280 On the Probable Errors of Freqrtency Constants 



Genekal Problem. Toßnd the prolxMe error of aiii/ coiiilanl i\ of a frequency distribiUion 

 and the correlation of any txro coMtants c< and C(.. 



Any coDstant will be a function of the mean h and the momeiits ;ij, ^13, ^J.^, ... /i, ... about 

 the mcau. Hence : 



Ci = (j){h, /lo, ^»3, f»4 .../i, ...), 



and if the errors be siuall as compared with the qiiautities in wbich thcy occur we cau write 



where i^n, </>^,... are the diflerentials of tlic known function with rcgard to the sulxscript 

 quantities. Hence thcir numerical vaiues are known and we may write, if X deuotcs a kiiowu 

 numerical quantity : 



5Ci = Xl8/4 + X28/'» + ^3ÄM3 + 



Square, suni and divido by tlio iiumber of ca.ses and wo have : 



+ 2\{K2<T^(T^,ri^, + 2X,X30-ft(7-^,rj^ + 



=x.w+.S(x,vv,) 



+ 2X10-4. S'CX^o-m/äx,) 



+ 2,5'(X,X,.<r^^>r^,,r^,^,.) (liv). 



Our forniulic (vii), (viii) and (xii) give all the Standard deviations and correlations required on 

 the right-hand side. Similarly 



5ci' = X i'8A + Xo' 8/i2 + ^3^t^3 + 



Hence : 



8ciSci=X,X,'SÄ 2 + Ä ((X,X,' + X'5X,) SÄ S/x,} 



+ 5 {(X,X V + X',X,.) V<z 8M- 

 Or: 



+ S {(X,X,' + X',X,) a^rr^^r^^^ 



+ 2Ä'{(X,XV+X',X,.)'^M,'^M,.'-M,'V( (sv). 



Equations (xiv) and (xv) give ono the mcnns of deterniining the iirol)able errors and the 

 correlations of the constants of any frequency distribution whatevcr. 



Theoretically the investigation is quite straightforward. Practically it is often very laborious. 

 Thus if our con.stants only go as far a-s the fourth monient, we .<<hall .still want all the nioments up 

 to the eighth to detormine their proliablc enors (sop equations (vii) and (viii)). Hut t<i calculate 

 the first i-iijht nioments i.s a laboriou.s bit of arithmetic. It is often conveniontand sutticient not to 

 go further than the third or fourth momcnt, and to exprcss by .sonio as.sumed form of frequency 

 curve, the remaining moments in terms of tho earlier. This can always be done* if the frequency 



curves eatisfy the relation — ^ = " '" ., for in these cases there is a rciluction fomiula 

 y dx Cj+c,x + e»r' 



giving the n"" moment about the mean in terms of the n - 1"" and n - 2"' nioments. For example 



we have : 



• Phil. Tram. A., Vol. 186, p. 381. 



