A. Khind 
387 
A distinction between the directions of the major and minor axes is readily 
made when we note that a contour of (A) is inscribed in a rectangle of sides 20-^, 
and 2ap^, and that the major axis of such an ellipse is inclined to the greater side 
at an angle less than 7r/4. 
For if aaf + 2hxy + by^=l be an ellipse inscribed in a rectangle sides 2q and 2p 
having the greater side 2(/ parallel to the axis of x, and centre as origin, then 
Again a cos^ 6 ■\- h sin 26 + h sin^ 0 = \ , 
ib - a) sin 29 + 2h cos 26' = - 4 • 
For maximum or minimum r 
t^n2e,= ~^\ 
b - a 
2 (b - a) cos 2^1 — 4/i, sin 29^ = r, ^ at a maximum or minimum, 
cos 29 
2 d? 
{b-af + U? __2^d?r_ 
~~(b - a) " ry^^ d9' ' 
dh' . . 
Hence is negative when cos 29 is positive, 
.". 9 for a maximum is < 7r/4<. 
2 Vl -ig^^,3,^(r^, o-p.^ 
If sin 7 
then ^1 = V o-^^_ + cr^^^ sin ^ 
X2 = V o"^^, + o''^02 cos 
From these formulae Sj and So were calculated and the values of l"l77Si and 
1"17722 tabulated in VII. and VIII. respectively. The number must of course be 
multiplied by the factor -67449/^^. 
Since the population outside an ellipse of semiaxes kSj and is Ve"^"', it 
follows that, if an ellipse of twice the above linear dimensions be drawn, the 
probability that a point lies outside it is 1/16. 
If an ellipse be drawn with semiaxes k%i and /cSo, the probability P that a 
point lies outside the ellipse is P = e^^'^ Let 
