26 Mr. H. Hennessy on the Attraction of Spheroids. 



and therefore the term containing a or u must be that corre- 

 sponding to the difference between the spheroid and the sphere. 

 If we make cos 0=/^, 



/»-i /*2<y»R pt J *dr'dw'dp' 



T/ + X J Q J o Sr^-Vr? t|*j*'T •T^VT^P* cos^eo -co')] H-/ 2 ' 



and 



8m being an element of the difference between the spheroid 

 and the sphere, and/' 2 the quantity under the radical. 

 After substituting for r its value, 



• du 



u+2a-j- 



dr 



=/(/ + *-i) *-''■ 



But if 



yss^'4- \/\ — jot. 2 \/ \ — 1«/ 2 COs(co — w'), 



4 



Hence 



But cW is a prism of homogeneous matter included in the 

 prolongation of r' between the surfaces of the spheroid and 

 sphere. Hence 



dm'=l-(R a -a a )d^dco', 

 3 



and therefore 



in which 7/ is the same function of {*.' and w' that 3/ is of a and 

 eo, or so that when 



y=Y(f*,co), y = F0*',«>'). 

 When the attracted point is at the surface of the spheroid 

 r = a(l +«j/), and unless y=l, 



^ du 

 u + 2a-j-=0. 

 dr 



Whence the theorem (1.) holds at least, unless 



/xju/+ V\— /** V'i — f*' 2 cos (eo— «/) = !. 



