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PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



Our object was then to find such pure umbral equations connecting all the 

 " physical " constants with the algebraic constants. Their products will then give 

 the error correlations of all the " physical " constants in terms of the correlations 

 already known between the algebraical constants. 



For example, multiplying the above equation for x$ by x/,, X> X" X- we nave > since 

 X*X = B 0> xX.- = R,-, &c., are already known, 



= T'969,2668, 

 =1-572,0115, 

 = T'608,8746, 

 = 1-925,8379. 



R /10 = '9317, actually log ( ] 



E,, = -3733, ,, log R ( ,^ 



R,^, = -4063, log R,^, 



R* = -8430, log R^ 



It was these logarithms, of course, which were used in the further calculations. 

 Since h is measured negatively (i.e., towards dwarfs, x is positive), we must write 

 for transferring origin to the mean 



x' = x + a tan <f>, 



where a tan ^> is the distance between the old origin and the mean, or if m be used 



to represent the mean we have 



m = h + a tan </>. 



Hence we find the umbral equation 



S,,,X lrt = Antl. '232,4493x4 + Antl. I-822,3l79x + Antl. -I29,4233x*. 



Hence we determine 



2,,, = -0549, 

 and the pure umbral equation 



X,,,= Antl. 1-492,5897x4+ Antl. T082,4583x + Antl. r389,5637x*. 



In precisely the same way all the other " physical " constants, i.e., the standard 

 deviation, <r, the mean frequency, ?/. : , the distance between mean and mode, d, and 

 the skewness, Sk., were found, and the uuibral equations investigated. It is only 

 necessary here to give the results. 



