[163 
164] 
25 
dÇ 
dt ’ 
ÿ 
dt ’ 
a dii 
— e~) 1 dr 
m dii 
— e'Q 1 dr 
been re- 
ON GAUSS’ METHOD FOR THE ATTRACTION OF ELLIPSOIDS. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 162—166.] 
The following is the method employed in Gauss’ Memoir . “ Theoria Attractions 
Corporum Sphseroidicorum ellipticorum homogeneorum methodo novo tractata,” 1813. 
Comm. Gott. recent., t. ii. [and Werke t. vi. pp. 1—22]. I have somewhat developed 
the geometrical considerations upon which the method depends. 
The attraction of the ellipsoid is found by means of the following theorems, which 
apply generally to the case of a homogeneous solid bounded by a closed surface:— 
M denotes the attracted point, P a point of the surface, PQ is the normal (lying 
outside the surface) at the point P, dS is the element of the surface at this point, 
MQ, QX, and MX denote angles at the point P, viz. MQ the Z MPQ, and QX and 
MX the inclinations of QP and MP respectively to a line PX drawn in a direction 
assumed as that of the axis of X, MP denotes the distance between the points M 
and P. And X is the attraction in the direction opposite to that of the axis of x; 
the integrations extend over the entire surface. 
dÇ 
dt ’ 
ÿ 
dt ’ 
a dii 
— e~) 1 dr 
m dii 
— e'Q 1 dr 
been re- 
Theorem. The integral 
has for its value 
0, — 27r, or — 47r, 
according as M is exterior to, upon, or interior to, the surface. 
This is obviously a purely geometrical theorem. 
Theorem. The attraction is given by the formula 
dS cos QX 
MP 
o. III. 
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