Theory of Probabilities* 357 



xsyt 

 ~AB1 

 _ xs'yt' 

 ""ABE 7 ' 



where, to avoid multiplication of symbols, I have used A'B'E, 

 &c. to denote the probabilities of the compound events A'B'E', 

 &c. if these probabilities should be sought for. 



We have thus five equations to determine x, y } s, t 3 u; these 

 equations give 



x's'ty xst'y' s't' 

 u—ap "~" u—/3q ~" 1— u 3 

 and 



dsH 1 s'y't' stxy 



a! + ap — u fi' + (3q—u up + j3q—u ' 

 and we have thence 

 (u — ap) (u — fiq) (1 — u) = (a! + up — u) (/3 r + fSq — u) (ap + fiq — u) . 



Putting, for shortness, 



ap — a, 

 /3q = b, 



cJ+apssf, 



ap + fiq = h, 

 the equation in u is 



— (u — l)(u—a)(u — b) 



+ (u-f)(u-g){u-h)=0; 



viz. it is 



u\l-{-a + b—f—g — h)~-u(a-\-b-\-ab—fg—fh—gh)-\-ab-fgh^O; 

 or, since h = a + b ) this is 



u*(l-f-g)-u(ab-fy + h{\-f-g)) + ab-fyh = 0, 

 which is easily transformed into 

 w 2 . («/>' + /3?'-l) 

 -u.{{*p' + P(f-\){up + Pq + \) + up(pq-p'(f)} 

 + a/3pq - (a' + ap) (/3' + 0q) {*p + £g) = 0. 

 And we then have 



_ (up' + (3q'-l)(cLp + (Bq + l) +u ${pq-p'q')+<j 

 U - 2(ap'+ffl-<l) 



