Crofton — Solution of Certain Functional Equations. 23 
7. To determine a function cf>, sucli that 
It is known tliat 
Hence <^ (;r + Z>) ^''^ = ^ (^) . e-h^' ; 
Hence me''''cl>{x) ^ e'^'eiD^^aD ^^^y 
Put y = <f>{x)y and we have, as above, 
where A, B stand for - a ± ^?^+log.m^. 
It is easy to verify the correctness of the result by means of the 
theorem 
which easily follows from (^8), above. 
I have thought it worth while presenting these few examples to the 
Academy, not as being themselves of much importance, but as illus- 
trating the use of symbolic operators in such equations. Yery little 
is known as to the management of these symbols; but there is no 
doubt that any progress in this direction will assist in the theory of 
functional equations, which, as may be seen from the above cases, may 
often be at once converted into relations between symbols involving 
one or more variables with the signs of differentiation. 
