208 Scientific Proceedings, Royal Dublin Society. 



to be worth ascertaining wliether diagrams with, hyperbolas or 

 ellipses instead of the parabola would agree with tiie 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 Eyd- 

 berg speaks of as satellites, and which Kayser and Eunge 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 passed through any three assigned points. 

 We have therefore to determine the ellipse or hyperbola that 

 passes through three of the observed positions, and then to ascer- 

 tain how far the other observed positions lie from that curve. The 

 following lemma makes it easy to do this : — • 



Lemma. — When the ?/'s of a number of points are given (in this 

 case the successive values of 1 m), so that the accurate values of y"^ 

 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 



and the hyberbola 



b^ ^' 





furnish as their derived curves 



— + - = 1, and —.- Tz = ^> 

 a^ ¥ a? b^ 



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 ^, under it when 

 derived from a 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 PL xvi., when we substitute 



