XXXV— XLVI] Introduction Ixiii 



Criterion Type Equation to Curve 



«.. = A = 0, /S.>3 VII y = y,—L^^ 



(l+J=) (>-")• 



«:, = /Si = 0, /3o = 3 Normal y = y,e '^<'' (lix). 



K, = A = 0, A<3 Ila 2/ = y„(^l--j (Ix). 



K, = o A = o, /3,<i-8 lib y = yo , -*,„, (ixi). 



~ c tan~i - 

 e n 



/^,= l* V y = y,e--^l-''x-P (Ixiii). 



/c, > 1 < 00 VI y = i/^{x-a)"'^lx"'' (Ixiv). 



K,= x, i.e.20o^-S/3,-6 = O III 2/ = ^oe"^ « fl H-^j" (Ixv). 



/ *'\"'' / x\"'' 

 /Co < 0, i.e. negative. Belo\v/=0 Ii y = yo(lH — ) (l ) (Ixvi). 



>c,<0. Inside/=0 Ji y = y,[l + -J /V'ttj ••■('^^")- 

 K,<0. Above /=0 Ui y = y„- — — -— ...(Ixviii). 



For /o., <0,/=0 represents the biquadratic 



A (8/3., - 9A - 12) (i^, - 3/3,) = (10^ - 12A - 18)= (/8., + P^f . . .(Ixix). 



Type I is thus divided into three subclasses, limited range curves, J-shaped 

 curves and U-shaped curves. 



Diagram XXXV enables the reader at once to find the type appropriate to his 

 distribution, and Diagram XXXVI gives the same figure on a much larger scale to 

 indicate the changes that occur with large values of /3i and /8.,. 



Knowing the values of j3i and ySa the computer can fix his point on the 

 Diagram XXXV, but he may come so near a critical point or line, that one curve 

 may appear as reasonable as another. It is clear, for example, that in the 

 neighbourhood of the Gaussian point G, he might possibly use II, Ij, III, VI, V, 



* The branch of the cubic k2 = 1 with which we are concerned passes through the Gaussian point, 

 at which y = 00, and along this branch p is always >5. 



