546 



NATURE 



[October 2, 1890 



additions of light to the constant light of the star. These 

 additions must be shown as such. 



It is very important that I should point out that for 

 this method of direct integration to be adopted a scale of 

 light units must be employed, for the reason that the 

 amount of light which is sufficient to produce a change of 

 a magnitude in a faint star would only produce a change 

 of a fraction of a magnitude in a brighter star. 



Taking the light of a star of magnitude m as a unit, and 

 using the formula 



Lm-« = {2-5i2)»L,„, 



in which Lm represents the light of a star of magnitude 

 m and L,„-„ the light of a star n magnitudes brighter, 

 we get — 



The amount of light to be added for the different 

 magnitudes will therefore be as follows : — 



Additions for one magnitude = ( 2'5i — i )Lm. = i'5l L„, 

 „ next „ = ( 6-31 - 2'5i)L„ = 3-80 L,„ 

 .. » M = ( 15-85 - 6-3i)L™ = 9-54 L„. 



M .. ,. = { 39-78 - i5-85)L„ = 23-93 L« 



„ M » = (I00-02 - 3978)L„ = 60-24 L,„ 



It is obvious that these differences are in exactly the 

 same proportion to each other as the numbers represent- 

 ing the light of the stars of different magnitudes, and if in 

 our diagrams we take a certain length of line to repre- 

 sent the added light equivalent to one magnitude, about 

 2^ times this will represent the added light for the 

 next magnitude, and each succeeding magnitude will 

 be represented by a line i\ times as long as the preceding 

 one. A scale of this kind must be adopted in integrating 

 the effects of two sources of added light, for the reason 

 already stated. 



06 12 18 



Fig. I.— Hypothetical curve in light-units. 



Fig. 2. — Hypothetical curve in magnitudes. 



Thus while the amount of light to be added to a sixth 

 magnitude star, to take an instance, to increase it to the 

 fifth is I5 units, the number of the same units to be 

 added to a fourth magnitude star to make it a third is 

 9|. Hence the 14 units which raise a star of the 6th to 

 the 5th magnitude — that is, one whole magnitude — would 

 only increase a fourth magnitude star by about one-sixth 

 of a magnitude. 



To graphically represent what happens when by 

 cometary action a star is raised three magnitudes above 

 magnitude tn^ we get, in the above light-units — 



I -512 additions for one magnitude 

 3-80 ,, ,, the next magnitude 



9-54 ,. „ „ 



The sum of these numbers = 14-85, represents the 

 added light. 



The plan on which the following curves have been 



NO. 1092, VOL. 42] 



drawn will be gathered from the table given below, which 

 shows how on the above basis the light-units and magni- 

 tudes correspond : — 



