56 ROYAL SOCIETY OF CANADA 
(e). The slope, P, of the (U, V) curve. 
From (c) and (d), the slope of the curve, 
CU) DT = MERE RER dened Re (18) 
: Ke 
ie. the function, — D een seeseeeseaeeenenen ene siennes (19) 
has a determinate finite value, viz., 
on (7, T,) except at 7 = 7,, T = y and the segments, «; A; and, it may 
be, 7 = 7, We define P (r) at these points as r (7). With this agree- 
ment, the slope, P, is continuous even at the points, 6, and, where the 
latter is defined, P coincides with Since r (t) >0 for 7, <1 <7, 
(see 12), P (r) > 0 for 7, << 7 <7; Hence P (7), = 1%, isoneclase Dy: 
for TS ENT Ste Now for 7, 0 ENT, = eee 
P: Py exists and is finite, being equal to r’ (7, + 0). 
/Toto 
(f). The double point condition. 
From (1: b), if r and @ be polar co-ordinates, we have r (7,2) = r (73), 
and 6 (73) = @ (73) cannot both be true if 7, + 7; Since P = r°, and 
i = cos @ tor 0< = z/, we have P (7) = P G,), Onde 
U (73) cannot both be true if 7, + 7;. 
(g). Collecting these results, our family of admissible curves, £, in 
the (x, y) plane transforms into the totality, £’, in the (uw, v) plane: 
Lau UT) v = Vi) AON te Se Ss. 
Oo — 
with the following properties : 
I’: General Characteris ies : 
(a) V (1) is of class D’ on (7, 7,); 
(b) U (2) is of class C on (7, 7,), and D’ forr, < r< 7; 
Il’: Slope Conditions : 
(a) V'and U’ vanish at most at a finite number of points, 
T = y, and on a finite number of segments, «; Aj. = 1, 2,3. 
(b) Pis of class Con (7, .7,), and: D) for"7, <7 Size 
= FX when this quotient is defined. 
U! 
(c) P (z,) = P (7) and U (7,) = U @,),; cannot both he 
true wih EE 
