TITK (lAMMA, OK 1- ACTi )RI.\L. FUNCTIDN. 



641 



Moreover any decrease in .v or ^ increases both 0' and (i —ft)" ; 

 therefore, when either (.v+i) or (>'+ 1) is negative, (x, y) is infinite 



and therefore meaningless. 



W(i —OyiiO, whenever it has a meaning, = 



n(A-+3'+r)" 



Second proof that B{x,y) = 



ux n V . 



n{x-\-y+iy 



This all-important Proposition can also be proved as follows : 



Since TLx 



= ( log- j cW, if .v+ I > o, change d to fl^ where V>o. 



Then 



n.v 



log ^ Vfl'T-^' —. 



*y/ ft 



Therefore, /( r) being an arbitrary function of F, 



n.v2/(T')== 



^ogl\'^l-:{AV).ft^V^^-\ 



Let V be log ~^J{V) ^ [^og '^\ and ^fV ^ C (lo^^y d.p ^ Uy. 



V 



If V + I > o, write / for log 



Then 



Ux Uy = /'■( - di) d<b { log - 8''^2 1 > 



)e=o Jo \ 0/ 



and 



0lngi;0 ^ ^-log l/f^ . log 1/0 



= G)-'-' 



(j=i 



Therefore n.v nv= I /^(-J/).| d^pllog-Y ^ \' 



V = o 







/ . 



in this equation, write 



,, ,1 1,1 



0'+' = u, log - = r"i — log -• 



