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

356

ANALYTIC GEOMETRY

We have now factored (1) into the transformations (2), (5),
(4), where k = a'/a, namely into:
f' = X2, [x2=axl, XI 1=x+ ~ Y,
(6) aA Y
a a
( ) pY = a S2 +^ 12, /2 =A Y1, Yl = Y.
The first and last of these transformations are simple shears.
The second can be factored into two one-dimensional strains,
or, in case a or A/a or both are negative, into these and a reflection in one or the other or both axes. Thus the theorem
is proved in this case.
The proof is similar in the case that b': 0. The factors of
'1) are
b A
7) X-2 + - Y2) X2-= l X = XI
bf b' I \ \ f
IY'=Y2, L2=-b'yl, Yi=a +y.
Case 2: a = b' = 0. Here (1) becomes
(8) I A,-byx' b lvf' = by,
<8) ( ^ - A= a-b ' 0.
lY~ = ax.
This transformation can be factored into a rotation about the
origin through 90~:
(9) = — y, A'XI
and the transformation
x' = — bU y' = a
It can be shown that the rotation (9) is the product of three
simple shears, namely,
f = X, -2 = - Y1, [f =,
Y= X2 +2, Y2 = Y, Y1 = - y,
and this completes the proof of the theorem.
Homogeneous Strains. The extension to space of the transformations (1) is given by the formulas
x'= a x+bb y+c z,
y'= a' x + b' y + c' z,
'z = a"x + b"y + c"z.