lxiv Tables for Statisticians and Biometricians [XXXV — XLVI 



IV or VIII, and as all these types at that point transform into each other, the 

 forms actually deduced will be almost identical, however different their equations. 

 But there will be other occasions when doubt as to the use of the simpler of two 

 curves may arise; for example if /Sj = '8, /3 2 = 4-15, are we justified in using 

 Type III as simpler than Type I ? 



Now we have to remember that the variates &, /3 2 form a frequency surface, of 

 which the equation is 



M - x * ( ftl . H. _ afiftftftflA 

 Z = - e 2(1-R%f>j\2tf + Xp* 2^ s ft J (ha) 



and that the contours of this surface projected onto the j3 1 , /3 2 plane of 

 Diagram XXXV form a series of similar and similarly placed ellipses. Within 

 any one of these ellipses a certain amount of the volume of the &, /3 2 - frequency 

 lies, and therefore if this system of contours were properly placed round the /Si, /S 2 

 point on Diagram XXXV we could tell at once the probability that the given point, 

 owing to random sampling, should fall outside a given elliptic contour. 



The ellipse which has for principal semi-axes 11772! and ri772 2 , where £, and 

 2 2 are the principal axes of the ellipse : 



i - 1 (§L+£L- ?^A&\ rl n 



R\p t VSft 2 2ft 2 2ft 2ft / 



covers an area on which stands just one half the frequency, i.e. it is the ellipse 

 determined by the generalised probable error of two variates (see Table X, p. 24). 



The semi-minor axis l'1772!and the semi-major axis l'1772 2 of this "Probability 

 Ellipse " multiplied by \/N are given in Tables XLIV and XLV respectively, and 

 Table XLVI gives the angle in degrees between the major axis of this ellipse and 

 the axis of /9 a . It is thus possible to construct from Tables XLIV — XLVI the 

 " probability ellipse" round a given point /3,, /3 2 > and to test the area within which 

 half the frequency lies. If the probability required be not \, but much less, then 

 we note that the probability, that a point will lie outside the ellipse with semi- 

 axes X2j and \2 2 is P = e ~ ^ x . 



,_ -67449 

 Let \2 2 =1177v / iV r 2 2 x ^p (lxxii), 



or 

 and 



Hence 



Accordingly 



