( 213 ) 
/<*>.=&■ 1 — 
or if we put a = 1 
log a 
log a 
log b(^) + —-T 
/w=- 
log a 
Out of (a) then follows for <p n (x) 
<pn (#) = 9 -1 ! «* 9 (*) 4* —T P 1 
. ( 9 ) 
• (c) 
By putting g {x) = S and therefore x = g -1 (§) — 0 (S) we can 
reduce the standard form (C) to the form: 
x=G($) , y=G(ag + 0)>.( C '> 
or written more symmetrically, 
* = <?(<.,? +a.) , y = G(b l S + b,y. ...(C") 
So if we succeed in giving to y = <p («) this parameter representation, 
the function can be built up by infinitesimal iteration. 
The expression for f{x) loses its signification when loga = 0, so 
when a = 1. We are then however in the case ( B), where f{x) — ^ 
which form follows likewise out of (7) when there c is put equal 
to log a = 0. The form for <p n (x) we can also obtain by putting « equal 
to 1 in (c). Then holds Lira = n, so that (p„ (a) assumes the 
form (6). 
§ 3. Before passing on to giving some examples we shall endeavour 
to deduce along the same way differential equations of a higher 
order, to arrive at more arbitrary constants in the standard form. 
By differentiation of (4) we arrive at 
„,)/%) /») , n 
or if we put: 
/(*) * f\y) ’/'*(*) y x y 
q\Y 2 —X t ) + q 2 {(Y s — 217)—(X„— 2XS)} 7= — {(li.—O—(§.—§**)!• ( u ) 
Now we have on account of (3) and (10) 
15 
Proceedings Royal Acad. Amsterdam. Vol. XII. 
