Karl Pearson 
481 
Lastly : o-;' = jx., ( 1 + q^) - A ^3/70 - 1 V /tx^Yi l 
„ > (vii')- 
Equations (ii), (iii), (iv), (v), (vi), (vii) and (viii) form the complete solution of 
the problem when we make no approximations whatever*. 
If, however, fi^ =^3 = 0, then, the two components being equal, we havef: 
„j = = -i iV 1 
7i = - 72 = ^//^2 \ (ix). 
It will be seen that it is needful in order that the solution may be real that ^1 
should be positive or ySo < 3, i.e. the total frequency should be platykurtic. Now 
let us suppose that the values given by (ix) are a first approximation and that we 
need a second approximation in which the two normal curves will be unequal in 
frequency, mean and standard deviation. Write : 
n = hN, 7 = V/L^A <7 = V;^,(1 - (ixf-, 
and suppose : 
?h = ?? + 8//] , = 11 + Sn.;, , 
7i = 7 + S71 , 72 = — 7 + 872 , 
cTi = cr + ScTj, 0"2 = o" + 8(7.^, 
where the differentials represent small quantities of which the squares and 
products may be neglected to a second approximation. 
Our equations arej : 
??1 + 112 = ^> 
"i7i + "272= 0, 
^1 (71" + o"i''^) + (7/ + = 
«i (71^ + 871 o-j^) + v., (72'' + •^720-2') = JSffJ-i, 
«j (7/ + 671 + + "2 (ji + ^>y2(^^ + -W) = Nix, , 
Hi (7,5 + 107iVi=' + 1071 o-/) + /i2 (72' + IO72V2' + 1 572O = Nfi,. 
We now differentiate these and after differentiation put 
rii = 11.2 = 11, 7i = — 72 = 7> o"! = c-i = o"- 
Hence we find : 
- — 8712 (x), 
11 (S7, + S72) + 278??! = 0 (xi ), 
2??7 (S71 — S72) + 2na (Sa^ + Scto) = 0 ( xii ), 
tin (S71 + 872) (7- + a^) + 2S»i7 (7' + -V') + 6nay (Ba, - 8a:,) = Nfx, (xiii ), 
7 (7= + 3cr2) (g^^ _ g.y^) ^ 3^ + (gg.^ gg. ) = 0 ( 
« (S7, + §72) (07^ + 30y-o-- + lr>a') + 28niy (7^ + IO7-0-- + loa-') 
+ n (8(T, - 8a,) 20ya (7^ + 3a-)=Nfi, (xv), 
* They are, in a somewhat better form, those originally given by me in Phil Tranx. Vol. 185, A, 
1894, pp. 71—110 ; see Equations (14), (15), (18), (19), (27) and (29) of that memoir, 
t Loc. cit. footnote, p. 91. 
+ Loc. cit. p. 82. 
