IQ22.] P. C. Mahalanubis : Analysis of Stature. 51 



In our case we have, for 50 mm. unit, of grouping, 



p 2 = 182 10 42 



f* 3 =- °'47 7^ 43 77 

 / u 4) = ii'94 16 22 08 



M 5 = - 77 8 2 3 

 Thus X 4 =- 5-97 91 20 55 



A 5=~ 2 75 S3 07 24 



After some laborious arithmetical calculations 1 we find the 

 fundamental nonic : — 



P^ + 6'97 56 40 642V + 0-34 24 99 3oA 2 6 + t.3'57 7 6 59 77Pz 

 + 7-82 66 28 242V + 17*63 90 36 20pi A — 117 49 41 i3/> 2 * 

 — 0*40 73 08 98^2 — o'oi 19 04 62 = 0. 



I next form the nine Sturm' s auxiliary functions, retaining 

 four figures in the decimal. 



fA*)=. 9P*+ 48-82 95^ 6 + 2-o5 5o/) 2 5 + 67-88 83/> 8 * + 31-30 65ft, 1 



+ 52-91 yipr?- 2-34 9 9p 2 -0-40 73 

 / 3 W=-i-55 i2^ 7 -o-ii 42^-6-03 46^-4-34 81^-1175 93^* 



+ 091 38^ 2 ' 2 + 0*36 20^2 +o-oi 19 

 /s(*) = -13-86 58^ + 2i-3i52/>2 6 -4'68 37^-36*21 8o/> 2 8 - 54*86 28^ 



+ 2-28 60^2 + 0*40 73 

 /*(*)= + i-66 93^2 6 -o-54 78^-0-90 53^ 3 ~9' 22 89^ + 0-09 56^ 



+ o-o6 15 

 fb( x )= + 6-70 i8/>2* + i03'78 45^ 2 3 -38'6i S4p 2 % -i'S^ 6yp% + 0'2i 04 

 /«(*)= -417*52 59pz 3 + 160-89 n^2' 2 + 7' I 8 96^2-0-89 03 

 fi( x ) =-2-48 49^2^ + 0-01 94^2 + 0-01 64 

 /*(*)= ~5"66 47^-0*15 

 U( x ) = -o'oi 41 



We can now find the number of real roots from the changes 

 of sign in the Sturm's functions. 





+ CO 







— c 



/(*) 



+ 



— 



— 



/.(*) 



+ 



— 



+ 



M*) 



— 



+ 



+ 



h(x) 



— 



+ 



— 



h(x) 



+ 



+ 



— 



h(x) 



+ 



+ 



+ 



W) 



— 



— 



+ 



fM 



— 



+ 



— 



/»(*) 



— 



— 



+ 



U*) 



— 



— 



— 



- My best thanks are due to Prof. j. M. Bose M.A., B.Sc. of the Mathematics 

 Department of the Presidency College, Calcutta for his kind help in checking the 

 arithmetic in many places. 



