[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 147 
FINAL EXAMINATIONS. 
NUMERICAL CALCULATION. 
Consider the differential equation 

ax: 
Find the smallest value to give to m in order that the equation admits 
a solution of which the representative curve, symmetric with respect to 
Oy is tangent to Ox at the points A, A’ of abscissae x= +1, x= -1, 
the value y corresponding to, x =0 being equal to 1 (point 8)—Determine 
the points of inflection between A and J’ of the representative curve of y. 
Find, as exactly as possible, the portion AB of this curve, the unit of 
length being supposed equal to 40 divisions of the square employed. 
DESCRIPTIVE GEOMETRY (Diagram).* 
An equilateral hyperbolic paraboloid has for vertex the point 
de cote 10 cm. and a’ éloignement 10 cm. projected on the major axis of 
the sheet; a principal parabola P is horizontal and its focus de cote 
10 cm. and d’éloignement 10 cm. is situated 1 cm. 5 m. to the right of 
the vertex. 
A second hyperbolic paraboloid has director plane, a plane of 
profile; of rectilinear generativers there are: 1°. the axis of the first 
paraboloid; 2°. a horizontal de cote 13 cm., of which the projection on 
the plane of the parabola P meets the axis of this parabola 3 cm. to the 
left of the vertex and the tangent at the vertex 3 cm in front of the 
vertex. 
Consider, on the one part, the region A of the space limited by the 
first paraboloid and which corresponds to the interior of the parabola ?; 
on the other part, the region B of the space limited by the second 
paraboloid and which corresponds to the part of the horizontal plane 
de cote 10 cm. situated in front of the axis of the first paraboloid. 
Represent the solid bounded by the two paraboloids, by the hori- 
zontal planes de cote 17 cm. and 2 cm. and by the plane of the profile 
situated 10 cm. to the right of the vertex of the first paraboloid, the 
solid part being always in the regions A, B. 
It is supposed that the planes of projection are transparent. 
(2) ORAL. 
MATHEMATIQUES ELEMENTAIRES. 
Supplementary trihedral angles. Applications. 
Symmetry with respect to a line, a point, a plane. (Programme of 
the Premiere). 
Relations between the coefficients and the roots of the equation of 
the second degree. Applications. 


*A solution of the problem in this paper is given in Revue de Mathématiques 
Spéciales Nov. 1910, XI, 42-45. 
