COR. I. From the point a (Fig. 3) draw any two lines ak' and am' and from c any line cl' meeting the first two in m' and l'. Join l'b meeting am' in n'. Produce cn' to meet al' in k'; then the quadrangle k'n'l'm' is the correlative of knlm, that is the corresponding apices k and k', n and n', etc. are in perspective as seen from the centre O, determined by finding the point of intersection of any pair of corresponding rays such as kk' and mm' (Art. 1). COR. II. Since (Fig. 5)

which shews that the line OC is harmonically divided in d and k; and, therefore, the pencil of rays projecting its divisions from any external point, and a fortiori from any apex of the complete quadrangle EgfF, must be harmonic. COR. III.-Bisect ac (Fig. 4) in O′; then, since

dc da
cb ab

or

do' - O'c do' + O'c
CO'-O'bc0'+ O'b

we have, after multiplication and reduction,

O'c2=O'b. O'd=0't2,

which shews that any circle passing through points b and

ас

2

d intersects the circle described about O' with radius ac/, at right angles.

Similarly, if bd be bisected in O", we have

O"b2=0"a. O'c.

COR. IV. The vertical through d is termed the polar of point or pole b, being the locus of intersection of pairs of tangents drawn past opposite ends of chords traversing the pole b in the circle whose radius is O'c.

That the polar dd' is vertical in this case can be easily shewn. Thus, let bb' be any chord traversing b and let d' be the intersection of the pair of tangents drawn past its ends; then the radius vector of d' is

Now, if dd' be vertical, we ought to have the relations

But, since bb' always passes through b and O'b' is always at right angles to the chord bb', cos will be always equal to

O'b'

and therefore dd' is necessarily vertical. Similarly

O'b' the vertical through a is the polar of point c in the circle whose radius is O"b.

which explains the origin of the term harmonic mean.

COR. VI. The intercepts ff, and OO, (Fig. 5) are perspective of the intercepts dd,; whilst the intercepts gg1 are

perspective of intercepts OO, and projective of intercepts

dd,.

COR. VII. If the line d'O' (Fig. 4) traversing the centre

O' and the intersection of tangents drawn through the ends of the chord meet the circle containing the chord in c' and a', the line d'b' will be harmonically divided in c' and a', as shewn by the relation

O'b'. O'd' = O'c'2.

If the curve about O' were an ellipse instead of a circle, this relation would still obtain (Drew, Prop. XVIII.); hence the terms of the corollary apply also to the ellipse.

COR. VIII. The lines bisecting two adjacent angles, such as FOE and EOO, (Fig. 5), are harmonically divided by the rays containing the angles, and are normal to each

other.

3. Given three points a,b,c, of a line u, (Fig. 6) and three homologous points on a second line; to find a fourth pair of homologous points of the same two lines.

CASE I.—When two of the given homologous points are coincident. Let us and u, be the given lines, a,b,c, and a,b,c, the given trio of homologous points, and a, a, the pair of coincident points. Join and produce b1b, and c12 to meet in O2, the centre of perspective of the given two lines (§ 1); and from O, project any fourth point d, of u, upon u, in d2, the required fourth homologous point of u

CASE II. When the second line has no homologous point in common with the first. Let a,b,c, and abc be the given trio of homologous points; then, to find a fourth point, pass an auxiliary line u through two of the points which are not mutually projective, such as a, and b; and, upon the line a.a produced, take any point O as pole, from which project the point c upon u, in c. Join b,b, and c,c, by lines intersecting in the second pole O,; then, to find a fourth point on u, corresponding to any assumed point d on u, project d from O upon u, in d1, and d, from O, upon u, in d.

COR. I. The auxiliary line need not pass through b. If, for instance, a,b,' be the auxiliary line; choose, as before, the pole O anywhere on the line a,a produced; and from O project b and c upon the auxiliary in b,' and c,'; and join bb and c,c' to meet in the centre of perspective O' of the lines up and u. Any fourth point can then be found in the usual way, making use of the centre O, instead of O..

DEF. I. Between the angles and intercepts, bounded by every consecutive four elements of homologous systems, there exists the following important relation. Let any consecutive four rays abcd of the pencil S (Fig. 17) traverse the points ABCD of the corresponding transversal u; then, since the triangles bounded by u and the rays abcd have the same altitude, their areas are as their bases; or

Now two pencils are termed anharmonic when the segments, made by the pencils upon two transversals, form equal and similar ratios. Thus, when in Fig. 7

the punctuated lines u and u' are divided anharmonically by the anharmonic pencil O. The anharmonic is the general ratio of homologous systems. A line divided anharmonically is termed a punctuated line or point

range.

DEF. II. Lines are said to be in perspective when they are divided by the same pencil of rays. Two pencils are said to be in perspective when they equally and similarly divide the same punctuated or anharmonic line. Two lines or pencils are projective when four anharmonic elements of one correspond to four anharmonic elements of the other. Thus the pencils O and O, (Fig. 6) being perspective of the same line u, are perspective of each other;