MATHEMATICS: A. DRESDEN 239 
suits had been used before, probably first by Poisson or by Jacobi,^ in 
the treatment of the integral (1) and also of more general integrals. 
These uses had been restricted however either to the derivation of con- 
ditions of the first order, or else to the study of special problems of 
variable endpoints.^ Now it is evident that the number of special prob- 
lems of this character increases rapidly as one proceeds to consider prob- 
lems involving more and more unknown functions. To secure a general 
mode of treatment for all these special cases, it has seemed desirable to 
secure formulas for the second derivatives of the extremal-integral in 
more general problems than the one treated in my paper cited above. 
3. The present paper derives such formulae for the integrals 
jf (x, yu-"'j yn, y['" •, yn) dx, where y = dy/dx, (2) 
and 
Jf (yi, • • • • , y„; y/, • • • yO dt, where y' = dy/dt. (3) 
In the case of the integral (2) the classical theory suffices to carry the 
work through and one secures expressions for the second derivatives of 
the extremal-integral in terms of two sets of conjugate solutions of the 
self-adjoint system of Jacobi differential equations: 
Denoting by Ujk and Vjk systems of solutions of Jacobi' s differential 
equations, which satisfy the initial conditions 
Ujk (xi) = djk, Ujk fe) = 0; Vjk {xi) = 0, Vjk fe) = bjk, 
and using the current abbreviated notation: 
/•o = ^,/„+,= ^, etc.; /o^^^=/o(^,),etc.; W = ^ 
^ by i dx ' 
we have 
- ^f):li,n+jyiiy2kV'ki{Xi) = ^fJUn+jy^i/lkU'kjM, 
bxibx2 ijk ' ijk 
( - 1 - ^ftUi y'ri - n+j y'n rj + 
»■ ij 
^fnli,n+jy'riy'rk[^lUkj{x,) -f B^^v' kj{Xr)]l 
ijk 
dXr^ 
= ( - DVr^ - Xf:lj, , - S/a,, „^k y'rj [Br^u',k{Xr) + b,,v\k ix,)]] 
OXrOyri j jk 
= ( - ^fnk n^y'rj " ^ A'U, n^j^ rk [^iW kj{x,) + 
j jk 
Br2v\j{Xr)]l 
ly-' ,^ky\j [BrWik M + d^u'ik {ocr)] 
^ysi jk 
= (- 1)^2/^1,, y\k [driUkj fe) -f- dr2v\j (x,)], 
jk 
