66o 



NATURE 



[February 19, 1920 



data, resolves itself into choosing between (a) the 

 absence of the Einstein effect, but the existence of 

 currents of absorbing solar vapours moving away 

 from the observer on the earth, and (6) the existence 

 of the Einstein effect, together with solar currents of 

 about the same magnitude as before, but in the oppo- 

 site direction. The question thus involves an exten- 

 sive knowledge of solar meteorology. There is, I 

 fear, no immediate prospect of a rigorous solution 

 of the problem of the displacement of the Fraunhofer 

 lines with our present incomplete knowledge of the 

 conditions necessary for the production of the 

 cyanogen-band spectrum and with our present limited 

 information regarding the circulation in the sun's 

 atmosphere. W. G. Duffiei.d. 



University College, Reading, February 8. 



Statistics of Valour and Service. 



In the Weekly Edition of the Times for Novem- 

 ber 28, 1919, the following statistics relating to 

 decorations awarded for services in the field are 

 detailed : — 



V.C. D.S.O. M.r, D.C.M. M M. 



Decoration 576 8,862 36,707 24,391 114,517 



ist bar ... 2 69s 2,932 468 5,719 



2nd bar ... — 70 167 9 180 



3rd bar ... — 6 4 . — i 



-An analysis of these figures, with a consideration of 

 the results arising from such an examination, may bo 

 not without interest. 



The figures may be reclassified as follows : — 



Number of Individuals who have Won Decoration 



V.C. D.S.O. M.C. D.C.M. M.M. 



With o bar ... 574 8,167 33-775 23,923 108,798 



„ I bar ... 2 625 2,765 . 459 5,539 



,,^ 2 bars ... — 64 163 9 179 



,,' 3 bars ... — 6 4 — i 



In making this reclassification I have assumed that 

 all the decorations were won subsequent to July, 1914. 

 There may be exceptions — as, for instance, in the case 

 of Capt. Leahy, V.C. — but as the number of such 

 cases must be small, their influence may be neglected. 



In analysing these statistics 1 shall employ a type 

 of method which I have applied with considerable 

 success to medical problems, chiefly of an epidemio- 

 logical nature {Science Progress, 1914; Proc. Lond. 

 Math. Soc, 1914; and various papers in the Indian 

 Journal of Medical Research). The argument as 

 applied to the present case is as follows : I^t us 

 assume the presence at the Front of a communitv of 

 individuals, initially undecorated, who were capable 

 of earning the decoration in question provided that 

 opportunity offered and recognition came. Let v^ be 

 the number of individuals who at anv moment were 

 in the grade x — that \s to sav, who had received the 

 decoration with x-i bars. Let <t>J(t)dt be the prob- 

 ability that an individual in grade x mav during the 

 time dt pass from the grade x to the grade x+i. For 

 such a pas.sage to occur, both ooportunitv must offer 

 and recognition must come. The function «, allows 

 for variations in the probability of further attainment 

 being dependent on the degree' of anterior attainment. 

 The function f(t) is unknown ; it describes the ebb 

 and flow of the conflict. Variations in the number 

 of individuals in the grade x are composed of influxes 

 and effluxes. The degree of the former depends upon 

 the number of individuals who have already attained 

 to the grade x-i. and the degree of the latter upon 

 the number of those who are in the grade x itself. 

 Thus 



dv^ = (<^:,_ , v^_ , - <l}^7/;) f{t)dt. 



Let us assume as an aporoximation that <t>,-h + rx. 



Let fi, denote the mean grade, and n^ and /i, the 

 second and third moments about the mean respec- 

 tively. By differentiating these values according to 

 the time, and by making use of the above differential 

 equation, we find 



Ml = ^(^' - I ), F2 = -/'i^'- I ), M,s = /"i^' - > ) {2e" - I ), 



where 6 is written iorff{t)dt ; whence 



and 



Solving the differential equation for successive 

 values of x, and eliminating the unknown function 6, 

 we have, if we remember that the participating 

 population N was initially undecorated, 



In the present instance, as in very many epidemio- 

 logical problems, we are ignorant of the value of N, 

 i.e. of the number of individuals in the participating 

 population. On the assumption that our hypotheses 

 are applicable, we can, however, calculate its value 

 by inaking use of the above eliminant between the 

 moments which results from our hypotheses, and b\ 

 taking advantage of the fact that, in calculating the 

 values of moments about any selected grade, informa- 

 tion regarding the number of individuals in that grade 

 is unnecessarv, as in each case this number is multi- 

 plied bv zero. In the present instance, as we are 

 ignorant of the number of individuals in the zero 

 grade (i.e. of the number of the undecorated), we 

 emplov moments about the origin. I^t us denote the 

 second and third of these by /i'„ and n', respectivelv, 

 then by introducing the values n, = Nju,, n^ = 'Nix'^, 

 )i, = Nu',, into the eliminant we find 



The results of these calculations are indicated in 

 the following table : — 



Number of Individuals having the Decoration 



The correspondence between calculated and actual 

 values is good, as was in some measure to be ex- 

 pected, since the number of grades in the statistics 

 m no case exceeds four, whilst the number of con- 

 stants at our disposal is three. We may, however, 

 conclude that, for particular values of these constants, 

 our assumptions are sufficient to account for the facts, 

 and proceed to examine the significance which 

 attaches to the.se values in the light of our assump- 

 tions. The final test of the adequacy of the assump- 

 tions must clearly depend upon the reasonableness of 

 such inferences as mav obtrude themselves. 



These values are as follows : — 



c/i 



V.C. ., 

 D.S.O. 



M.C. .. 

 D.C.M. 



M.M. .. 



-0-5 

 +296 



— 0005 



+0-735 



— 0096 



0014 



NO. 2625, VOL. 104] 



N 



41,763 

 215,498 



240,477 

 1,103,730 

 1.077,444 



In the values of N we have the values of the par- 

 ticipating populations. To, recapitulate : They denote 

 the numbers of persons at the Front who were capable 

 of earning the decoration in question if opportunitv 

 offered and recognition came. The standard of the 



0042 

 0166 

 0023 

 01 13 



