OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 385 
If now e be some constant, then 
v—e 
*, (y- Y) 2 
t 2 ~^ b 2 
1 )'A»—x'v= — 
Choosing the constant e= 
become exft y respectively. 
2/ 2 (a 2 + 5 2 ) 
a 2 5 2 
Therefore 
the right hand sides of these equations 
« 2 + 6 2 
a 2 b 2 
a 2 + b 2 fx 2 
a 2 b 2 \a. 2 
(y-Y ) 2 
b 2 
_d_( , 0< ,, 9 a 2 + & 2 
V ~dy\ X ^ ' a~b 2 ^ 
■ 2 / 
, a 2 -\-b 2 (x 2 
07 Q 1 0 
a 2 o" \«. 2 
(y~ Y ) 2 _-,W 
b 2 Jdy 
Putting in for y' and t/; their values, it will be seen that for the y of Clebsch’s 
forms of expression for the components of the velocity, it is necessary to take 
A//%-Y)+Y(2,-Y)-2/ 
-\t\ of 2 d" b I ctb . _-i 
- Y '-^^t|‘- 2? Sm 
The terms, containing t only, may be omitted. 
Thus if p be the density of a distribution of matter of which x is the potential 
P=~ 
1(*X , d \\_ 1 ZM-V) m(y- Y) 
ATr\dx 2 dy 2 ) Air a 4 6 4 (x 2 (?/—Y) 2 \~ 
+ P / 
therefore 
c o? — b 2 x(y—Y) 
P= 47t‘ ~cdb 2 ~‘ (o? , (y -Y) 2 J 
a 2 ^ 
The potential of this density (see Art. 15) at an internal point is 
f^- Y )+f|sin- 
V x? 
a 2 
b 2 
sin 1 
3 D 
MDCCCLXXXIV. 
