[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 143 
DIFFERENTIAL AND INTEGRAL CALCULUS. 
[Time, 7 hours ; 7 a.m.-2 p.m. ] 
Ox, Oy, Oz being three given rectangular axes, consider a surface 
S, of a single sheet. Suppose s any portion of S, without any common 
8 PP yp 5 
point with Oz and not having a tangent plane parallel to Os. 
I. Suppose À the area of the projection of s on the plane of xy; 
B the volume bounded by the area s, its projection A and the projecting 
cylinder; C, the volume bounded by the area s and the cone having 
this area for base the origin for vertex ; D, the volume bounded by the 
area s and by the conicoid which has the contour of s for directrix, Oz 
for axis and xOy for director plane. 
The quantities B, C, D representing the volumes in question with 
suitable signs, show that 
(1) 3C=B-2D 
as long as the areas is not cut by certain lines situated on S. Show 
also that the formula is still true without this last restriction, if the 
elements of 2, in magnitude and sign, be always such that 
B=ff2dx dy 
(the double integral being applied to the area A), and if at the same 
time the elements of volumes C, D, are also affected by suitable signs 
depending on x, y, p, q (-- = g= 5 . Indicate as far as possible 
the geometrical conventions of sign to which we are thus led. 
II. The cone (supposed reduced to a single nappe) which bounds 
the volume C, determines, on the cylinders of revolution of radius 1 
which has Os for axis, an algebraic area of which the elements will be 
affected by the same signs as the corresponding elements of C, in con- 
formity to the preceding conventions : suppose Æ this area. 
On the other hand turn s about Oz and designate by / the volume 
of revolution thus generated ; by G, the area of the meridian section of 
this volume, an element of Æ# or of G being equally affected by a sign 
(the same in the two cases) according to suitable convention. 
Determine the surface S such that, for every portion s (without 
point common with Oz or tangent plane parallel to Os) taken on this 
surface, we have the relation 
(2) aA+8B+3cC+eE+ SF F+gG=0 
27 
where a, 3, c, e, f, g are constants. Show that S will verify a certain 
partial differential equation of the first order of which the coefficients 
are rational functions of x, y, z, p, where p= nl x8 +4” (the radical being 
taken as positive). Indicate (again geometrically) the determination 
of the common sign to give to any element of # and to the corresponding 
element of G such that this equation is the same for the whole surface 
under consideration. 

