the calculus of functions. 
397 
Assume $ x — q{fx,fx j then it becomes 
i 
*{fx>fx} = q{fccX,fax} 
i i 
this equation will be satisfied if we determine/ and f so that 
tiie following equations may be fulfilled. 
f x x —fx and/a x =/ x 
i i . 
from these result/a* x =fax = fx • 
, i 
but we have by hypothesis a particular solution of/a* x =fx-, 
therefore the general solution of 4^ x = ip « x, is 
+ •* = <?> {fx,f*xj 
If the function a, should be of such a nature that k x —x, or 
even if c£ n x = x, f x will then become x. 
for example, suppose ^ x = [~~l) 
X 
, . X 1 I x \ ax — 1 
then since ai = , a x = a — — - — — — = x 
ax — i \ax—il x 
a — — . — i 
ax~~ i 
and we have v | j x = q t 
L ax— i J 
a particular case of this is when a = o } ^ ( <r ) = ^ ( — ^),its 
solution is 
4/ x = q | x, — x^ = q (x 2 ) 
the same as in the last Problem ; as another example take 
then since ux = — , a £ = ~ = x and i}/ x = q 
X 
This affords a solution to the following question : 
Required the nature of a curve, such that if any two ab- 
scissae, whose rectangle is equal to a given square be taken, 
their corresponding ordinates may also be equal. 
