( 552 ) 



So 



„ , , , 16 du dïi dv bv 



{v — uydg d»; 05 ö»j 



§ 6. Let lis now return to equation (VII). We see immediately that a 



8 bv bu 



solution, which does not cause h z=z -t^-t- to vanish, is 



given by 



" = 95 (>i) > V — -^ (^), 

 where </) and if? are respectively functions of >; and § only. 



It is clear that equation (IX) is satisfied, when /i (§) := ƒ, (t^) = 0, 



so when i)= 2)"= (§4)- 



It is. worth noticing, that when m = ^ (tj) and u = i|j (§) are sub- 

 stituted into the equation for F, this form becomes a solution of 



4 bl\F'b^) 



and so this tallies perfectly, because we have here the differential 

 equation of Liouville. Indeed, the problem of the surfaces with 

 constant mean curvature always leads to an extended equation of 

 Liouville, as (IX) does, in whatever way we treat it. 



That we really find a sphere here must follow from (VI). These 

 equations give for u-=.if) (?^) and v = ip ($), 



\ V -\- u 



Q V 



the wellknown formulae for the sphere in minimal coordinates. 

 We find : 



2 

 i.e. a sphere with radius — , as is necessary. 



Now that we have regarded the special case /\ (|) = ƒ, (t]) ^= 0, 

 we can put both functions equal to 1 by introducing new functions 



/: (b) = §1 and ƒ, in) = r^i, 

 which we shall again indicate by ^ and tj. This is of high importance, 

 if eventually the solution of equation (VII) were to be found. 



