[114] PROFESSOR DE MORGAN ON SOME 



one direction of PQ for which T does coincide with P, namely, that determined by m = y. 

 But if P be on that curve, there is only one direction (and there may not even be that one) 

 in which T does not coincide with P, namely, the direction determined by m = — y x -4- -vy. 



And this exceptional direction is precisely that of the tangent of the curve. For y x =a 

 has the direction of its tangent defined by y = - X*-*"XH* and Xy =ah y V = ~ Xty + Xyr 

 Both of these, when y x and y^ become infinite, take the same value as — y^-i-Yj,. But if 

 y, be infinite without y y , it is through x = const., giving y also infinite : and if y^ be infinite 

 without y„ it is through y = const., giving y m ; so that the exceptional direction is tangent 

 to the curve in those cases also. 



If then, at a point on the curve which gives y, r or y infinite, we find v +y v finite, it 

 follows that y = y on that curve, or the curve is a singular solution of y' = y : while it does 

 not follow that such is not the case, even if y x + y^ y be infinite. In this last case there is 

 a singular solution if, as in the former, y = — y* H- y y be found true. 



I think the preceding detail will be necessary to prevent suspicion from attaching to the 

 following short, and I believe conclusive, account of the matter. On a curve on which y r 

 and y y are both infinite, y is determined by - y^-j-yrj, or - X*y"^"Xyy k°th fractions of the 

 form co -f- 00 > and both having the same value, or extreme form of value (0 or co) with 



— Xr-r-yy. Now if y x + y^y = P, we have y = - — + — , and y is the value of y' on the 



Xv Xy 

 curve, whenever P-Hyj, = 0; that is, certainly when P is finite, and as it may happen 



when P is infinite. But when y^ = CO and not y^, we have y = const, and y = on the 



curve; so that y = y is satisfied when P-^yj,= 0; and similarly for the case in which y, 



is infinite without y^. The reason why this proof is never given, probably lies in the want 



PTi 



of the habit of applying the determining process for - and — to functions of two variables 



co 



so called, in the case in which one of them is constant. 



Nothing which precedes necessarily applies to any case in which <v = co or y = co is the 

 equation to be examined. And I have not made any examination of the cases in which 

 y = co j since this is only an accident of the position of the axes. 



As an example, let ?/' = « + y/{% — y). Here y* = co , y y = co are satisfied only when 

 y = as. And y" = — 1 f 1 (l — a) {x — y)~^ which is only finite for x = y in the case of a = 1. 

 And in this case y = x is obviously a solution. When a is not = 1, there is no singular 

 solution, unless x = co or y = co , should give one. But y = x is nevertheless a straight 

 line which passes through cusps of primitives, having infinite curvature. The primitive 

 equation is 



x = c + 2 (1 - a) log (1 - a - \/x — y) — 2 -y/C* - #)• 



Let yy + x =f(y). This is the equation of the family of curves which always cuts at 

 right angles the family of straight lines y — yx = — y f(—y'~ l ) '■ so that the first equation is 

 that of the involutes of the singular solution of the second. The singular solution, if any, of 

 the first is found by eliminating y between it and y =fy : and this singular solution is the 

 evolute. 



