56 Records of the Indian Museum. [Vol. XXIII, 



This equation involves /**. We can however transform this 

 equation to the ^-variables. 



Dividing throughout by /V, we get 



(ft-3)-S*-(^-^)T-^(^-3^)=o 



V / /V 5 vv f'z'v-i 15 V ^2 i>2 / 



But /? £ = m/ /<2 2 and 04=^/^3 



Changing to the ^-variables and putting * = w ^ we get 



(*-s) 



5 15 



Thus — ^ _ i(^-5fe)±V / V5(^-5^)^A(^- 3)(5 ^ 2 -3^ t ) 



/*2 2(02-3) 



The condition for a real solution is that 



!( ft - 5&) > A/,V(ft - 5ft) z + T + 5 (& - 3)(5/5 2 a - 3ft) 

 Squaring and substracting 



o>^(ft-3)(5ft 2 -3ft) 



Pearson has shown that it is necessary that w l and w. z should 

 be of the same sign. 



The necessary condition for real solution becomes : — 



For lepto-kurtic curves, ft - 3>o or ft>3, it is necessary 

 that 3ft should be greater than 5ft*. 



For platy-kurtic curves, ft — 3<o i.e. ft < 3, the condition 

 is that 5ft a must be greater than 3ft. 



With ungrouped distribution it is almost impossible to find ft 

 directly. We can however find ft in terms of ft and ft, from 

 Table XLII (b) } p. 78 of Tables for Biometricians and Statisticians. 1 



We have, ft = *o6 87 56 



ft = 3-50 46 



,* o /-> 68 756 _ 



For ft = 3*5 ft = 23-72 89 + ^-x [2-0142] = 25-11 37 



100 000 



A = 3'5 



• 



4 =2372 89 + 



4'G 



31-00 + 



ft = 3'5^ 46 



4 = 25-11 37 + 





= 25-23 60 



68 756 _ „_ 



x [10-766] = 28-40 23 



100 000 

 46 



5000 



x [13-28 86] 



Wc have greater than 3, and 30 4 greater than 50 8 * hence 

 hall obtain a real solution. 

 The quadratic is 



•50 46**.— 1*54 26# + o*95 31 27 = 



I . K Pearson: "Skew Correlation and Non-Linear Regression", p. 8 



l)i pei < ompany Research Memoirs). 



