70 ROYAL SOCIETY OF CANADA 
$8. Reduction to an Integral on a Path without Points of Regression. 
(a). Construction. 

U-; LU 
(Fig. e) 
We now proceed to show that the value of J” taken along Ç,, (see S6) 
exceeds that along any other path of the set, £’. For this purpose con- 
sider any particular curve, £. If it have a point of regression we proceed 
to construct an associated curve, C, without such points, which gives J” a 
greater value. We then show that 7” along ©, exceeds J” J along C or 
along any curve, £, without points of regression. Since w (t) is con- 
tinuous (1”: b),* there must be a point, ¢, on at which wu (t) reaches a 
minimum value, uw, say. Ifu_, < us, we adjoin to £ the line, U_y U ON 
Ou. The resulting curve we name £*, and the parameter of the end-point 
we still callt,. Sinceonu, Uy uw =0ifu, < uo, it follows from $6 that 
the integral, 7”, taken along £* is equal to Z” along £, although if w, < Up, 
£* does not belong to the set, £”. 
As in §5 (a) we divide £* into arcs on which » = f, (u). The num- 
ber of arcs and points of regression on £* can be made to exc2ed the 
number on £” by not.more than one. We draw ordinates, u = 4, ,u= 1, 
and through each point of regression of £* after the manner of the 
preceding paragraph, and propose to consider the contributions to J” of 
the arcs between two successive ordinates, v, and v,. 
(b). Intersections of an Ordinate with £*. 
Consider any ordinate, u = v, where v, <v < v, Since uw’ (t) = 0, 
within Lj and we can certainly choose a parimeter, ¢t, on u_, U, so that 
u’ (t) > 0 on w, u, wis either an increasing function of ¢, or a decreasing 
function of t in some vicinity of (7’ 7),7’ <7; and likewise in some 

1 See p. 14. 
