the Spectrum of Sodium. 507 



ends of the horizontal lines in the other monads should be 

 some curve of which the parabola is a particular case. This 

 may happen in different ways, but the simplest hypothesis is 

 that they are hyperbolas or ellipses. It appeared therefore 

 to be worth ascertaining whether diagrams with hyperbolas 

 or ellipses, instead of the parabola, would agree with the 

 observed positions of the lines in one of the other monads. 

 Sodium is the monad selected ; chiefly because of the peculiar 

 pair of lines that present themselves in the spectrum of this 

 element — which Rydberg speaks of as satellites, and which 

 Kayser and Bunge regard as probably belonging to a fourth 

 series of lines, of which they are the only term that has been 

 found. There seemed some ground for hoping that the 

 inquiry would reveal the true significance of these lines. 



A curve of the second degree, with the axis of x as one of 

 its principal axes, can be drawn through any three assigned 

 points. We have therefore to determine the ellipse or hyper- 

 bola that passes through three of the observed positions, and 

 then to ascertain how far the other observed positions lie from 

 that curve. The following lemma makes it easy to do this: — 



Lemma. — When the y's of a number of points are given 

 (in this case the successive values of m), so that the accurate 

 values of y 2 for the successive points can be obtained, we may 

 use, instead of the ellipse or hyperbola, the curve derived 

 from it by making the new ordinates proportional to the 

 squares of the old ones. Thus the ellipse 



2 T 7 2 — -L, 



and the hyperbola 



a 



d 2 b 2 



furnish as their derived curves 



-5 + 72=1, and -s — T5=h 



a 2 b 2 ' a 2 b 2 ' 



in which z must be positive. In other words, the derived 

 curve is the portion on the upper (i. e. positive) side of the 

 axis of x, of a parabola with its axis vertical. This parabola 

 passes through the ends of the axis major of the ellipse or 

 hyperbola. When derived from an ellipse its vertex is above 

 the axis of a } under it when derived from an hyperbola. The 

 parabola degrades into a straight line, if the curve from which 

 it is derived is a parabola instead of an ellipse or hyperbola. 

 Thus fig. 1 of Plate VI., when we substitute its derived 



