MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
117 
points whose boundary is defined by those values of x i . . . . x n which make a certain 
function or functions vanish; as, for instance, all those sets of values which satisfy the 
two sets of inequalities 
x\ + x 2 : + . . . -j-x 2 m — r 2 < 0 
and 
( x 1 — cq)" fi- (x 2 — a. 2 )~ (x m — a m )' — r < 0. 
The dependent variables being y x ... . y n , the word “ surface ” will be used as an 
abbreviation for the term “ set of equations expressing the dependent variables in 
terms of the independent ones.” Not only is there much saving in labour, both to 
the writer and to the reader, in the adoption of these terms, but there is the 
additional advantage that the same explanation is applicable alike to the most 
general case and to that in which geometrical conceptions enable the argument to be 
grasped with a clearness unattainable in reasoning of a purely analytic character. 
The reason for adopting the word “ surface ” instead of “ curve ” is that, as the expla¬ 
nation for the curve has been already given, it would be superfluous to repeat it, 
while it seems a real advantage to give the investigation for the case of one dependent 
and two independent variables. 
It will be necessary, in the first place, to examine the conditions under which the 
solution supplied by the rules of the Calculus of Variations is applicable. 
Let the function to be made a minimum be 
U = | dx 1 . . . | dx m f(x i, . . . x m y lt . . . y n ,), 
where f(x±, . . . x m y x , . . . y H ) includes the fluxions of y x . . . y n with regard to the 
independent variables. 
To find the variation in this expression, let us increase y x , y 2 , . . . y n to 2/i+/q, 
y 2 -{-h 2 , . . . y n -\-h n (/q, &c., being functions of x x , x. 2) . . . x m ); then the fluxions of 
y v &c., will be increased by the corresponding fluxions of h v &c. 
For the purpose of ascertaining the limits within which the quantities /q, h 2 , &c., 
must be confined in order that our reasoning may be valid, let us write, in place of 
/q, /q, &c., a.&yi, aA y 2 , &c., where a is a constant, the same for all the variables, and 
A y x , A y 2 , &c., are finite functions of aq, x 2 , . . . x m , or, more accurately, are functions 
which, though they (and their fluxions contained in the integrals) may vanish, neither 
become infinite, nor give infinite values to these fluxions, for values of the independent 
variables included in the region of integration ; they are not-injinite functions. If we 
represent by z, z, &c., any of the dependent variables or their fluxions, we may 
represent the corresponding change by aA z, aAz'. By Taylor’s theorem we may 
write the new value of U corresponding to the new values for the dependent 
variables in the form 
U y+ £ — U y -j- 
+ la 3 2 
dzdz' 
AzAj 
