10 CHARLES CARPMAEL ON THE LAW OF FACILITY OF 
Subtract unity from each of the 7 variables in turn, we obtain » integrals each equal to 
+} x n 
fine v Me. Jan Citas dz, [2 a2<s5+ y—1]. 
Of these » integrals 7, and r only, contain the part R; namely, those obtained by subtracting 
unity from one of the r chosen variables. 
Similarly if we subtract unity from two of the variables, and take every possible com- 
3 : : ; nn VL. 
bination of the 7 variables two at a time, we obtain —~—— integrals each equal to 
? 1 o 5 
0 Lee ee 
and of these ” SS contain the part R; namely, those obtained by subtracting unity from 
any two of the r chosen variables. 
So ifs be any number not greater than #, if we subtract unity from s of the variables, 
à 5 à É . n 
and take every possible combination of the # variables s at a time, we shall obtain AE 
LS |n-s 
integals each equal to 
ay fa eooasonge fa, xp. ax, [2a <5 +y—s] 
0 0 
la 
and of these, ifs be not greater than 7, bee will contain the part R of the integral (ii), 
whilst if s be greater than 7, R will not appear in any. 
Hence the series 
HA cine Sane Jan dr ener cie |e nc = st yl—nf ne res fan av, Seseee qe, ee all 
0 0 -_ 
aaa, Roue PR as ar... dm [Er <T +y—s] 
ie 0 2 
RE Fe a ie ae Jan an. dn, [Sect y 6] GD 
r (1— 1)" times, ze, it will not contain it at all. 
Now this is true for any 7 particular variables, and for all values of 7 from = 1 to 
y=t. The series (ii) then, contains those portions, and those only, of (ii) which are due 
to values of the variables all less than unity, it is theretore equal to 
ies Rae e(ais > far ds. DATES ata. 
