[545 
545] 
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 
531 
n — 1) fold 
i, that is, 
f to be 
y is, that 
function 
y, of the 
either for 
,1 values), 
tsively to 
regard to 
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for which 
:en to be 
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tonal and 
to 0, y). 
ire of a 
equation 
we have 
equations 
ented by 
osing the 
breaking 
) distinct 
h system 
.fferential 
;r system 
equation 
2 - c = 0, 
nsidering 
ever, we 
r (faulty) 
-xy) = 0, 
= 0; and 
3 are in 
iposable ; 
in the 
5°. It is necessary to consider a form f— 0 involving the m constants connected 
by the (m —l)fold relation. For taking such a system of constants, imagine an 
equation f{x, y, c 1; c 2 ,..., c m ) = 0 rational and integral in regard to {x, y), and representing 
an indecomposable curve; such an integral equation leads to a differential equation 
of the form </> {x, y, p) = 0, rational in regard to (x, y); whence, conversely, a differ 
ential equation of the form last referred to may have for its integral the equation 
f{x, y, Ci, c 2 ,, c m ) = 0. And we cannot in a proper form exhibit this integral in terms 
of a single constant. For first consider for a moment c a , c 2 ,..., c m as the coordinates 
of a point in m-dimensional space; the curve is not in general unicursal, and unless 
it be so, we cannot express the quantities c 1} c 2 ,..., c m rationally in terms of a para 
meter; that is, we cannot in general express c 1} c 2 ,..., c m rationally in terms of a 
parameter. Secondly, if by means of the {m — 1) fold relation we sought to eliminate 
from the equation f = 0 all but one of the m constants, we should indeed arrive at 
an equation F (x, y, c) = 0 rational and integral as regards x and y, and also as regards 
c; but this equation would not represent an indecomposable curve. 
6°. It is important to remark that, even in the case where <£ {x, y, p) is one 
valued in regard to {x, y) (2°), there is not in every case a form /=0 one-valued 
in regard to (x, y). A simple example shows this; let a, fi be incommensurable 
(e.g. a = e, /3 = w), then the equation </> = /3xp + ay = 0 {ay dx + fix dy = 0) has for its 
integral c = x a yP, where x a y& is not a one-valued function of x, y, and we cannot in 
any way whatever transform the integral so as to express it in terms of one-valued 
functions of x and y. But taking x a yP to be a one-valued function—if e.g. for all 
real values of x, y we consider x a y& as representing the real value of {±x) a {±yf— r 
we shall have, without any loss of generality, the integral of the differential equation; 
the whole system of curves c = x a y?, c any value whatever, is the same whether we 
attribute to x a y? its infinite series of values, or only one of these values. 
7°. The meaning is, that the equation /=0 is not satisfied irrespectively of the 
values of c 1} c 2 ,..., c m by any relation between x, y only. 
8°. The meaning is, that the function f is not the product of two factors, rational 
or irrational in regard to c x , c 2 , ..., c m , but each of them one-valued or taken to be 
one-valued in regard to x, y. Thus the function 
/= x*y 2 -o, = \xy + f{c)} [xy - V(c)}, 
is decomposable; but if we do not take any one-valued function, then f=xy — c is 
indecomposable; if we take f{xy) to be one-valued, then it is decomposable. 
The case /= x a y? — c {a and fi incommensurable) is to be noticed; starting from 
the differential equation ay dx + fix dy = 0, there is no reason for writing the integral 
— £ 
f= x a yP — c—0 rather than in either of the forms f=x ma y m P — c=0, f=x m y m — c=0, 
{m an integer), {a, fi), {ma, m/3), (—, —\ are, each pair as well as the others, two 
777// 
— — 
incommensurable magnitudes. If we choose to take x a y& but not x m y m as one-valued. 
67—2