(51) 



40 . . c. V. L. Charlier. 



Choosing the mean of the given frequency curve as the origin of the coor- 

 dinates, we obtain through muItipHcation by 1, and x^ and adding the 

 equations of condition 



!J'U = '''1 + ^2' 



0 = X ,r'K {■'■ - + Ic, X x% {c, - .t), 

 [x, = Ic^ Ï x^ {x — c J + Ic^ I «l;« — .r), 



= i: ^i^j (,T — c J + /.-^ X ,7.-^ '1^3 (C., ~ X). 



Now we have 



= c;ï'K + (î)cr'i:!/^i + a)cr^i:2/Hi + - ■ -, 



and in lilîe manner 



But 



S']^ =1, 



i:/-]> = x^ + x, 



and hence we have 



Y.x''^,{x - cj = cf + 2c,X, + XÎ+X,, 



v,^3,pj,, _ _ ,3 3,0 + 3., (Xf + X,) + X? + 3X? + X,, 



and corresponding expressions for ^x"<]^.^{c,, — x). 

 The equations (51) thus take the form 



'•''0 = ^''i 



[X, = \ [c\ + 2c, X, + X^ + XJ + K [cl - 2c, X, + X| + XJ, 

 '^-3 = ^4 + 'ic\ K + 3q(X? + XJ + X? + 3Xf + X/l 

 + h [cl — 3o| X^ + 3c,{Xl + X^) — X3 3X| — X,]. 



From the first two equations we get 



(52) — ^1 — — ^2) ^'1 = + l^'o (('2 — K)^ 



(Cg — Xj - X,) = — ix„ (cj + Xj), 



whicli expressions substituted in the latter two equations give us the relations 



V2 (^^2-^1 - K-K) - ic--2-K) + ^c, X, + XJ + XJ - (c, + XJ [ci -2c,X, + X| + XJ, 

 ^3 (^^'2 - ^1 K - K) = {c.2-~K) [c? + X, + 3c, {X'i + \) + X? + 3X? + XJ 



+ y [c|-36iX, + 3c,(X| + XJ-X3-3X|-XJ. ■ 



I do not know, if these equations can be algebraically solved (h. e. reduced to 

 the 4"' degree). They may be numerically discussed, though somewhat laboriously. 

 It seems, however, advisable to take another course. 



