Contents. xxiii

CHAPTER XIII.

THEORY OF DUALITY AND RECIPROCAL POLARS.

PAGE

Principle of duality,......... 382

Reciprocation defined, ......... 384

Substitutions to lie made in any theorem in order to got the reciprocal

theorem, ........... 384

Reciprocal of Pascal's theorem, ....... 385

Properties proved by reciprocation, ....... 385

Special results when reciprocating conic is a circle, .... 386

Equation of the reciprocal of any conic, ...... 387

To iind the centre ol! reciprocation so that the polar reciprocal of a given

triangle may be given in species, ...... 388

Lionnot's triangle, .......... 389

Tangential equation of conic given a focus and circumtriangle, . . 390
Equation, of the Brocard ellipse, . . . . . . .391

Metapolar quadrangles and their metapolos,..... 392

CHAPTER XIV.
RECENT GEOMETRY.
SUCTION I.—OK A SYSTEM OF THREE PIGUEES DIRECTLY SIMILAK.
llomothctic figures ; double point or centre of similitude of, . . 393
To (hid the double point of two polygons directly similar, . . 394-
Three directly similar figures; corresponding lengths, angles of rota-
tion, double points, triangle and circle of similitude of . . 395
In ditto, triangle formed by three homologous lines is in perspective
with triangle of similitude and locus of centre of perspective is
the circle of similitude,........ 396
Invariable points and invariable triangle, ...... 397
„ ,, form a system of three corresponding points, . . 397
,, triangle and triangle of similitude are in perspective, . 397