OF TESTIMONIES OR JUDGMENTS. 621 



and hence we are to seek, as before, the value of 



Prob. xyz 

 Prob. xyz + Prob. xyz 



Assuming then as before, 



wz=s, vz=t, xyz = <p 



we find, by the calculus of Logic, the following expression for ^ as a developed 

 logical function of x, y, w, v, s, t, and z, viz. : 



(p=xywszvt^-xyvtzws + xyzwvst 



+ 0{xyvws t z + yvztxws +yvxws z t+ xyw s z vt + xwzsyv,t 



+xwyzvst+xyzvws t+xzyvwst+yzxsvwt+xyzvws t 



y xzvws t + zx y v ws t + x y z v iv s t) 



+ terms with coefficient ^ (2) 



Hence, adopting the simplification of Art. 21, we have 



V = xyivsz + xyvtz + xyz + xyv + yvzt + yv + xyw + xwzs 

 + xw + xy + xz+yz + x+y + z + l 

 =xw(y + l)(zs + l)+yv(x + l)(zt + l) + (x + l)(y + l)(z + l) . (3) 



whence we form the algebraic system 



xw(y + 1) (zs + 1) + xyv(zt + 1) + x(y + 1) (g + 1) 



_ xivy(zs + 1) + yv{x + 1) (zt + l) + (x + V)y(z + 1) 

 a 2 



_xiv(y + l) (zs + 1) yv(x + l) (zt+l) 



€t+ C-i (Xn Co 



xio(y + l)zs _ yv(x + l)zt 



~ a i C 1 p i <*>2 C 2P2 



__xw{y + l)zs + yv(x + \)zt + (x + 1) (y + l)z 

 r 



_ xyzws + xyzvt + xyz 



= xw(y + l) (zs + l)+yy(x + l) (zt + l) + (x + l) (y + 1) (s + l) _>, 



1 



X being a subsidiary quantity introduced for convenience. 

 From the above we find 



_ xyw(zs + 1) + y{x + 1) Q + 1) 



w 2 2 2 — Ty 



xiuzs(y + l) 



«i c iPi= — x — 



i „ _ xw(y + l) (gg + l) + (a? + l) (y-f 1) Q + l) 

 VOL. XXI. PART IV. 8 E 



(4) 



