638 ON THE SURFACES WITH PLANE OR SPHERICAL CURVES OF CURVATURE. [886 
In conclusion, I remark that I have throughout assumed Serret’s negative con 
clusions, viz. that the several cases, other than those considered in the present memoir, 
give only developable surfaces, or else surfaces having circles for one set of their curves 
of curvature. These being excluded from consideration, there remain 
PP, Serret’s two cases PP, 1°, PP, 3°; 
PS, his three cases PS, 1°, PS, 3°, PS, 4°; 
SS, his two cases SS, 2° and SS, 4°; 
but PP, 1° is a particular case of, and so may be included in, PP, 3°; and similarly 
PS, 1° is a particular case of, and may be included in, PS, 3°; the cases considered thus 
are 
PP, 3°; PS, 3°, PS, 4°; SS, 2° and SS, 4°. 
It would however appear by what precedes that the case SS, 4° includes several cases 
which it is possible might properly be regarded as distinct; and the classification of 
the surfaces SS can hardly be considered satisfactory; it would seem that there should 
be at any rate 3 cases, viz. the surfaces which are the Inversions of PP, 3°, PS, 3° 
and PS, 4° respectively. 
I regard the present memoir as a development of the analytical theory of the 
surfaces PP, 3°, PS, 3° and PS, 4°.