6 CARL STØRMER. M.-N. KI. 
We attain our end most rapidly, however, in accordance with Pro- 
fessor Runge’s suggestion, by regarding $ as a function of U, and con- 
sidering the corresponding curve, 
f= f(0) 
where U is an abscissa and & an ordinate!. Let A, B and C be the 
: — I I 
three points whose abscissæ are =, å and as and let AA’ be 
parallel with the axis of ‘abscisse ; then 44! = = and A’ C= 
(AU) 
I Bs 
Thus if, instead of the direction of the tangent in the point B, we 
take as approximate direction AC, we obtain 
n 
tang=-—D 
p="D, 
ß being the angle made by the tangent with the axis of abscissæ. 
On the other hand, tan =f’ (U), where the value of U på 
nN 
Now since 
7 å EON 
f (U) F4 aU 2U 
oN 
1 I am indebted to Professor Runge for information respecting this method of procedure. 
At the beginning of March, 1907, I had sent a description of the method with proofs 
to Professor Mittag Leffler, asking if it were known. He sent it to Professor 
Runge in Göttingen, who drew attention to the above simpler proof. 
