50 ROYAL SOCIETY OF CANADA 
tion. Assuming the admissible curves in the form, y = f (x), he obtains 
for the attraction the definite integral, (laying aside constant factors) : 
_ pe PON an PI 
1 = i EE hae NL MEN (1) 
and for the mass 
The application of the ordinary rules of isoperimetric problems leads 
immediately to the equation of the meridian curve of the maximizing 
solid in the form, 
22 (#+y) A Os) NOM RSRRR (3) 
the constant, À, being determined by the mass. 
(d). Moïgno-Lindelüf, (1861), using polar co-ordinates, and assum- 
ing the admissible curves in the form, r = f (0), obtains for the two 
integrals the values, 
(el do 
f= fr sin 6 cos 0,0 01) Ky in la ESS (4) 
0 
and, again by the ordinary method of isoperimetric problems, for the 
meridian curves, 
Tr COSMO, NN PR moe menace tases niScctinos (5) 
(e). Finally, Aneser (1900), in his Lehrbuch der Variationsrechnung’, 
reduces the problem to a non-isoperimetric problem by means of Euler’s 
artifice, which we have discussed in the preceding chapter. Starting from 
polar co-ordinates, and assuming the admissible curves in the form, 
r = f (4), he transforms the integrals, (4), by the‘substitutions, 
0 
u = cos 0, v = | rs MB Os di Ole Re (6) 
Under this transformation, 
if _ f (# ALU, Popo ommieececeakaet feel CAR (D) 
uo du 
and the isoperimetric condition reduces to an identity. He finds as the 
-olution in the (u, v) plane, 
to which corresponds in the (7, #)—plane the curve, (5). 



! Moigno-Lindeléf—Calcr d. Var., p. 244; reproduced by Dienger (1867) 
Variationsrechnung, s. 61), who also discusses the Legendre condition. 
2 Kneser, L. c., s. 28. 
