..--A ì 
886] 
SPHERICAL CURVES OF CURVATURE. 
637 
which belong to the case PS, 3°. Hence, in this general case, the Inversion is a 
surface PS, 3°. 
I have spoken above of the particular case e = 0, f= 0: here the equations of 
the two sets of spheres are 
a; 2 + y 2 + z 1 — Zby — 2 cz — 0, 
¿c 2 + y 2 + z 1 — 2 cnx — 2 yz = 0, 
which have the origin as a common point. Taking this as the centre of inversion, 
or writing 
K*X K*Y K*Z | _ _ a 
x = —, y = vv > 2 = n > where 12 = X 2 + Y 2 + Z\ 
12 ’ * 12 
the transformed equations are 
or, interchanging X and Y, say 
which are of the form 
12 
bY + cZ — — 0, 
a A + yZ — \X 1 — 0, 
bX + - \K* = 0, 
aY + yZ — = 0, 
x +tZ-P = 0, 
Y+ez-u =o, 
belonging to a surface PP, 3°. Hence, in this case, the Inversion is a surface PP, 3°. 
It thus appears that the surface SS, 4° has an Inversion which is either PS, 3°, 
PS, 4° or PP, 3°. The inversion has in some cases to be performed in regard to an 
imaginary centre of inversion. 
It was previously shown that the surface SS, 3° had an Inversion PS, 3°, and 
we thus arrive at the conclusion that a surface SS, with its two sets of curves of 
curvature each spherical, is in every case the Inversion of a surface PS with one set 
plane and the other spherical, or else of a surface PP with each set plane. Serret 
notices that the centre of inversion may be imaginary: this. (he says) presents no 
difficulty, but he adds that it is easy to see that the centres of inversion may be 
taken to be real, provided that we join to the surfaces thus obtained all the parallel 
surfaces. 
It seems to me that there is room for further investigation as to the surfaces 
SS: first, without employing the theory of inversion, it would be desirable to obtain 
the several forms by direct integration, as was done in regard to the surfaces PP 
and PS; secondly, starting from the several surfaces PP and PS considered as known 
forms, it would be desirable to obtain from these, by inversion in regard to an 
arbitrary centre, or with regard to a centre in any special position, the several forms 
of the surfaces SS. But I do not at present propose to consider either of these 
questions.