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ENGINEERING: J. C. HUNSAKER 
planes 02 = 0 and 23 = 0 cut the surface in orthogonal coincident 
directions. 
Particular interest attaches to the tangent hyperplane perpendicular 
to the mean curvature h. This cuts the surface in the asymptotic di- 
rections and the axes of the (degenerate) conic made up of these two 
directions are the principal directions. The intersection of the sur- 
face and this hyperplane has therefore the fundamental properties of the 
Dupin indicatrix. 
For the proof of the geometric results here stated and for the proofs 
and statements of a large number of others, many of which are entirely 
new, some only new statements of the results of Levi, Kommerell, or 
Segre, reference must be made to our complete memoir ^ Differential Ge- 
ometry of Two-dimensional Surfaces in Hyperspace' which will be 
pubHshed in the Proceedings of the American Academy, Boston. 
1 Kommerell, Die Kriimmung der Zweidimensionalen Gebilde in ebenen Raum von vier 
Dimensionen, Dissertation, Tiibingen, 1897, 53 pp; E. E. Levi, Saggio sulla Theoria delle 
Superficie a due Dimenzioni immersi in un Iperspazio, Pisa, Ann. R. Scu. Norm, 10, 99 pp; 
C. Segre, Su una Classe di Superficie degl' iperspazi, Torino, Att. R. Acc. Sci., 42, 1047-1079 
(1907). 
2 Ricci, Lezioni sulla Theoria delle Superficie, Padova, Drucker, 1898. (Lithographed, 
edition exhausted.) 
3 The vector analysis used is a modification of the Grassmannian system; see Lewis, 
Proc. Amer. Acad. Arts Sci., 46, 165-181 (1910), and Wilson and Lewis, Ihid., 48, 389-507 
(1912). 
^ The corresponding indicatrix for Vi in Sz is not Dupin's but the range of points upon the 
normal described (twice) by the normal curvature vector a. 
DYNAMICAL STABILITY OF AEROPLANES 
By Jerome C. Hunsaker 
U. S. NAVY AND MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
Received by the Academy, April 6. 1916 
The first rational theory of the dynamical stability of aeroplanes is 
due to Bryan^ whose work was extended and applied by Bairstow^ with 
wind tunnel tests on models. 
The utility of such tests in predicting the aerodynamical properties of 
a full size aeroplane is now well understood and the validity of this appli- 
cation has been repeatedly demonstrated. The late E. T. Busk of the 
Physical Staff of the Royal Aircraft Factory, England, appHed Bairstow's 
model tests to the design of an aeroplane and recently succeeding in per- 
fecting an inherently stable machine which could be flown 'hands off.' 
Neither the details of Busk's experiments nor of the type of aeroplane 
developed by him have been disclosed by the British War Ofhce. 
