MATHEMATICAL SECTION. 61 
d+r 
Bey fee oy LCE = Tbe = d=r))] 
Ane Sp ae 
d—r 
Ve—h 
h do (a? — po — i?) (P aay Caen B) 
= 2m0 =i : Lo ee 7 
Ve—F 
where a= 7/7) +H; andb=Y@—r/ +h (5) 
Pe ey ae 
Assume tan ¢ = > \ ——— then ¢ has the limits _| and 
a — p 
a b? 
0. Wehave then p? + ? = Few ae ae where Ag 
Se 
is an elliptic A function to the modulus | jee S Transform- 
ing we have, expressing in A functions 
? 
(1 — d¢’) (a¢’ —-, 
A= amo 1 ab? ay _ 
A¢g ?— hag’ 
0 
ab? te dg = dg (#— iy. dy | 
amd a} at) apt fa a ae 
eh ; 9 Um 49) a9} 
b? | , dg Rh 
= 2md77 | fx-Bs¥ Ag 
Lo 0 
ft] } 
(d + r) dg | 
mae f 4rdh? 
0 (1 + a (d— ry sin *0) Ag | 
= amo | Fe EH (4) (S45) fe 
where fF, E, and J/ denote quadrantal elliptic integrals of the 1st, 
2d, and 3d species, according to Legendre’s notation. If our object 
was to obtain here a formula for convenient computation of the 
attraction the third term could still be expressed in integrals of the 
1st and - species, which have been tabulated by Legendre, 
