238 
MATHEMATICS: A. DRESDEN 
country, and to obtain a collection of material for illustrating prehis- 
toric aboriginal pathology and surgery. Considerable success was met 
with in both directions. The anthropology of the coast was mapped 
out for the distance of approximately 600 miles, and some insight was 
obtained into that of the highlands. It was ascertained that important 
separate political and cultural coastal groups, such as the Chimu and 
the Nasca people, were no special units, anthropologically, but belonged 
to the same physical type as the rest of the coast population. The 
collections made on this trip, being selections from nearly 5000 burials, 
are especially valuable. Finally, the exploration made possible rich 
original exhibits at San Diego, covering practically the whole field of 
pre-Columbian Indian pathology, to which are added 60 crania showing 
all the forms of ancient Indian trephining. The general results of this 
expedition have already been published,^ but the material collected 
offers a rich opportunity for further investigation. 
1 Hrdlicka, A., The most ancient skeletal remains of man, Smithsonian Rept. for 1914, 
pp. 491-552, pis. i-xli. 
2 For preliminary reports on this work, see Smithsonian Inst., Misc. Collect., 60 (1912); 
Compte-Rendu XIV Cong. Intern. d'Anthropologie et d'Archeologie PrShist., Geneve, 1913; and 
Trans. XVIII Intern. Cong. Americanists, London, 1914. 
^ Hrdlicka, A., Anthropological work in Peru in 1913, with notes on the pathology of 
the ancient Peruvians. Smithsonian Inst., Misc. Collect., 61, no. 18 (Publication 2264), 
t914, pp. i-v, 1-69, 26 pis. 
THE SECOND DERIVATIVES OF THE EXTREMAL INTEGRAL 
FOR A GENERAL CLASS OF PROBLEMS OF THE 
L In an earlier paper, ^ I obtained expressions for the second derivatives 
of the extremal-integral for the problem of minimizing the integral 
in terms of fundamental solutions of the Weierstrass form of Jacobins 
differential equation for that problem. ^ In the same paper these ex- 
pressions were used for deriving necessary conditions which must be 
satisfied by a curve which is to minimize the integral (1), if one or both 
endpoints are allowed to vary along a curve,' or if curves are admitted 
whose slopes possess a finite number of finite discontinuities.^ 
2. The method of differentiation employed in arriving at these re- 
CALCULUS OF VARIATIONS 
By Arnold Dresden 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF WISCONSIN 
Presented to the Academy, January 18, 1915 
(1) 
