62   Life Contingencies

6. We now have Dx =vxlx

Cx =vx+ldx =vx+1 /lx —ix+1) =z,Dx —Dx+1

Nx+1 = Nx = Dx+1 +Dx+2 +Dx+3 + [~l1x=Cx+Cx+1+Cx+2+ •

=vNx —N,+1 =vNx_1 —Ni.

I f we write R,= 11Ix +21x+1 +Mx+2 +

=Cx+2Cx+1+3Cx+2+ .. . and either   Sx = N +Nx+1 +Nx+2 + .. .

= Dx+1 +2Dx+z +3Dx+3 + .. • or   Sx = Nx +Nx+1 +Nx+2 +

=Dx +2Dx+1+3Dx+2 + ..

we have   Rx =vSx —Sx+1 =vSx_1 —Sx.

7. The arithmetical values of these commutation symbols are tabulated, but in using such values the student should be careful to see whether the sum of the D., values is put down as an N-value or as an N-value. If the N-value is used and these values are summed the column of sums will be an S-value. If the N-value is used it will be followed by an S-value.

8. We have seen that Ax = -

x

1

n Ax D ICx+n+Cx+n+1+Cx+n+2+

x

Mx+n D+

n Ax+n

Dx   Dx

I

Similarly   = n'4x = — [Cx+Cx+1+ . . . +Cx+n_1] Axn

Dx

Mx — Afx+n

and hence

Dx