MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
129 
Y 1} Y 2 , &c., we must have, considering only fluxions of y x , oq -f- [1, 1 ] < Yj — 1, 
cq + [1, 2] < Y : — 1, and so on, one inequality from each equation. Again, 
from considering the order of differentiation of y. 2 , cq -f- [2, I] < Y 3 — 1, cq + 
[2, 2] < Yo — 1, and so on. Since every differentiation gives us a new equation, 
among the quantities in question we get altogether n -fl eq + cq + &c. -f- a„ 
equations ; hence the difference between this and Y T -j- Y 3 + &c. must be equal to 
the number of quantities to be assumed in order to solve these equations. It is easy 
to see that the most favourable way in which the differentiations cq, a. 2 , &c , can be 
disposed consistently with the inequalities to be satisfied will give the number stated 
above as the least number of this difference. Now suppose the equations obtained 
in the Calculus of Variations from making the coefficients of 8y x , 8y 2 , . . . 8y u 
vanish are denoted by (1), (2), . . . ( n ), and let the [l] denote the highest 
differential coefficient of y 1 occurring in the second or in a higher degree in the 
function to be made synclastic, [2] that of y 2 , and so on ; then it is easy to see that y r 
cannot enter into the equation (p) by fluxions of order higher than jYj -f [p], and that 
y p will enter into it in the order 2 p. Hence in this case [p] -+- [r] = [pr\, [r] + \_p~] 
= \rp~], and we have for this case to find the set for which SQp, r] is greatest. Now, 
since we are to take one index for each equation, in our %{(p) -f- (r)} we are to take 
only one term from equation (r), and hence (7*) on the right-hand side is to appear only 
once. Moreover, we are to take only one term from each variable, and therefore we 
are simply to take S{(_p) + S ( 7 ’)}, where each refers to the values (1), (2), (8), &c.; and 
hence, in all, double the sum of the order of the highest of all the fluxions of each 
variable in the expression to be made synclastic. But this is exactly the number of 
the conditions supplied by equating the limiting terms of SU to zero in the expression 
to be integrated, except in the case we are at present discussing (where some of the 
highest of all the fluxions do not appear in the second degree, but in the first only) ; 
and therefore in this case the limiting conditions cannot be satisfied, and the problem 
becomes, in general, incapable of solution. 
MDCCCLXXX VIT.—A. 
