118 
MR. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
( 2 .) Suljstitiiting ( 1 —po) 
^3 = 2/3 = Ih - Ih 
H = ^ - 2h - h = ^ - Pi - Pi + 2j- 
(3.) 8 iil).stitiiting (I— 2p) foi' 
2 C^ = 1 - 2^. — Xg = 1 — — p, + Xj 
- Pi - Vi = ^ - Pi - Pi + 2/i 
^4 = 2:3 = I ~ 743 - p .^ + 2^. 
The second and third cases are obtained, like the first, by simply expanding. Thus 
% = (/37)/(^0’ 
iM = (r) - (By) = (U) - (G) - (By) 
= (U) - (B) - (C) + (BC) 
or, dividing by (U) 
23 ^ 1 - Pi - 23 = 1 - P3 - Pi + 2i 
as al)ove. 
§ 24. The correctness of the transformations given may, of course, be verified 
directly. Thus suppose 
Pi < Pi < Pi < <^>'5 
then the equations to the bounding planes of the congruence-surface are 
x=0 y = 0 z = 0 
X ^ 2 h y = Pi « = Pi 
+ V = Pi -Y- Pi ^ Pi - ^ .p 
X y — z = 2h 
X — 2/ + 2 = 7.73 
- X + 2 / d- « = 2h 
If (I — pP) be substituted for 773 , ami (1 —p-^) for pg, then 
Pi <^ - Pi < 1 ■- Pi^ 
and the equations must l)e 
x = 0 2/ = d z = I - p. - 7)3 ) 
X =: 2h y = Pi 2 = 1 - 2h 
A- + y + s = Pi - 2h - Pi - ^ 
X -f 27 — S = Pi . 
X — 2 / + - = 1 — Pi 
- X y z ^ I - 773 
V 
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11. 
