MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
101 
this continuity of 8y would require a large amount of work, but it will be sufficient for 
our purpose to take the simple case where the function to be made synclastic contains 
only x, y, and y. 
Let us apply his cjdneral method to the integral * 
therefore 
integrating by partSj 
Hence 
U = J 1 fix, y , y) dx ; 
SU = r^Sy] l + \(y-yy]8ydx. 
dy Jo \dy dxclyj 
S 2 U = 
d?f 
jdydy 
* 9 , ^J 1 , f fd 2 f * , <vf ^ d 
• v + dy ,*!j § 2/J 0 +) + j-a/v - dx 
It will save trouble if, in what follows, we use the abbreviations 
|= Y ». | = Y„ and, in general, -~ ) = Y r , 
d?f dy _ .... . d~f 
— = Y oo5 yy — J-on anc b in general, 
dy 
dy dy 
dy {r) dy is> 
= Y„. 
Then we may write 
S 2 U = 
‘ + [( You Sy + Y 01 Sy - | (Y w Si/ + Y u Sy) $y dx), 
of which the part under the integral sign is 
f (Y w Si/ + Y 01 sy - y 01 Sy - Sy - % - Yn *y ) ^ 
f{( Y » 
dY, 
ai 
dx 
8y 
dY 
dx 
pgy- Yu3y)j S ydx. 
( 5 ) 
Now let z be a solution of the equation 
/ dY, 
I J. nn 
oi\ 2 _^lL^_Y n y=0. ..(6) 
^ ' 00 dx l ' dx 
* Jacobi treats this case by a method not identical in form with the general method he gives. 
