MR. E. P. CULVERWELL ON DISCRIMINATION OE MAXIMA AND 
106 
11. It is important to observe that this transformation is only valid, provided none 
of the quantities z 1} z 2 , . . . z n used in the transformation vanish for any value 
of x passed through in going from x 0 to x v For, suppose z r vanishes, then, as 
y = z r h r y, the corresponding value of y must be infinite. Now, since % . . . z„ 
and therefore 
AoU = L +Ja(y 0 -^i + . . ?)Bydx-, 
and this can only be independent of the form of By when 
i ( T »-f 1+ '-')= 0 ' ' • • 
that is to say, when y + Ay satisfies the equation satisfied by y, i.e., 
Y„ 
dYy 
dx 
+ &c. = 0. 
(a) 
(&) 
If, then, the solution of (5) be y = / (x 
. . . c 2n ), that of (a) is obtained in the well-known form (see 
Todhtjnter, ‘History of the Calculus of Variations,’ p. 271, or Jellett, ‘ Calculus of Variations,’ p. 84)— 
A F f-A Cl + rf/ 
dc-, 
A + 
+ 
AL 
dc 9 _„ 
Ac. 
'2n' 
00 
'] dCq 
It follows that sq must be (q df/dc 1 + C 2 dfjdc 2 + &c., or, shortly, z 1 = y x ; y lt y 2 , . . . y 2 » being indepen¬ 
dent solutions of (c). 
In finding z 2 , we employ a similar process. Since the part of c 2 U under the integral sign has been 
expressed in terms of the fluxions of f$ 2 y, that of A eU can be expressed in terms of the fluxions of B 2 y 
and A 2 y. Hence, if we choose Ay, so that A 2 y = constant, the integral vanishes identically, and the 
whole variation A tU depends only on limiting values of By, and not on the general value. Hence, as 
before, Ay must be a solution of (a), and evidently it must not be the y 1 solution. Let us denote it by 
y 2 (observing that y 2 , however, cannot be quite arbitrarily chosen from the remaining 2 n — 1 solutions 
because the equation for Ay is only of the (2 n —2) th order, and can only have 2 n — 2 solutions). Then, 
since z 2 = A 1 y/A 2 y = 1/A 2 y . djdx (Ay/%), it easily follows that z 2 = c d/dx (y 2 /y{)- Similarly, when we 
come to the third transformation, we have %A 3 y = A 2 y, whence 
_ A 2 y Id 
A 3 y A 3 y dx 
‘A/A y\~ 
dx \ y 2 / 
AfuA) 
. dx Vy-, / _ 
and, when A 3 y = constant, Ay is a solution y 3 of (a). 
~d 
d 
2/i' 
Hence, 
Zo = 
dx 
1 (y*\ 
± (n\ 
_dx \ih /. 
and similarly for the remainder. As nothing hangs on the discussion of these quantities, it is not worth 
writing out any more. In fact, the only application of the investigation to the present case would be to 
show that the transformations in Art. 9 are always possible, provided a sufficiently short length of the curve 
be taken; that is, to show that it is always possible to choose sq, z 2 , z s , &c., so that none of them vanish 
for any value of x between the limits of integration. But it is easier to see that this is, in general, the 
case from the equations obtained for z v z„, z 3 , &c., in the course of the transformations themselves. For, 
z r being a function of x and arbitrary constants, it follows that, if we put z r = 0, we can solve for the 
value of x in terms of those arbitrary constants, and hence, by taking suitable values for the constants, 
we can, in general, ensure that z r does not vanish for a value x = a: 0 . 
