John Blakeman 337 
Adding together equations (xiv), Cxv), (xvi) and (xvii) multiplied by — 2, 
we get 
'"'^ O-iE^ PnO-x 
_ ^'^p,,'^''M^p„'' 3i ^^'^x^'^iiz-^'^x'^J/ (xviii) 
But we have as well-known results 
V 2 _J P22 — -9 2 _ PiO~P-20^ ('YiY^ 
"^^"x^p^.-^^xv^r jvv^ vxx;. 
Hence, substituting in equation (xviii) the values given by equations (viii), (ix), 
(x), (xix) and (xx), we get 
S 2 ^ ^4 - ^2^ 1 - P22-P 1I j^40-P2 o' Psi-P20Pn K-Q-x''^ -. 
N't)' ^W' ^Np,,^ Np.^.pn 2Na^'\, 
- {^13 -puK + 2iJ„(r/ (1 - X,] (^xi), 
which may obviously be re-written 
V 2 ^ 1 /I _ 1 ^ 4. P^o - ^P-io" JPai - 3^20^1 1 , [ P22 - 'ipn _ 1 - r^ ] 
This is the complete formula for to obtain which has been the first object 
of the present investigation. The formula has been thrown into the form given 
as equation (xxii) because each terui as there exhibited vanishes for a normal 
distribution, since for normal surfaces we have 
Again for a normal distribution 
_ ray. _ 
Hence N\, = S (3/. - y)*] = . S {n, (w, - x)'} 
'TO-. 
a-x* 
,3_. 3 
= "iN . r-a,f X raxOTy = 3iV^ . \o . Xn , 
NX.^ = S (x, - xf (y. - = . max' 
ax 
= Wr-'ay-ax" = SNX, . a^". 
Further %i = %:) = 1, while rj = r. 
Biometrika iv 43 
