By Robert Lachlan

This quantity is made from electronic photographs from the Cornell college Library historic arithmetic Monographs assortment.

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D Ex. 1. Show that if A, B, C, 63. be any four points in a plane, and if the six lines joining these points meet in the points E, F, G then the two ; POINTS CONJUGATE TO TWO POINT-PAIRS. 86 any one of these points are harmonically conjugate with EFG which meet in the same point This follows from § 60. lines which meet in the two sides of the triangle Ex. Deduce from 2. corresponding theorem when four straight § 62, the lines are given. Ex. two of the point of intersection of Ex. two straight lines be drawn intersecting and C, D respectively, show that the locus and DC is a straight line.

ABC) - (BCD) + (CDA) - (DAB) = Therefore (ii) AB, let AB cut D C and If the points CD is triangles (AHD) and (HCB). 0. on opposite sides of the the point H. in (ABCD) That lie line Then the expn clearly equal to the difference of the areas of the (ABCD) = (A HD) - (HCB) = (ABD) - (DCB) = (ABD) + (DBC). is Similarly we may show that (ABCD) = (ABC) + (CDA). Hence, as before, (ABC) + (CDA) = (ABD) + (DBC) that is, Ex. 34. 1. If a, b, c, d be any (abed) that Ex. ; (ABC) - (BCD) + (CDA) - (DAB) = 0.

Sin A OD + sin CO A . sin 110 1> + sin AOB. sin COD -0. O Let any straight in line the points A, B, C, D. be drawn cutting the rays of the pencil Then, by § 25, we have BC. AD + CA But if ON on the line . BD + AB CD = 0. AB = 0A. OB. sinAOB, for NO AD, NO CD, &c. . , above relation, we obtain the relation BOC . sin A OD + sin CO A This relation is . of great use. sin BOD + sin A OB . sin COD = 0. It includes moreover as particular cases several important trigonometrical formulae. 31. Ex. If 1.