40 Sven Wickseil 



TABLE V. 



Direction cosines of the squares 





Tu 



Ï12 



Tis 



fäi 



T32 



T28 



f 8 2 



T33 





— 1.0000 



0.0000 



0.0000 



0.0000 



— 0.9848 



4 0.1736 



+ 0.1736 



+ 0.9848 



A., 



+ 1.0000 



0.C000 



0.0000 



0.0000 



4 0.9848 



— 0.1736 



+ 0.1736 



+ 0.9848 



B, 



— 0.3090 



— 0.6737 



4 0.6714 



+ 0 9511 



— 0.2189 



4 0.2181 



4 0.7059 



+ 0.7083 



B 2 



— 0.8090 



— 0.4163 



4 0.4149 



4 0.5878 



— 0.5730 



+ 0.5711 



4 0.7059 



+ 0.7083 



ß, 



— 1.0000 



0.0000 



0.0000 



0.0000 



— 0.7083 



4 0.7059 



4 0.7059 



+ 0.7083 



ß 4 



— 0.8090 



+ 0.4163 



— 0.4149 



— 0.5878 



— 0.5730 



4 0.5711 



4 0.7059 



+ 0.7083 



ß. 



— 0.3090 



4 0.6737 



— 0.6714 



— 0.9511 



— 0.2189 



4 02181 



4 0.7059 



-f 0.7083 



ß 6 



+ 0.3090 



+ 0.6737 



— 0.6714 



— 0.9511 



f 0.2189 



— 0.2181 



4 0.7059 



+ 0.7083 



ß 7 ' 



+ 0.8090 



4 0.4163 



— 0.4149 



— 0.5878 



4 0.5730 



— 0.5711 



+ 0.7059 



+ 0.7083 



B* 



+ 1.0000 



0.0000 



O.oooo 



0.0000 



+ 0.7083 



._ 0 7059 



4 0.7059 



+ 0.7083 



ß. 



4 0.8090 



— 0.4163 



+ 0.4149 



+ 0.5878 



4 0 5730 



— 0.5711 



+ 0.7059 



f 0.7083 





I fi Qi tQft 



-p U.oUtfU 











— 0.2181 



+ 0.7059 



4" 0.7083 



c, 



— 0.2588 



— 0.2415 



4 0.9353 



+ 0.9659 



— 0.0647 



4 0.2506 



+ 0.9683 



4 0.25C0 



c 2 



— 0.7071 



— 0.1768 



+ 0.6847 



+ 0.7071 



— 0.1768 



-f 0.6847 



+ 0.9683 



4 0.2500 



c 3 



— 0.9659 



— 0.0647 



4 0.2506 



-f 0.2588 



— 0 2415 



+ 0.9353 



+ 0.9683 



f 0.2500 



c 4 



— 0.9659 



+ 0.0647 



— 0.2506 



— 0.2588 



— 0.2415 



4 0.9353 



+ 0.9683 



4 0.2500 



c 6 



— 0.7071 



+ 0.1768 



— 0.6847 



— 0.7071 



— 0.1768 



4 0.6847 



+ 0.9683 



4 0.2500 



c a 



— 0.2588 



4 0.2415 



— 0.9353 



— 0.9659 



— 0.0647 



4 0.2506 





4 0.2500 



c 7 



+ 0.2588 



+ 0.2415 



— 0.9353 



— 0.9659 



4 0.0647 



— 0.2506 



+ 0.9683 



4 0.2500 



c„ 



+ 0.7071 



4 0.1768 



- 0.6847 



— 0.7071 



+ 0.1768 



— 0.6847 





4 0.2500 



c 9 



4 0.9659 



+ 0.0647 



— 0.2506 



— 0.2ä88 



4 0.2415 



— 0.9353 



+ 0.9683 



4 0.2500 



C M 



+ 0.9659 



— 0.0647 



4 0.2506 



+ 0.2588 



+ 0.2415 



— 0.9353 



+ 0.9683 



4 0.2500 





4 0.7071 



— 0.1768 



+ 0.6847 



+ 0.7071 





— 0.6847 



+ 0.9683 



4 0.2500 



c„ 



4 0.2588 



— 0.2415 



4 0.9353 



+ 0.9659 



4 0.0647 



— 0.2506 



+ 0.9683 



4 0.2500 



In the cited work Charlier has published the values of x 0 , y 0 , v 20 , v u , v 02 , 

 v 30 , v 03 , v 40 , v 0i for each of his 48 squares. The values of the remaining moments 

 v 2i- v i^> v si' v 22' v i3 nave a ^ s0 Deeu computed and, though unpublished, kindly placed 

 at my disposal. All this work being already done, the most convenient way to 

 get at the moments of the compound squares, is to compute by the formulae giving 

 the moments of a frequency distribution that is made up of two component distri- 

 butions added to each other. 



We will give those formulae as they are of a very general interest. But as 

 they depend on algebraical developments of some length we give them without 

 demonstration. 



Denoting by index (n) the moments of a northern square, by index (s) the 

 moment of the diametrical square, and by n l and n 2 the relative numbers of the 

 stars in the two squares, we have:* 



vy = n t Vf/'" 4- n 2 v^ w , 

 n i + n 2 = 1- 



Applying here the formulae (45), (45*), (45**) we get the equations 



* Charliers values for v$ must he taken with negative sign if i is an odd number. 



