Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

TRANSFORMATIONS OF THE PLANE

339

A transformation, as (c), which arises as the result of two
successive transformations, as (a) and (b), is called the product of
these transformations. Similarly, (a) and (b) are spoken of as
the factors of (c); or (c) is said to be factored into (a) and (b).
Let the student verify the fact that, if the circle i) is subjected to the transformation (c), it is carried over into the
same ellipse iii) into which i) was carried by the successive
applications of the transformations (a) and (b).
Properties of the Transformation. One of the most important properties of one-dimensional strains is that, like the transformations previously studied, they carry straight lines over into
straight lines.
This was proved geometrically on p. 304. The transformation considered there is given analytically by (1). It was
proved also that, if L is a line of slope A, the slope of the line
into which L is carried by (1) is *
(3) X' = X.
From the theorem contained in formula (3), it is seen that
a one-dimensional strain carries parallel lines into parallel lines.
Consider an arbitrary curve, C. Its
slope at any one of its points, P, is y
X = lim MQ
Q=P PM
Perform the transformation (1) on C. c
Then PM remains unchanged in
length; but MQ goes over into -.M' Q'=I MQ. ________
Hence the slope, X', of C' is FIG. 9
X' = lim MQ lim MQ =l lim MQ
OQ-P P'M' Q-P PM QP 'PM
or X' = IX.
We have thus extended the validity of formulas (3).
* The proofs on p. 304 were given for compressions, but they are valid,
also, for elongations.