146 ROYAL SOCIETY OF CANADA 
Find the characteristic curves of the partial differential equation thus 
obtained, by employing the semipolar coordinates p, w (polar coordinates 
of the projection of the point on the plane xOy), s (coordinate of the 
point). 
Study the projections of these characteristic curves on the plane 
xOy. Show that there exist characteristic curves which are situated on 
a cylinder of revolution with the axis Os, and discuss their form. 
III. Suppose the constants J, c, bound by the relation 
(3) b+3c=0 
and consider the curvilinear integral, 
[LC SJ snee (se) 
taken from a point # to a point A” of the surface S, along the path Z 
situated entirely on this surface. Show that if S satisfies the condition 
which has been imposed on it in the II. part, and if, under the sign 
the sign of the term in dz has been suitably chosen, the integral 7 does 
not change its value, when 47 and 7 remaining fixed, we change in 
a continuous manner on the surface, the line Z traced between these 
two points. 
If, instead of the relation (3), the constants à, c are connected by 
the relation 
(4) 64+ 6c=0 
a property ur to the preceding appertains to the integral 
p° 
= fs pee (xdy — y0x) — egs- are (xdx + yoy) + P[efp*èz — a(xdy — vex) | 
p 
where es is a suitably chosen polynomial in p and € is one of the two 
quantities +1, —1. 
IV. Suppose further that the surface S contains a circumference of 
which the plane passes through Os and which has no point common 
with Os, or with the cylinder of revolution considered above (end of 
II. part). 
On each of the characteristic curves for the different points of this 
circumference take a finite arc, such that the portion X of S thus de- 
termined does not contain any singularity. 
Supposing given the value of the integral Z (in the case of relation 
(3)) or J (in the case of the relation (4)), the length of a certain path Z 
joining JZ and JZ’ and situated on X, what are the other values that this 
integral can acquire when Z is scccessively replaced by all the. other 
paths which can be traced between the same points on & ? 
Indicate the relation which exists between the radius of the circum- 
ference the distance of its centre to Oz and the coefficients of equation (2) 
in order that the integral considered be unique under these conditions. 
5 July. 
