146 Mr. S. H. Burbury on the 



Let it now be required to find the chance that 



, ^11 + ^21 + • • • +%j 



#1 = ;= snail = c, . . . Ci+aci, 



^12 + ^22+ • • • "T^H-2 in ,7 



# 2 = 7^ Snali = C 2 . . . C 2 + <%C 2 , 



&c. &c. &c. 



T ' -• a? l«* + g2n+ ■ • • + dfr» v Tl _. ,7 



#» — 77^ Sna11 — G n ' ' * C n + " C n, 



and let this chance be F(c x . . . cj<&?i • . . dc w . 



In the same way as for one variable x we find that 



/•CO /*CO , 



F( Cl . . . c„) = K Ml. A M n XV*** ■ ■ ' +0 "'" ) V_1 , 

 in which K is constant and 



x = j|. . . m • • • •> ^ ,9,+ ' • • + *" J v=I ^x • • • *»„• 



— 00 



That is, 



X = J J . .f(x x . . . x n )dx x . . . dx n 



+ l /^ 1 • • • ^,/(^i • • • a n )dvi • . . dx n +&c. 



— ^^A J ) • • • ®i*af(®i • • • ^J^i • • • da? n 



— &c. 



+ terms involving powers and products of 1 . . . 6 n above 

 the second degree. 



All terms of the form JJ . . . xf(x x . . . xjdx x . . . dx n vanish 

 by the conditions oif{x 1 . . . x n ). 



The higher powers and products will be neglected for the 

 same reason as in the case of one variable ; and therefore 



10i 2 - 2 1 



X=l — j^—X 1 2 —^ O^X^ + &C. 

 1 ,— 0,* 



— fj-^i 2 T + X ^A+ • • •) 

 = e 



—0 2 _ ' 



N __ - (*i 2 ^ + «i*a«A+ • • • ) 



