A Treatise On The Principles And Practice Of Harbour Engineering

210 HARBOUR ENGINEERING. , A+ D D 2A-D and y FI 6 ’ whence we can obtain the equation for the locus, viz :— 36xy + 18Da: - 18Ly - LD =0, which indicates a hyperbola. Transfer the origin to xr yr Then 36(æ + x^y + yj + 18D(a; + x^ - 18L(y + y^ - LD = 0. If the coefficients of x and y in this expression be made equal to zero, the values of x^, corresponding thereto, will give the centre of the curve. Thus 36^+180=0, and 36^— 18L=0. Therefore ^i= - TV’ , L and Xj=—• Accordingly, the point N is the centre. Referring the equation to N as origin, with NH and NP as axes, it becomes K’+IX'' - y) * ““(**r) - “(» -<) - w-0‘ whence ^+|ld=0. Therefore NH and NP are the asymptotes of the rectangular hyperbola constituting the curve. Further, since xy= ~yLD’ we see that the locus lies on the conjugale hyperbola. If the transverse axis of a rectangular hyperbola be 2a, then ixy = - 2a2 is the equation of the conjugate hyperbola. This may be written a2 Comparing it with the preceding value of xy, we see that ±=ÄLD; 2 9 ’ that is, a=4 VLD- Turning again to the general equation, 'i^xy + 18Da:- 18Ly - LD= 0, it will be observed that the curve passes through the point ~ , — G 6