MINIMA SOLUTIONS IN THE CALCULUS OP VARIATIONS. 
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for the region S fi- f/S can only differ from zero by a quantity infinitely small compared 
to a 3 , and therefore the least positive value of S 3 U must vanish for the region S. 
It has now to be proved that 3 2 U must be capable of a negative value of order a 3 
when the region of integration is extended beyond the region S. It sometimes seems 
to be considered that this may be inferred from the fact of S 3 U changing sign when 
§ 3 U = 0. But, as S 3 U does not represent the increment of S 3 U due to extending the 
region of integration beyond S, this is not admissible. If we draw an imperfect 
analogy from algebra, we may say that what we have to prove is that in no case does 
S 3 U behave as if it had a square factor the value of which, after vanishing, remains of 
the same sign, but that, if S be a region of integration of the character supposed in 
Proposition 2, for which it is possible to make S 3 U zero, then for any region including 
S it is possible to make S 3 U take either sign, the limits being in each case supposed 
fixed. 
To prove this, let us suppose that y x , y 2 , . . . y n represent values of the 
dependent variables which make the first variation vanish. Let aA^j, aA^ 2 , &c., 
represent the variations for which § 3 U = 0 (that is, A = 0) when the integration 
is extended over S. The limiting values of all variations, except those of the highest 
fluxions, are zero. Let S' be a region including S, and let a A 2 y 1; a A 2 y 2 , &c.,be varia¬ 
tions having at all points of S the values a a 1 y 1 , aA^, &c., for all the variations, 
and having at all points of S' not common to it and S the values zero for all the 
variations. 
Now take a third region S", wholly included in S', and of which a portion is 
included in S, and the remainder S 2 excluded from S ; and let a A 3 y u a A 3 y 2 , &c., be a 
variation having at all points of S' — S'' (representing in that way the points contained 
in S' and excluded from S") the values aA 2 y l5 aA 2 y 2 , &c., while, over the region S", 
a A s y l} aA 3 y 2 , &c., are determined by the conditions that y Y + a a 3 2/i> V- 2 + a A 3 y 2 , &c., 
are the values which make U a true minimum when the integration extends over the 
region S", and the limiting conditions are that, for all except the highest fluxions 
of the variations, a 3 z = A 2 2 all over the boundary of S", z representing, as before, 
any dependent variable or fluxion. (By taking the region S" small enough, these 
conditions can always be satisfied.) 
Then the variations represented by A l5 A 2 , and A 3 are admissible ones (Art. 18). 
Then A 1 S XT = 0 when the integration extends over S, and A 3 3 U, when the integra¬ 
tion extends over S', is identically equal to it, and therefore vanishes (for the a 2 
variations are zero except over S, where they equal the A , variations). Again the 
A 3 variations are the same as the A 2 ones, except overS" ; and over S ' the functions 
V\ + a A 3 y Y , &c., give a smaller value to the integral than do y 1 -f- aA.^ (because 
they make it an absolute minimum compared to all near values), and therefore A 3 2 U 
is smaller than a 2 3 U, the integration being extended over S'. But A 2 2 U = 0, and 
therefore A 3 3 U is negative. But clearly S 3 U may be positive, and we have now shown 
that it may be negative, for one value is a 3 A 3 3 U ; hence it is capable of either sign. 
R 2 
