88 M. M. SLOTNICK 
at the point in question. The first pair of equations deals with the 
gradient of gravity and those of the latter pair are referred to as the 
curvature quantities. 
Our problem is to find the values of p, g, r, s, and ¢ for our surface 
defined in the form (21). This equation defines z implicitly as a func- 
tion of the two independent variables x and y. Recalling that the 
center of gravity of the suspended system bears the same relation 
to the equipotential system through it as does the point P to the 
surface S previously described, we conclude that 






0z 02 
— = p=o, and —=gq=o0 
Ox y 
Since, however, 
dU aU az dU Vd) az 
(25) —+——— =o, —+4+— — =o" 
Ox Oz Ox oy oz OY 
aU oaU 
it follows that at the origin, —-= ——=o. 
Ox oy 
Similarly, since at the origin, p=q=o0, we find that 
@7U dU az 
a. sass = 0, 
Ox? oz Ox? 
(06) b a?U i aU d% 
2 - — = 0, 
oy? ez Oy? 
@U aU a% 
Cc —S>S ———_- = 
Oxdy 2 Oxdy 
By definition 
072 072 072 aU 
Sa DS . ie and) — 
Ox? Oxdy Oy? Oz 
Hence 
Us 
(27) peg are sa 
§ 
and 
* The method of obtaining these derivatives of s with respect to x and y when s 
is defined implicitly, as in (21), is fully discussed in Goursat-Hedrick, Mathematical 
Analysts, Vol. 1 (Ginn & Co., 1904), p. 42. 
418 
