128 
MR. E. P. CULYERWELL ON DISCRIMINATION OF MAXIMA AND 
functions arising from the solution of the partial differential equations. This excep¬ 
tion is well known in the simpler cases, but I am not aware that it has been generally 
discussed. 
A partial differential equation between independent variables aq, . . . x m and 
dependent variables y x , . . . y n will, in general, require for the complete deter¬ 
mination of y x , . . . y H several sets of limiting conditions ; for instance, one set 
when x m has its limiting values x m = ffx x , . . . x m _ x ) and f x (x x , . . . x m _ x ), and 
another set when x„,_ l = f 0 (aq, . . . x m _ 2 ) and f x (x x , . . . x m _ 2 ), and so on until, 
finally, there is a set derived from x 1 = c 0 and x 1 = <q. But in the particular 
case in which the conditions at the limits are the values of y x , . . . y tl , dy x jdx m , 
. . . dy n /dx m , d^yjdadr,,, &c., for a single surface f(x q, x 2 , , . , x m ) = 0 the 
functions y x , . . . y n will be completely determined without any further limiting 
conditions (provided the proper number of conditions be given), and when the 
limiting conditions are of this character there is no difficulty in finding the 
requisite number of conditions relative to each variable. The limiting equations 
furnished by the Calculus of Variations are not, however, so simple as this, being of the 
dual character above. But, as this does not affect the number of the conditions at the 
limiting values of x„„ it will serve our purpose to find the number of conditions neces¬ 
sary in the simpler case. Let there be n equations, represented by (l), (2), . . . (??.), 
between the n dependent variables y x , y 2 , , . . y n , and let the highest order of 
differentiation with respect to any independent variable in which y r appears in (s) be 
[r, .S']. Then, provided each dependent variable appears in each equation, it can easily 
be shown that the number of conditions necessary to determine the dependent vari¬ 
ables is the greatest of the sets of numbers % [r, s], so chosen that in each set there is 
one term corresponding to each variable and one to each equation. For from § 22 it 
is evident that the number of functions required is the same as if there teas hut one 
independent variable, for the values of the single set y, dy/dx, dryjdad, &c., at the 
boundary determine those of all the other fluxions of y. We may, therefore, discuss 
the question on the supposition that there is but one independent variable. Now it is 
evident that, if we could determine all the successive differential coefficients of each 
function for each point of the bounding surface, we could, by Taylor’s theorem, 
expand the function in a series ; and we know the limiting values of the differential 
coefficients of the y functions with regard to all variables aq, aq, &c., wffien we know 
those for any variable (Art. 22). Hence the problem will be solved if we show how 
many of the limiting values we must assume in order to determine all the rest. But we 
can show that it is possible to find the first Y x quantities of the series y x , dy x /dx x , . . . 
d Y '~ 1 y x /dx x Yl ^ 1 ; the first Y 3 quantities y. 2 , dyfdx x , . . , d Y -~^y 2 /dx Y ~~ l ; and so on for 
the rest, provided we assume 2 [r, 5 ] of these functions; Y 1 and Y 3 , &c., being, if 
necessary, indefinitely large. For differentiate the equation (1) a x times, the equation 
(2) <x 3 times, and so on. Since we are not to introduce any differential coefficients of 
orders higher than Y x — 1, Y 3 — 1, these being the orders of the highest fluxions in 
