( 387 ) 
the equation of the surface ean be written in the form: 
zee Hy) de? + d,a*y + day? + dy? Heet +. (1) 
By a slight deformation we arrive for the new surface at the 
equation : 
2—@- ia - B + em + Yao Hd (Cr HY) y° + d,2° + d,a*y . (2) 
where the Greek letters represent small quantities, which can all be 
regarded as of the same order, namely of the order of the small 
variation which an arbitrary parameter appearing in the coefficients, 
has had to undergo. Also the Latin letters must be regarded as 
having been varied somewhat, which is however immaterial. 
Let us now calculate by means of the wellknown conditions: 
dz 22 072 
Ou? Ow Oy Oy? 
ee (3) 
1 02\? Oz Oz ' 02\2 
+(5 De "dy He 
the position of the displaced umbilie ; then we shall find after neglecting 
all terms which are small with respect to those which are retained, 
the two linear equations : 
y, +2d,a+ 2d,y=0; y, + (d, — 3d,)e + (8d,—d,)y=0 . (4) 
from which in general we deduce without difficulty the sought for 
displacement. 
This however is different when the determinant 
d, — 3d 3d, — d, | 
KS Sata 8Gatad) ®) 
L a 
3 3 
disappears. In that case no finite values satisfy the linear equations 
(4). This proves, however, only that the displacement of the umbilic 
has become of a lower order than the quantities indicated by the 
Greek letters and that therefore the terms of the second order in & 
and y must be included in the equations (4). If we do so we obtain 
by comparing the two new equations and eliminating the linear 
terms the new equation: 
(d, —3d,)y,—24,y, +[12d,¢, +3(d,—3d,)e,—2d,e,—8c,2d,Ja? + 
4+ [6d,e, + 4(d,—3d,)e, — 6d,e, —2c,°(d, —3d,) vy +[2d,e, +3(d, —3d,)e, — 
— ie Calta Oh ise ate (ants) ah EE msi ge (0) 
which must be combined with one of the equations (4). 
This equation (6) is of order two in w and y, from which therefore 
ensues: 1st that the displacement becomes of order } with respect 
to that of the Greek letters used in (2), 24 that the umbilic originally 
situated at the origin of the system of coordinates on the surface (1) 
Ars 
