c nxP . For positive values of x the expression if = x m is a 

 series of parabolas for positive values of m, and a series of 

 hyperbolas for negative values of m as shewn in the follow- 

 ing figures, viz., curves 1 to 7, Fig. 1, and curves 8 to 14, 

 Fig. 2. 



Whatever the value of m, if it be positive 1 the curves 

 y = x m will all pass through the points .v = 0, y=0, and 

 •v = l, u = l ; but negative 1 through the points .t'=0, /y = 3c; 

 *=1, u = l; x = co $ y=0. 



If y is to be positive throughout, then for negative values 

 of a, the ordinary convention which assigns a different 

 quadrant for the representation of i/ = .v 2r and y=x lr + 1 , (r 

 being any integer) must be abandoned, and the curves 

 y=x m for negative values of .r, will be symmetrically 

 situated on the opposite side of the Oy axis, i.e. will be the 

 images of the curves for positive values. Thus the quad- 

 rant xOtj is the rer/.o/r of >j for positive values of .v, and 

 the quadrant ijO.v that for negative values. See Figs. 1 

 and 2. 



In general m, n and p will be regarded as susceptible of 

 increase by indefinitely small quantities, and it is for this 

 reason also that the field in graphical representation does 

 not necessarily follow the usual convention, viz., that 

 which places say (- a-) 2 in a different field to (- a?)». 



If y may also be negative, then the ordinary convention 

 may be followed when necessary. In all cases the conse- 

 quences of the particular convention adopted must be 

 carefully attended to. 



The curve y = e nxV illustrated in Figs. 3, 4, 4a and 4b is 

 less simply described. Differentiating this expression 



1 Thus if the value be -'- dm the curve will not he identical with the 

 curve for the value - 8m, where ± 8m are values of m slightly greater 

 or less than zero. See heavy lines on Figs 1 and 2. 



