori- 

 zontal line ECF, and through B, the lowest point of the curve, draw a vertical 



0' k: r q r f 



line CBD, meeting EF in C. From any point P in the curve draw PQ perpendi- 

 cular to CF, and PK touching the curve at P and meeting EF in K ; then PKQ 

 will be equal to the angle which an element of the curve makes with a horizontal 

 line at P : Put CQ=^ and PQ,=j/ and the angle PKQ,=0. 



d 2/ 6? (V 



In all curves tan = -~ , and (making d oo constant) d tan = — r^ 



we 



have therefore 



<?2.y 



m 

 c 



and 



d^y 



m 



dx c ' d.x^ cdx 



Now, if the vertical pressure on the curve at P be a column of matter whose base 

 IS, d X and altitude y, we have m=ydoo ; and, on this hypothesis, 



d"^ y _ ydoc _ y 

 dx^ cdx c ' 



and 



do6^ 



y 



= a constant. 



This differential equation is identical with that deduced from the functional 

 equation 



the latter must therefore express a property of the former. In this case, a; and 

 y increase together, and the value of 7/ when 07=0 is the perpendicular BC from 

 the lowest point of the chain, this is the quantity equivalent to C in the functional 

 equation. We have now (independently of the integral equation deduced from 

 the differential equation in article 12) this elegant proposition in statics. 



26. Theorem. — Let ABH be a perfectly flexible chain of uniform thickness, 

 and composed of infinitely small links suspended, in a vertical plane, from 

 two fixed points A, H : Suppose that an infinite number of infinitely thin 

 columns or rods, PQ, &c., are attached to the chain, and hang fi-eely, and 

 quite contiguous to each other, with their lower ends in a horizontal 

 straight line ECF, thereby forming a continuous plane sm-face between 

 that line and the plane curve ABH. Assuming now the straight line ECF 



