MINERALOGY: R. A. DALY 
659 
Now the sum of any number of terms of a series S (sin mrz)/n is known 
to remain uniformly bounded,^ and the series is known to converge every- 
where, uniformly save in the immediate vicinity of the values z =0, 
± 27r, ± 47r^ etc. Hence the series displayed above converges uni- 
formly for all X, y save in the immediate vicinity of x = y, where, how- 
ever, the sum of any number of terms of the series is bounded. Hence 
all six types of series will converge to values small numerically f or | o- — o- 1 
small, and, when multiplied through by any continuous function may be 
integrated term by term as to x, yielding uniformly convergent series in 
y. Thus the H series will have the stated properties for | o- — o- | ^ 5 > 0. 
v3. Generalization. — The theorem suggests at once a theorem in Gen- 
eral Analysis as defined by E. H. Moore.^ If we employ a quasi-geomet- 
rical terminology this generalization may be stated as follows : any set of 
orthogonal vectors in a functional space lying near enough to a complete 
set of orthogonal vectors in that space is itself complete. Another still 
wider generalization suggests itself: any set of vectors in a functional space 
lying near enough to a complete set of vectors admitting a reciprocal set is 
itself complete and admits a reciprocal set."^ This second generalization 
evidently plays the same part in relation to biorthogonal sets that the 
first does for orthogonal sets. 
1 See Kneser, Die Integralgleichungen und ihre A nwendungen in der mathematischen Physik, 
Braunschweig, 1911, pp. 84-95. 
2 See D. Jackson, Rend. Circ. Mat. Palermo, 32, 1911, (257-262). 
3 See Bull. Amer. Math. Soc, New York, 18, 1911-1912, (334-362). 
^ For a theorem of this type see an article of mine, Paris, C.-R. Acad. Sci., 161, 1917, 
(942-945). 
LOW-TEMPERATURE FORMATION OF ALKALINE FELDSPARS 
IN LIMESTONE 
By Reginald A. Daly 
DEPARTMENT OF GEOLOGY AND GEOGRAPHY. HARVARD UNIVERSITY 
Communicated by W. M. Davis, October 5, 1917 
Recent monographs by A. Heim^ and D. Triimpy,^ dealing with certain 
rock formations in Switzerland, put new emphasis on an important prob- 
lem in minerogenesis. At different horizons of the Jurassic limestones 
of the Churfirsten-Mattstock mountain group, Heim has found abun- 
dant crystals of albite which have evidently developed m^i/w and are not 
of clastic origin. The crystals are automorphic, with maximum lengths 
of 0.2 mm. and average lengths much less (see fig. 150 in Heim's memoir). 
Minute crystals of ankerite are associated. Both albite and ankerite 
are regarded by Heim as due to crystallization on the sea-floor, during 
