94 C. V. L. Charlier 



N 



N N 



(89) 



N N N ■ 



N N 



where N^^. - N. 



Eventually we may use, instead of iVg^ and iV^g, two moments of the fourth 

 order, say N^^^ and N^^, or, better, iV^^ — ^^ao^ — ^iV^g^. 



If the moments are taken about the means — m and n — of F{x,y), we 

 evidently have 



Multiplying (87) by x'^y' and integrating, we now get the following 8 equations: 



N = N, +N„ 



0 = N^{m-^^ — m) -\- N^{m2 — tn), 



0 = N^ {n^ — n) + iVg {n^ — n), 



NN,, = N, (a,^ + [m -mr) + N,{a,'^ + 



NN,,=N,{^,' + (n, ~ nr) + N,{.-/ + {n,~n)% 



(90) 



iViVj^ = N^ (m.^ — m) [n^ — n) - -\- N, (m^ — m) {n, - 



NN^Q = N^ [3-^1^ (w^ — m) -\- (m^ — mY\ + N^ [Sct.,^ {m,—ni) + (m 

 iV^iV^o3 = N, [S^.-^v, -n)^ (n, - n)'] + N, {3y./{n, - n) + {n, - n 



2 



These equations are to be solved regarding the 8 unknown quantities (89). 

 The equations may be eoinewhat simplified through introducing the difference 

 — mg and — instead of — w, — m, — n and — n. 

 We have, indeed, 



— m = ^ (Wj^ — mg), 



(91) 



N 



n^ — n^ (»^ — Wg), 



(91*) 



N 



n, — n = — ^(n^ — n,). 

 Substituting these values in the 5 last equations above, we get 



^20 



(92) 



-^1 2 



— - a 

 N ^ 





N, . 





N,N, 

 NN 



^1 2 



— i a 

 N ^ 



+ 



^,2 



N ' 



+ 



N,N, 

 NN 



N N 



(92*) N^, = [m, - m,) (n, - «g), 



