Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, ETC. 609 



The old words univocal and equivocal may be revived, though not precisely in their 

 old meanings, to signify functions of one value only, and functions giving choice of more than 

 one value. Thus e* is univocal or equivocal, according to the value of x. A univocal 

 function which is always continuous and finite has a differential coefficient of the same 

 qualities. 



Let y 1 strictly denote the limit of dy : dx. When dx and dy are infinitely small, I do not 

 say that dy = y'dx, but that dy = pdx, where p is infinitely near to y'. There are cases in 

 which y is real, and p imaginary, of the form y + adx m + bdx n ^/ - 1. 



Let the words before and after relate to value progressing from -co to + co : thus 

 x is — 8 before it is - 7. Let the phrases 'just before' and 'just after 1 relate to infinitely 

 small differences of value. 



Let the axes of x and y be called horizontal and vertical. And, (x, y) = being the 

 equation of a curve, let dcp : dx = or X = 0, and = 0, be called the horizontal and 

 vertical subordinates of m 0. When a curve meets one only of its subordinates, its tangent 

 is of the same name with that subordinate, either horizontal or vertical. 



Let (x, y) be a function which, for all real and finite values of x and y, is real, finite, 

 and univocal, and such that = does not always give X = 0, d> = 0, as would happen, 

 for instance, if were a positive power of another such function. The curve = then 

 divides the whole plane of co-ordinates into regions in which is always positive, and regions 

 in which is always negative. And every passage over = is a change of sign in (p, 

 unless it be also a passage over one of the subordinates. Thus, if (x, y) be a point of <p = 0, 

 <p (x, y — dy) and <p y (x, y - dy) differ in sign, and <p {x, y + dy) and d> t (x, y + dy) agree in 

 sign, if dy be positive : whence either <p or <p y changes sign in passing through (x, y). Again, 

 x being constant, y cannot change from one point to the next nearest of <p = 0, without 

 passing over <p y = in the interval. Accordingly, when y = is not a separator of positive 

 and negative regions, as when <p y = (w + y) 2 , it follows that one value of x gives to <p = only 

 one real value of y, at most. A curve which is not a separator must be regarded as having 

 its branches repeated an even number. of times: and then we may say that y (or rather the 

 curve x = const.) does not pass from one point of = to the next before or after without 

 crossing y = an odd number of times. 



When (x, y) and (x + dx, y + dy) are both on 0=0, we know from <p x dx + (p v dy 

 + 2 (sPxr^®* + •••) + ••• = ° that (p x dx + <p y dy is of the second order, at lowest. Hence the 

 closest straight line to the curve" is that on which, (x+dx, y + dy) being on the straight line, 

 <p x dx + (p y dy is absolutely zero: and thus we get the tangent and the usual forms connected 

 with it. Of the two regions divided by this tangent, dx = and dy > point out the one in 

 which <p x dx + (pydy has the sign of (p y , or, just after (x, y), the sign of 0. Proceed to 

 (x + dx, y + dy) on the tangent. Then 0, at (x + dx, y + dy) has the sign of 



0« + 20 w y' + <p m y'\ or of - <p y . y", 

 unless this last vanish. That is, dy being > 0, agrees with y at (.17, y + dy), and with 

 - <p y -y" at (x + dx,y + dy) on the tangent. If then <p y and - <p y . y" agree in sign, and if (x, y) 

 be not a multiple point, the curve «= does not fall between the positive continuation (or 



78—2 



