DIFFERENTIAL EQUATIONS 
345 
A/s can be set =1, and the remaining m —1 set =0. We are 
thus led to m linearly independent solutions of (1).* 
We see, therefore, how in all cases to derive from (8) and (7) n 
solutions of the form (6), out of which an arbitrary solution of (1) 
can be constructed by means of (5). 
The variables p k are known as the normal coordinates of the sys 
tem. They are uniquely determined, save as to their order, when 
the r k are all distinct; but when some of the r/s are equal, an infi 
nite number of different choices is possible. 
III. Geometrical Interpretation. Singular Solutions 
15. Meaning of a Differential Equation. Just as, in Integra 
tion, our first object was to discover the devices by which the 
integrals we meet in practice can be evaluated in terms of the ele 
mentary functions, so here we have studied in this chapter analogous 
devices for solving differential equations such as occur in physics 
and geometry by means of explicit formulas in the elementary 
functions. We came, however, to see that an integral can be consid 
ered from a higher point of view and that the integral of any con 
tinuous function always exists, regardless of whether it can be 
evaluated as above ; namely, the area under the curve yields precisely 
the integral. Moreover, this area may in any case be approximated 
to by Simpson’s Rule, Introduction, p. 344. 
In the case of the differential equation 
(!) t =f{X ' V) 
the situation is similar. Suppose /(x, y) to 
be continuous throughout a certain region S 
of the (x, ?/)-plane. Then the equation (1) 
assigns to each point (x, y) of S a definite 
direction, namely, the direction of the line 
whose slope (dy/dx) is f(x, y). We can think of these directions as 
indicated by short vectors drawn at the points. 
To integrate equation (1) is to find a curve drawn in S, such that, 
* Here, as in so many other cases in physics, a thorough knowledge of Linear 
Dependence is indispensable for an understanding of the subject in hand ; cf. 
B6cher, Algebra, Chaps. 3, 4, or better still, B6cher, Annals of Math., ser. 2, 
vol. 2 (1901), p. 81.