DR WALLACE ON A FUNCTIONAL EQUATION. 



639 



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 



line CBD, meeting EF in G. 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 

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

 line at P : Put CQ,=.r and PQ=2/ and the angle PKQ=<^. 



In aU curves tan (p=-f- , and (making dx constant) dtaa (j> — —f^ 



we 



have therefore 



dx 



: — , and 

 c 





cdx 



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



\s, d X and altitude y, we have ni=ydx; and, on this hypothesis, 



d^ y _ ydx _ y 

 da? cdx c 



and 



rf2l 



da? 



■ —=—, = a constant. 



This differential equation is identical with that deduced from the functional 



equation 



/ W •/ W = { /(Xo + x;) +f{x-x) } : 



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

 y increase together, and the value of 3/ when cr=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 ilexible chain of uniform thickness, 

 and composed of infinitely smaU 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 freely, and 

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

 straight line ECF, thereby forming a continuous plane surface between 

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



