1080 Proceedings of Royal Society of Edinburgh. [sess. 
The largest of the eight curves shown in fig. 32 has been 
described according to values of x, y calculated from these two 
equations, by giving to - lvalues tan 0°, tan 10°, tan 20°, . . . . 
tan 90°. The seven other isophasals partially shown in fig. 32, 
all similar to the largest, have been drawn to correspond to seven 
equidifferent smaller values, 19 A., 18A . . . . 13 A, of the parameter 
a, if we make the largest equal to 20A. 
§ 88. It is seen in the diagram that every two of these isophasals 
cut one another in two points, at equal distances on the two sides 
of O W. If we continue the system down to parameter 0, every 
point within the angle COC is the intersection of two and only 
two of the curves given by (127), with two different values of the 
parameter a. If we are to complete each curve algebraically, we 
must duplicate our diagram by an equal and similar pattern on 
the left of O : and the doubled pattern, thus obtained, would 
show a system of waves, equal and similar in the front and rear, 
which (§77 above) is possible but instable. We are, however, 
at present only concerned with the stable ship-waves contained in 
the angle ± 19° 28' on the two sides of the mid-wake ; and we 
leave the algebraic extension with only the remark that all points 
in the angle C O C of the diagram, and the opposite angle leftward 
of O, can be specified by real values of the parameter a : while 
imaginary values of it would specify real points in the two 
obtuse angles. 
§ 89. By differentiation of (127), we find 
^—=-t—-tdca\l/ 0^8); 
dy 
which proves that tan -1 £ is the angle measured anti-clockwise 
from O Y to the tangent to the curve at any point ( x , y), in the 
lower half of the diagram. Elimination of t between the two 
equations of (127) gives, as the cartesian equation of our curve, 
{x 2 + y 2 ) z + a 2 (8y 4 - 20a; 2 ?/ 2 - a 4 ) + 16a 4 y 2 = 0 . . (129). 
But the implicit equations (127) are much more convenient 
for all our uses. It is interesting to verify (129) for the case 
-t=± in (127), corresponding to either of the two cusps 
shown in the diagram. 
