[651 
igeneous 
n three 
irface of 
e second 
function 
N) and 
A, fX, V, 
and so 
651] 
or say this is 
ON A SPECIAL SURFACE OF MINIMUM AREA. 
65 
2 VV 2 (A + fi)(\ + v) 
- 2A" (ji + v) {(A + v) 2 fi'- + (A + fx) 2 v' 2 } = 0. 
We have to write in this equation A' 2 = aA 4 + 6 A 2 + c, and therefore A" = 2oA 3 + 6A, 
&c.; the left-hand side, call it Q, is a symmetrical function of A, fx, v, and is con 
sequently expressible as a rational function of 
p, = A + [x + v, 
q, = /xv + vX + A fx, 
r, = X/xv. 
We ought to have il = 0, not identically, but in virtue of the equation 1+^ = 0, 
that is, il should divide by 1 + q; or, what is the same thing, O should vanish on 
writing therein q = — 1. 
To effect the reduction as easily as possible, observe that we have (A +/x) (A 4- v) = A 2 -f- q ; 
and therefore 
2//V 2 (A + /x) (A + v) = 2X 2 /x' 2 v' 2 + q2p! 2 v\ 
Similarly, in the second term, 
(/x + v)(X + v) 2 = (v + A) (v 2 + q) and (/x + v) (A + fx) 2 = (/x + A) (fx 2 + q). 
The complete value of il thus is 
£l = 2{Aq + B)-[(C + D)q + E + F], 
where 
A =2 AVV 2 , B=Sfx' 2 v' 2 , 
G = 2XX" OV 2 + D = ^A" (¡V 2 + 
E = SAA" (fx' 2 + v'% F = 2A" { Vf x' 2 + fxv' 2 ). 
We find without difficulty 
A = a 2 ( q i — 4>q 2 pr + 4>qr 2 + 2p 2 r 2 ) 
+ ab (- 2q 3 + q 2 p 2 + 4qpr - 3r 2 - 2p 3 r) 
+ ac ( 4g 2 - 8<?p 2 4- 8pr 4- 2p 4 ) 
4- b 2 ( $ 2 - 2p?’) 
4- 6c (- 4g 4- 2p 2 ) 
+ c 2 ( 3), 
B = a 2 ( § 2 r 2 + 2pr a ) 
4 ab (- 4^r 2 + 2p 2 r 2 ) 
+ ac (- 2q s + ? 2 p 2 + 4gpr - 3r 2 - 2p 3 r) 
+ b 2 ( Sr 2 ) 
4- be ( 2q 2 - 4pr) 
+ c 2 (-2q +p 2 ), 
9 
c. x.