68 ROYAL SOCIETY OF CANADA 
points on £”’ (see § 4 (a)). Points of regression, £.e., points at which 0” 
and therefore U’ change sign, will correspond in the two curves. An 
ordinate, 6 = const., of the (7, 4) plane, by (3), maps into an ordinate, 
U = const., in the (U, V) — plane. In § 4, we have merely made a 
translatiou of the parameter to eliminate the stationary points, and have 
the relations 
u(@)=U(), p@) =P ©). 
Again when @ is a decreasing function of r, U (7), (= cos @) is an in- 
creasing function of 7, and hence also u (t) of t (see § 4a), and vice 
versa. Hence from (e) it any ordinate, u = const., be drawn, not through 
a point of regression or through the homologue of a stationary point, 
the points of intersection of the ordinate with the curve are finite in 
number, and if these be arranged according to increasing values of the 
slope, p, u is alternately an increasing and decreasing function of ¢. It is 
not difficult to prove from the continuity of the slope on each arc, £,”, 
that the same holds at the homologues of the stationary points which are 
not also points of regression. We have finally therefore, f any ordinate, 
u =v, be drawn, not through a point of regression, the points of intersection 
of the ordinate with the curve are finite in number; and if these be 
arranged according to increasing values of the slope, p, u is alternately an 
increasing and decreasing function of t ; and for all values of u, u is an 
increasing function of t for the intersection with the greatest slope, p. 
(f). Suppose now that we have drawn the ordinates, u = u,, and 
u = 1, and also through the points of regression of £”. The number of 
such points of regression <r + 1, (see § 5a); and is therefore finite. 
Let uw = v, and w = v, be any two adjacent ordinates among these, 
v, < V2 and let u = v, v, << v < v2, be any ordinate not meeting a dis- 
continuity of p. Then by (Il : €, p. 13), the value of p at the 
intersection of u = v, and £’’ are all unequal. We name the arcs of £” 
9 
between v, and v,, 1,2, 3 . . . . in such a way that 
PSL pS pee Sn Cee eu) 
Then this naming is independent of v for v, <v < v, For suppose, 
if possible, that for any other ordinate, u = v', v, < v' < Vz, not meeting 
a discontinuity of p, the order of magnitude of p,, p,, ps. + «+ . à 18 
different from that given by (4). We have then for some z and j. 
p, (v) > pj; (v), and p, CU) << pj (UD es... (5): 
We suppose that v <v”. Since by hypothesis, p; and p; are continuous 
in the vicinity of v and v’’, we have: 
PAG} Gers a0 for u > v in some vicinity of v 
pi (u) — pj (u) < 0 for u < v'in some vicinity of v } 
Consider the set of points on (v v’’) such that there is no point, w, between 

