r, 
PROFESSOR K. PEARSON ON THE INFLUENCE OF NATURAL 
must be the correlation surface for and for all values of x^, ;r^, ... x„. AVe may 
write it in the form 
y , — 2C,2 .Yj«2 + C\|Y2^ 
2 = Co expt. - J- A ph 
'^'12 
. . (foii.). 
Comparing this with the known form of the correlation surface for two variables,'^ 
+ ^V • • 
^ expt. - i ^ 
27rcr^o-2\/1 — 
yp - ^’i . yp 
1 O \ O I ( 
1 — 5 12 " \cri" cTtO-, cr.v 
' 1 “ 2 
we have at once 
T' r< _ f' 2 
n- 2 O _ r 21 — kllkgg _ klA 
0-1 fool- , 
cr, 
fo(l-fop) = 
n n T' 2 
'•-^11'^00 '^■[2 
AC„ 
Whence 
Or, generally : 
'OsO’iCTo — Oi'^/A .......... ^x. 
” ^is/v^CiiGoo.(xi-)’ 
o-i" = 0 - 2 " = Coo/A.(xii.). 
^J}J ^pq!^pp^qq .(xUl.), 
(jp — \/V>ppI^ .(xiv.). 
Thus correlations and variations are fullv determined in terms of the discriminant 
and its minor for the constants c^, c^o, C\q. - ■ ■ <~‘pp, Cqq-, <^pq, • • • Cm- 
We have next the inverse proposition to find the c’s in terms of the r’s and cr’s. 
We have, by well-known propositions in the theory of determinants : 
CiOii + C12C12 + C13C13 -f . . . + Ci„Ci« = ■A. 
+ C’r’Ofo + C 13 C 23 + . . . -h Ci„C 2 „ = 0 . 
Cllf'31 + O12O32 + C13C33 -}-...+ ei^C'S/i = 0. 
Or, 
+ 2 + C 13 C + 
Cl 1 erf” + Ci2^ \ocr^(y'o + Cis-l’isO-iO-s + 
ClpwiCriCTo + Ci2Cr2‘' + Ci^)‘2s(r2(X3 + 
Clir3iCriCr3 + Cio')'s20'20'3 4“ CisCTs" -|- 
+ Cl,(C,„; - 0. 
• • “h Ci„ri„crio-„ = i, 
• • + Ci;,r.2„cr2cr,, = 0, 
• • + Ci„r3„o-3cr„ = 0. 
Hence solvino- 
o 
CiP^jCTicr,, -{“ Cior^oCTocr,, -p Ci3'?’„3(T;jCr,; + . . +Ci„cr„'— 0. 
Cll — ►^ii/S, Cirj — felj/b. 
* Pearson, ‘Phil. Trans.,’ A, vol. 187, p. 264. 
(xv.). 
