78 
[729 
729. 
ON A THEOREM RELATING TO CONFORMABLE FIGURES. 
[From the Proceedinqs of the London Mathematical Society, vol. x. (1879), pp. 143—146. 
Read May 8, 1879.] 
Consider two plane figures, say the figure of the points P referred to axes 
Ox, Oy, and that of the points P' referred to axes Ox 1 , Oy' ; and let x, y be the 
coordinates of P, and x', y' those of P'. If the figures correspond to each other in 
any manner whatever, P and P' being corresponding points, then we have x', y' 
each of them a function of x, y\ and we may consider the second figure as derived 
from the first by altering the distance OP in the ratio Vx' 2 + y 2 4 Va? 2 + y 2 , and by 
rotating it through the angle tan -1 — tan -1 - ; say by the Extension V«' 2 + y'- 4 \fx 2 + y 2 , 
and by the Rotation tan -1 tan -1 - ; where the Extension and the Rotation are each 
of them a determinate function of x, y, the coordinates of P. 
Passing from the point P to a consecutive point Q, the coordinates of which 
are x + dx, y + dy (the ratio cly 4 dx being arbitrary), then the coordinates of the 
corresponding point Q' will be x + dx, y + dy, where 
, . dx' 7 dx 7 7 . dy' 7 dy' 7 
dx = —j— dx + dy, dy = dx + dy. 
dx 
Writing < ~ / and instead of dy' -r- dx' and dy 4- dx, the expressions 
Vdx' 2 + dy' 2 -4 Vdx 2 + dy 2 , and 
tan - 
— tan - 
i fy 
dx ’ 
will in general have values depending upon that of the arbitrary ratio dy : dx. But 
they may be independent of this ratio; viz. this is the case when x', y' are functions 
of x, y such that 
dx _ dy' dy' dx' 
dy dx ’ dy dx ’