L. ISSERLIS 
409 
The case \' = fi' = v' = \ = 0 is proved in the same way. There are three cases 
of this type as well. 
Next take the three cases of type 
X = /X = 0, 
\' = fM'= 0. 
Equations (79) — (81) become 
(79) R=R', i.e. v"- = v'-, 
(80) y^'v'ry-^ - 4.r,,y (v.YrV'^ (- v')}, 
(81) y,'v*r,.J^ = 4>{- r.,,yv) {- y/y.^v- (- v)}, 
(80) leads to (7/''V/ + ^^xyli) l/'i''' = 0. 
We have already shown that the first factor is positive, 
and hence i> = 0, 
and these values satisfy (81). 
The three cases of type \ = /j, = /j,'=v'=:0 lead to 
(79) v'r%, = 0, 
(80) X''r\, = 0, 
(81) (72-2^2 _ ,y.,'2;,^^'2) ry," = 4 {r,i^\' - r,„jv) {- y.!y.?v" (r„X' -v) + y^y^-X''^ {r.x,v - \')], 
whence \' = v = 0. 
There remain the three cases of type 
\ = v= 0, 
=v' = 0. 
Here (79) is satisfied identically. 
(80) becomes (V- - 7,>=)= = - 4/>A* {7/ (^'' - 7i>') (- m)!- 
(81) - y;^-\'-^Y r\, = 4/vV (- 7, ifi^ - 7,A'^) ( - X')}> 
which reduce to 
(V- - y^-fx-) {r-y^X'- - (7,-r-,,,, + 47/ r.,.^) = 0, 
(/A= - 7;^ \'') [ r\,,j? - i^:^ + 47, t'y,) \'-^\ = 0. 
The only common solution of these equations is 
V = ^ = 0. 
We have thus accounted for all the fifteen possible cases, 
(iii) Three regressio7i curves linear. 
In six cases out of the possible 20 cases the linearity of three only of the 
regression curves involves the linearity of the remaining three. 
