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For by art. 15., the angle <p when t is aminimum with respect to y = 56°„28'.

But by construction the angle DAB =56°"28'; consequently DAB = tp.

Hence, in the triangle DAB we have: rad.: AB: : sec. DAB : AD : : tang. DAB: BD, or, radius being unity, 1: c: : sec. <?:AD—c sec. <p; and 1 : c :: tang, p : BD=ctang. <p.

But by Case I. art. 9., t=c sec. <p, and z—c tang. <p. Hence it follows that AD = t andBD = s’.

Again, because AC =AD=£ and AB=c,BC=z t—c=x by Case VI. art. 9. Moreover,because in the triangles ADB ACE theangles ABD, ACE are equal, each being aright angle, and that DAB and AEC arealso equal, each being the complement ofthe angle CAE ; these triangles are equi-angular, and consequently DB : AB : : AC :

CE or z:c::t: CE= ^ which by art. I6.=y.

Another method of construction. — On anypoint in the vertical line KC erect the