MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
105 
When we substitute for B n y its value in terms of By and its fluxions, we must clearly 
have an expression of the form 
S 2 U = L + JN u {/o (x) By (n) +f x (x) By + &c. + f n (x) By} 2 dx ; 
and, as it is evident that the coefficient of S// ( " )3 is unaltered by the transformations, 
the expression for S 2 U may be written in the form 
L + | Y nn ( By w + By (n ~ l) + &c. + %) 2 dx, 
where L contains By and its fluxions up to, but not including, By' 1 . 
10. This is the same form as that in Jacobi’s method, and it has been obtained on 
the supposition that integrations may be performed on By, By, .. . By (n) ; this requires 
that By . . . 8y°' -1) should be continuous in every sense, and that By {u) should not 
become infinite, but it may suddenly change from one finite value to another. It is 
very important to note this in connection with the condition determining the point at 
which a curve ceases to fulfil the synclastic condition. 
As yet, the equations determining the quantities z x , z. 2 , . . . z n used in the trans¬ 
formation have not been given, and it might be supposed that they would have to be 
found in order to determine the part of S 2 U depending on the limits. It will, 
however, be seen that in this as well as in every other case under the Calculus of 
Variations it is only necessary to use these quantities to show that the reduction can 
be made. As a matter of interest, however, a short discussion is given below.* 
* It may be interesting, though not required for our immediate purpose, to examine the relation 
which the coefficients z lT &c., in the transformation bear to the value for y which gives the synclastic 
value to the integral. For this purpose it will be convenient to use A oU instead of c 2 U. 
In the first place, it is evident that, if we have any expression for o 2 U as a quadratic function of Sy 
and its fluxions, we shall get cAU by taking the polar, with respect to the quadratic function, of the 
point whose coordinates are Ay, Ay, Ay, and so on, using the word “ point” in an extended sense. If a 
proof of this is desired, write (S + KA) for S in 
3 2 U =/(oy, Sy, . . . Sy M ), 
and get 
C + KA) 2 U = f{(S + KA) y, (S + KA) y ...(() + KA) y»j, 
and, after expanding, equate coefficients of K, which is quite arbitrary. Hence, from equation (7) we 
obtain 
r n n 
A f U = L + S 2 A,. 4 Aj y r Syj s dx ; 
and, since the integral contains only fluxions of A 1 y 1 , A <5U will depend only on the limits, whatever be 
the form of Sy, provided A x y, that is, A yjz^, is constant; and therefore z x is one of the values of Ay for 
which A MJ is independent of the form of MJ. Now, 
*u=L + [(r t -5 + ^- fa.)** 
P 
MDCCCLXXXVII.—A. 
