ON A THEOREM RELATING TO CONFORMABLE FIGURES. 
79 
[729 
729] ON A THEOREM RELATING TO CONFORMABLE FIGURES. 79 
and the two figures are then conformable (or conjugate) figures; that is, figures similar 
as regards corresponding infinitesimal elements of area. We have, in this case, 
V dx' 2 + dy' 2 -f- V dx 2 + dy 2 , and tan -1 ^ — tan -1 ^ , 
dx dx 
S. 
each a determinate function of x, y, the coordinates of P; and we pass from the 
element PQ to the corresponding element P'Q' by altering the length in the ratio 
Vdx' 2 + dy’ 2 -7- Vdx 2 + dy 2 , and rotating the element through the angle tan -1 _ tan -1 ^ ; 
CLOC CLOG 
say, this ratio and this angle are the Auxesis and the Streblosis respectively, these 
being, as already mentioned, functions of x, y only. 
Considering now any two conformable figures, say the figure of the points P, 
and that of the points P' ; we have the theorem that we can from the first figure 
obtain a third conformable figure by means of an Auxesis and a Streblosis which 
—146. 
are respectively equal to the Extension and the Rotation by which the second figure 
is derived from the first. 
) axes 
se the 
her in 
x\ y 
ierived 
In fact, if in the three figures respectively we take x, y, x', y', and x", y", for 
the coordinates of the corresponding points P, P', P", the first and second figures 
are conformable : and we have therefore 
dx dy' dy' dx' 
dy dx ’ dy dx' 
tnd by 
the third figure is to have the Auxesis ^x' 2 + y' 2 — ^x 2 + y 2 , and the Streblosis 
'x 2 + y~, 
tan -1 ^7 — tan -1 - ; 
X X 
e each 
viz. writing r for Vx 2 + y 2 , we ought to have 
which 
of the 
dx „ = ^+y^ dx *1 -fy dy , 
d ,/ = *y'-*y dx+ ™' + yy' dr , 
But 
notions 
and it is therefore to be shown that there exist x", y" functions of x, y satisfying 
these relations ; for, this being so, we have 
dx" dy" dy" dx" 
dy dx ’ dy dx ’ 
and the third figure is thus conformable with the first. 
Writing, for shortness, 
1 xx +yy' B _ xy - x'y 
r 2 ’ r 2 ’