Researches into the theory of probability. 35 



The solution of the above equations is dependent on a certain nonic, given 

 by Pearson. 



We commence with the elimination of the quantities x^, and by means 

 of the relations 



X^ Î 



(39) {ill — ih)'h = — y2^ 



(?/i — :'/o)-^'2 = Vi- 



We then obtain the equations 



?/,?/,[3x-, — 2(?/j + 2/2)] = ^.3. 

 y\y, [3,ii - 2 {y\ + y^y^ + = C„ 

 yi//2[ï5>^1(//i 4-^/2)- 20,i;j(//3 + yj,^^ + y/^) + i\{y\ ^ y\y^ + y/,?/^ + ^^)] = 



Putting 



« = ?/2 1 



(40) 



= ;'/iJ/2(:'/i -f ?/2). 



we obtain the fundamental equations 



(41) \ 2«3 + 3C,?« + 4C2 

 [2(i,;-C3)'-6H=' + 3C,« + 3Cg. 



Eliminating w between these equations we obtain the nonic of Peakson: 



0 = 24;*" + 84Q u'^ + 36C| + T2'C^ C, + ÙOÇ^ 



(42) — 18C| + 444C^^ C.n* + (2S8C^ — lOBCC^Cs + 27Ct) ?r' 

 -((33C2CI + 72C3gn" — 96Ctq« — 24C«. 



When a root of this equation is found, we may calculate the corresponding 

 value of ni from either of the equations (41). The values of y^ and y.-^ are then 

 equal to the roots of the equation 



(43) ^^-l^y + n = 0. 



The value of x^ = x.^ is found from the equation 



(44) -dux^ = '2/0 + C3. 



Finally we get the values of ,îj and from (39). Tliese equations are all 

 linear with exception of (43). For obtaining real solutions from this equation it is 

 necessary that the inequahty 



W^" — 4:U^ > 0 



is fulfdled. It may also be observed that for the reality of a solution it is neces- 

 sary that the resulting values of of and a| — obtained through the first tw,o equa- 

 tions (36) — should be positive. 



It is here supposed that we have solved the nonic (42). The solution of an 

 equation of the ninth degree, where almost all powers, to the ninth, of the un- 



