404 Proceedings of Royal Society of Edinburgh. 
(i) 
into two regions, one of which contains the ^-discriminant and one 
which does not. There are three species of tac-loci according 
as the direction of the integral curve lies in the former or the 
latter region or along the isoclinal : (i) if it lies in the region not 
containing the ^-discriminant the 
two integral curves have opposite 
curvature ; (ii) if it lies in the region 
containing the ^-discriminant the 
curvature is the same for both in 
direction hut not in general in 
magnitude ; (iii) if the direction of 
the integral curve lie along the 
isoclinal, there is, as will be shown 
later, an inflexion on one of the branches of the integral curve. 
Figure 3 gives a geometrical representation of the three cases. 
If the double point is a point of the first order, the directions of 
the tangents to the isoclinal are given by the quadratic 
Fig. 3. 
^ 2 ^+ 2 '^ + ^ = 0 
( 4 ) 
therefore in any particular case it is easy to decide to which species 
the tac-locus belongs. (See examples (1), (2), (3) and (7).) 
If, however, the roots of (4) are equal, i.e., if 
(5) 
the p-discriminant is a cusp locus for the isoclinal family. 
Similar reasoning to the above shows that it is also a cusp locus 
for the integral family ; and in every case, except when the 
direction of both families is the same, the curves contiguous to y 
passing through Q both lie on the same side of P, and therefore the 
curvature of both branches is in the same direction, i.e., the cusp 
is a ramphoid cusp. (See fig. 4.) 
If, however, at any point the direction of the integral curve is 
the same as that of the cusp locus, there is in general a tac-point on 
the integral family, the contact being of higher order than the first. 
(See examples (4) and (5).) 
If the roots of (4) are imaginary, the ^discriminant is a locus of 
conjugate points for both families. 
If at any point P, which is not on the envelope locus of the 
