V 



568 Prof. K. Pearson on Lines and Planes of 



Hence : 



V <Tx\ 2 <?X 2 CT;r 3 * (Tx q 



which may be written : 



( q"2 1- 1 J os x gx x + r 12 x 2 <rx a -r r l5 x 5 crx 3 + . . . + r x qX q a Xq = 0. 



Or system (xvii.) may be replaced by : 



I _ ____ \ + r ]2 x 2 o-x 2 crx l + r n x z <r X3 (Tx l + . . . + r^Xqaxq^ = 0, 



(ABA 

 1 — ~ 2 — \ + Tm^^o^ + . . . + r 2 qXq(Tx q (Tx 2 = 0, 



rtfXiiTxqVxi + r^XzVxjTxq + riqXtfrxjTx. + . . . +a^ 2 o- 2 o: ( 1 5 — 1 =0 



\ & Xq / 



. . . (xviii.) 



For multiplying (xvii.) by x 1} x 2 , . . . Xq respectively and 

 adding, we find 



or if R be a maximum or minimum value of u, Q= — 1/R 2 . 



Now compare these equations with those we obtain for the 

 directions and magnitudes of axes of the ellipsoid of residuals: 



g\xi 2 + cr\x 2 ' 2 + . . . + (T* Xq Xq n + 2r l2 ax l (Tx 2 x 1 'x 2 l + ... 



-f 2rq-iq<TXq- 1 (Tx q X ! q-iXq =€ 4 . (xix.) 



These are : 



xj<r% ( 1- ^ a <i x J+r 12 x 2 'crx 2 <Tx l + . . .+r iq x q l <Tx q <rx l = Q, 



r n x i 1 <Tx l o-x 2 + x 2 'a*x,(l— ft>2 G * x )+ • • • + r%a:q ! <rx q <rx t = > 

 r 19 ^ 1 V« 1 o-.r 9 + r n x 2 ] <Jx 2 <Jx q + . . . + xq'o^xql 1 — -^ 2q _ 2 ) = 



..." (XX.) 



Now eliminate the x's from (xviii.) and the x r s from (xx.) 

 and we have precisely the same determinant to find 



