I 95 ] 
IY. On the Discrimination of Maxima and Minima Solutions in the Calculus 
of Variations. 
By Edward P. Culverwell, M.A., Fellow and Tutor, Trinity College, Dublin. 
Communicated by Benjamin Williamson, M.A., F.R.S., Professor of Natural 
Philosophy in the University of Dublin. 
Received June 5,—Read June 10, 1886. 
The criteria for distinguishing between the maximum and minimum values of 
integrals have been investigated by many eminent mathematicians. # In 1786 
Legendre gave an imperfect discussion for the case where the function to be made a 
maximum is \f{x, y, dy/dx) dx. Nothing further seems to have been done till 1797, 
when Lagrange pointed out, in his ‘ Theorie des Fonctions Analytiques,’ published 
in 1797, that Legendre had supplied no means of showing that the operations 
required for his process were not invalid through some of the multipliers becoming 
zero or infinite, and he gives an example to show that Legendre’s criterion, though 
necessary, was not sufficient. In 1806 Brunacci, t an Italian mathematician, gave 
an investigation which has the important advantage of being short, easily compre¬ 
hensible, and perfectly general in character, but which is open to the same objection 
as that brought against Legendre’s method.t The next advance was made in 1836 
* This sketch is founded on Todhunter’s valuable ‘History of the Calculus of Variations.’ 
f Bbunacci’s method may be explained as follows: Let 
. U =JK'*'£■!)** 
be the function to be made a maximum, and let us denote dzjdx and dz/dy by p and q respectively. 
Thus, the limits of z being supposed fixed, 
au = 
dz dp 
bp + y- d* dy 
+ 
bz 2 + 2 
d n ~f 
dz dp 
dq 
bz bp + 2 
Tf , dlf „ 2 , „ dlf 
dz cq dp A dp dq 
op bq + bcp\ dx dy. . . (1) 
dq 2 / 
The first integral must be made to vanish by a relation between z, x, and y. Eliminate z for the 
coefficients in the second integral of (1) by means of this relation, and let it be wi’itten—■ 
jj (A os 2 + 2B bz bp + 2C bz bq •+- P bp^ + 2R bp Sq + Q b <? 2 ) dx dy. ..... (2) 
Now, remark that, the limits being fixed, (& 2 (a dx + fi dy) vanishes, whatever a and /3 may be, for 
4.6.87 
