Vol. 8, 1922 MATHEMATICS: 0. VEBLEN 193 
3. The differential equation (1.1) has a unique set of solutions de- 
termined by the initial conditions, = and dx^/ds = when s = 0. 
These solutions may be written in the form : 
x' = p'' + - ^, (ri^)p r i's^ - i (ri,,)p i^fes' - ... (3.1) 
in which the subscripts mean that the parentheses are evaluated for x^ = p\ 
i.e., at the point P. 
Let us now substitute / = ^^s and solve (3.1) for y\ obtaining 
/ = rP\x\x\..,x'') (3.2) 
This determines a transformation of the x's into new coordinates y'^,y'^...,y^. 
The y's are normal coordinates. They are determined uniquely by the 
x's, the point P, and the differential equation (1.1). They have the char- 
acteristic property that every curve 
y = (3.3) 
is a path, i.e., a solution of the equation, 
|^Vci/ff! = 0 (3.4) 
as^ ds ds 
into which (1.1) is transformed by the substitution (3.1). Moreover every 
path through P is given by (3.3). 
4. Substituting (3.3) in (3.4) we obtain 
CUee = 0 (4.1) 
in which the functions are evaluated for the values of y such that y = 
^V. This is more simply written in the form 
CL,y"/ = 0 (4.2) 
which is an identity in y\ Differentiating this with regard to s, we obtain 
y"/ + 2C,y"^^' = 0. (4.21) 
Since is arbitrary this gives 
ds 
+ 2Ci^ = 0. (4.22) 
by"' 
Multiplying this by y^, summing, and using (4.2) we obtain 
^^y'/f^Q. (4.3) 
