Mr. Herapath on Functional Equations. 443 
As an example will best illustrate the propriety and neces- 
sity of the above restriction, let us take our old equation 
bes fa.yer + fia, (1) 
supposing the only condition for the present to be @x = x, and 
leaving fx and f,x entirely unlimited. By changing in (1) 
into «2 and eliminating between the resulting equation and 
(1) Wax, there is obtained 
UT Sufi ar 
wa = eee 2 (2) 
which is the complete solution of (1) without an arbitrary 
function. To prove this, let us add some function ¢2 to the 
solution (2), which for brevity we denote by Px; and it is 
evident that if Px be not the complete solution, 9 may have 
such a form that 
Pr + Gx 
shall contain it, or at least some one of the other solutions. 
Substitute Px + gx and Pax + ga for Pa and Yaz in (1), 
and we get 
Pre+ or =fr, Pexr+fa.par+ fie; 
or since by (2) Pace fit. et, Jie 
gr=fr.panr. 
Changing in this equation 2 into av it becomes gaw = fax. ga, 
and of course gives 
Ce=S 2 far Ox; 
an equation in which, if fv be as we suppose it unlimited, ¢ x 
must be null, and consequently Pz or (2) is the complete so- 
lution. But if gv be assumed to have a real value, then must 
fx.f«ax«=1,and as aconsequence by (2) fiw#= —fxifiar; 
which are what I have termed in my above-cited paper the 
conditions of possibility. 
The trouble of working out and printing the solution of a 
periodic of a higher order than the second, is the only reason 
why I do not here avail myself of it in corroboration of the 
position advanced. 
From what has been shown, there is manifestly a marked 
distinction between periodical equations considered indefi- 
nitely, and those of which I have treated in my paper. In 
the former, single solutions flow from the given functions 
being indefinite ; in the latter, innumerable solutions are the 
consequences of certain restrictions in the functions proposed, 
For the sake of distinction I shall hereafter call the former 
simply periodic functions, and the latter porismatic periodical 
Junctions. 
It would be hazardous in subjects so very general to pro- 
$K 2 nounce 
