Vol. 8, 1922 
ZOOLOGY: W. H. COLE 
29 
must be satisfied, and inequality (4) implies that L{v) = 0. But since 
L{u) must satisfy equation (6) which is now equivalent to 
L{u)= ^^^u{x){^i{x)-^2{x))dx^ j\{x)^^^^ (10) 
it is necessary that 
j^^v{x)d{\i{x)-\2{x))=^' 
Then equation (9) impHes that Xi(6) — X2(6) =Xi(a) -X2(a). 
If in the above proof the functions m„(x), Vj{x), and v{x) had been de- 
fined as they were on the interval (a, y), where y is an arbitrary point on 
(a, h), and as identically zero on the rest of (a, 6), the same argument 
would prove that Xi(>') —\2{y) =Xi(a) — X2(a). That is, Xi — X2 is a constant, 
and equation (10) is equivalent to equation (7), with ^ = ^1 — ^2- This 
completes the proof of the theorem. 
If L also depends on a parameter, y, and if instead of assuming that 
equation (1) is satisfied for every value of y, it is assumed that L{un{x),y) 
approaches L{M{x),y) in the mean, the theorem does not apply. For 
instance the functional 
L{u{x),y)=u{y) 
satisfies this condition, but cannot be represented by an integral analogous 
to (7). 
1 Levy, Bui. Soc. Math. France, 48, 1920 (20). 
2 Hart, New York, Bui. Amer. Math. Soc, 27, 1921 (308). 
3 F. Riesz, Ann. Sci. Ec. Norm., Paris (Ser. 3), 31, 1914 (9-14). 
^ Frechet, New York, Trans. Amer. Math. Soc, 15, 1914 (152), or Evans, Cambridge 
{Mass.), Colloquium Lectures (60). 
^ Young, Proc. London Math. Soc, 13, 1914 (137), or Daniell, Princeton, Ann. Math., 
19, 1917 (288-93). 
" Bliss, New York, Bui. Amer. Math. Soc, 24, 1917 (24, corollary). 
7 Ihid. (38, Lemma 8). 
^ Ihid. (36-40, Theorems, 18 and 19). 
SKIN TRANSPLANTATION IN FROG TADPOLES ■ . 
By William H. Colk 
Z061.OGICAL Laboratory, Harvard University, and Department of Biology, 
Lake Forest Coi^IvEge 
Communicated by G. H. Parker, December 21, 1921 
1. The experimental inhibition of vision by means of opaque grafts 
over the eyes of frog tadpoles (from 20 to 100 mm. in length) should con- 
tribute evidence as to any regulatory interaction between the graft and 
the eye, tending to restore the function of the eye. For this purpose 202 
operations were performed upon two species of Rana, catesbeiana and 
clamatans, which had been collected from three widely separated sources. 
