MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
103 
S’U = [ (Y 00 8 f + 2Y 10 8y 8 \j + Y u Sif) dx. 
Writing in this 0 8 x y for 8y, where 0 is a function of x at present unknown in form, 
and expanding the result, we get 
S 3 U = | (Y^ 3 S,y 3 + 2Y 1O 0 S l2 , (0 S,y + 0 %) + Y u (0 S l2 , + 0 S,y) 3 ) dx 
= j (sy (Y w 0 » + 2Y 10 ee + Y, A) + 28,2, % (Y,„0 3 + y n ee) + Y u e° 8#) dx. 
Integrating by parts the term involving S^S^, 
s s u = [(¥,/> + Y u ee) 8,2/ 3 ]; + [ (Y m p + 2Y vl e0 + y u * - £ (Y,/ 3 + Yje^/Sx 
+ \(Y u 0 1 S 1 f)dx. 
The multiplier 0 is here quite arbitrary, and can therefore be determined so that 
the quantity multiplying 8y 2 under the sign of integration shall vanish, and hence, 
when this value of 6 is chosen, we have 
S 3 U = [(Y,/ 3 + Y u e&) + [ YJ-- 8 dx. 
This is the same form as that in the other reduction, and it is easy to prove that 6 
is a solution of the equation (6) used in finding z in Jacobi’s method. 
9. We have now given examples of the simplest cases. The method of treating¬ 
s', where 
U = f f{x,y,y,.,. y U) ) dx, 
j 
will now be given in its shortest form. 
Using the notation already adopted, we may write 
S 3 J f {x, y, y, . . . y {n) ) clx — j 2 % Y rs 8y (r) 8y (s) dx. 
If we transform this by writing z x 8 x y for 8y, and expanding the fluxions, we shall 
get an expression which may be written 
. ....... ( 7 ) 
J <0)(0) 
where 8 x y, for instance, means d/dx, 8 x y and A rs contains z 1 as well as x. This 
expression consists of terms in which 8 x y itself enters, and other terms in which only 
its fluxions appear. Integrate by parts the terms containing 8 x y ; thus, in the 
previous expression 
