MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
127 
but, as dz/dx 1 vanishes, so do all the other fluxions. It follows that, if the boundary 
conditions are Ay l = 0, d/dx r .Ay [ = 0, d/dx 3 . djdx t . Ay v = 0, and so on up to 
d a i+ a -2 • dx^' 1 " • . A y y , all the fluxions whose order does not exceed a x -f- a 2 • • • + a m 
must vanish at the boundary. For, as Ay l and dAyJdx r — 0, so do all other first 
fluxions, and therefore dAyJdx s = 0 : but this, with d/dx t . dAyJdx s , shows that all 
second fluxions with dx s in them vanish. Hence any fluxion d/AyJdx a dxi, — 0 for 
dAyJdxa = 0 and d 2 AyJdx s dx a = 0, and hence all second fluxions of dAyJdx a vanish, 
and therefore, &c. 
It follows from the preceding that in the examples in question all fluxions up 
to and including those of the 7th order must vanish for points on the boundary, 
and if we assume for Ay l an algebraic form we must write 
Mi = (</> { x i> x 2, • • • Yf{ x y)> 
where <f> = 0 is the equation of the bounding surface. Suppose we adapt this to the 
last example, the origin being taken as the point P in Proposition 1, and the 
bounding surface as 
x i x % — ?' 3 — 
so that the integration extends over all values of x 1 and x % which make the left-hand 
negative. Hence r is to be a quantity of the order /3, and, to adapt the expression to 
the preceding formula, we must write 
/3 s / 
P 1 
+ cos ( ~ ) + /3 
x a 
?15 
cos 
A'o 
where, however, cos (x 2 //8 2 ) and cos (cc 2 //3 5 ) are to be considered as abbreviations for any 
fluctuating functions of periods /3 3 and /P respectively. 
Similar assumptions can easily be made when other fluxions appear. The convenience 
in choosing p r == 1 is now evident, though, as far as the equations were concerned, it 
made no difference, as only the ratios of p, p l5 and p % entered into them. 
The limits within which the property holds are evidently given by the discussion of 
Art. 21. 
23. If all the highest of all the fluxions of any of the dependent variables appear in 
U in the first degree only, the foregoing reasoning would not hold, as cdl the varia¬ 
tions of that dependent variable appearing in S 3 U would then vanish at the limits, 
and therefore the variation S 3 U when taken over a small region would be zero com¬ 
pared to quantities of the order a 3 . (When there is only one independent variable 
there must be some one highest fluxion, but in general there is a group of fluxions 
higher than any others, not usually identical with what have been called the “highest 
fluxions,” Art. 20, but included in them ; it is only when each member of this 
group vanishes that the exception occurs.) It is known, however, that in this case it 
is, in general, impossible to fulfil the limiting conditions by means of the arbitrary 
