MR. G. UDNY YULE OX THE THEORY OF COXSISTEXCE 
] 16 
Illustrations of such surfaces will he found in figs. 16-23 (§ 36, p. 130-131). The 
inferior instead of the superior conditions then predominate in importance. 
Two of the planes of superior congruence—but not more than two—may simul¬ 
taneously pass outside the planes of inferior congruence and so disappear. Thus to 
determine the planes of infeilor congruence suppose 
Pi < < Ih Pi < P-i < 
The plane a. vanishes if 
Pi + + ih < 1- 
The plane ^ cuts the plane ^ = 0 in the line 
X -P ij — iw 
the limiting values of x and y being x — y = Pi it cannot disap})ear. 
The plane y cuts the plane 2 = (the limiting plane) in the line 
X — y = 0, 
and therefore it also cannot disappear. 
Finally, the plane 8 cuts the plane 2 = p^ in the line 
y - ^ ^ Pi - Pi‘ 
Fut tire greatest possible value of y — x is p^. Therefore the j)lane 8 disappears if 
Pi - ih > Pv 
An illustration of a surface of this character will be found in fig 19, § 36, p. 130. 
In any case there are four “ syllogism points ” (like KEGF of fig. 4) to the surface. 
§ 22. In all the preceding examples of figs. (l)-(4) the values assigned to p^, p^, 
and ^>3 have been less than 0‘5, so that the edges a^S, ay, a8 of the primitive tetra¬ 
hedron were truncated, if truncated at all, by the co-ordinate planes. If 
Pi + ih > 1 
ih + ih > 1 
Vi + Fi > ^ 
this ceases to be the case (rd/c equations A § 15), the truncating planes then move 
inwards, and the congruence-surface stands clear of the co-ordinate planes. 
A little consideration will show, however, that no new features are introduced into 
the surface itself Thus su 2 )pose that 2^nih) Pi 1®®® than 0’5, but that we then 
substitute {^—pi) for s® ns to make one ratio greater than 0’5. How is the 
original surface altered ? Substituting (1 f^^^’ih amounts to substituting y for C. 
Then if rc^, Zy Ije the original co-ordinates, aq, //^, 2 o the co-ordinates after the 
suljstitution, we must have 
