ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 145 
Establish the relations, 
Rr Vda 
costu= (1+ 5 ) cost x, cos 7) — G+ -) costy, 
RR R' 
Étant ea 
Tétany RP 
Verify by means of these relations, that the tangents RM, RN and 
RM’, RN’ form a harmonic pencil.—The point À of the line 7 can also 
be replaced by a point S of the line s. 


2 July. 
MATHÉMATIQUES SPÉCIALES. * 
[Time, 7 hours; 7 a.m.-2 p.m.] 
Given a parabola (P) and a line (D) of which the equations with 
respect to a system of rectangular coordinate axes, are: 
—2 px=0 =0 
ue nes ee 
and, suppose the surface (S) generated by a variable line (A) which 
meets (P) in A and (D) in a point B, such that the distance AB is a 
constant /. 
1°. Construct the projection on the plane XOY of a section of the 
surface by a plane parallel to the plane XOY; construct the tangent in 
a point of this projection and show that the curve obtained can be re- 
garded as the locus of the middle points of the chords parallel to OX and 
limited, on the one hand by a parabola of vertex O and axis OX, on 
the other hand by an ellipse of which the axes are in the direction OX 
and OF. 
2°. Two kinds of lines À can be distinguished, according as the 
abcissa of À is superior or inferior to that of B; in the preceding 
sections separate the arcs which correspond to the generatrices of the 
one system or the other and find the locus of the points which limit 
these arcs. 
3°. Consider the solid limited by the surface (.S) and by the planes 
s+a=0s%-2a=0; find its volume and construct its apparent contour 
on the plane ZOX. 
4°, Determine the orthogonal trajectories of the lines (A). Through 
a point A, two lines (A) can be drawn to meet an orthogonal trajectory 
in two points C and C’; show that this trajectory can be chosen such 
that the sum AC+AC’ is proportional to the abcissa of 4—Can the 
given constants be chosen such that only one orthogonal trajectory 
meets all the lines (A) between their points situated on the parabola (?) 
and on the line (D)? 
3 July. 
*The solutions of the questions in this paper are given in Revue de Mathématiques 
Spéciales Juin, 1910; X, 532-540. 
Sec. III., 1910. 10. 
