346 G. H. KNIBBS. 



dyjdcc = npa*- 1 e- p (8) 



from which it is evident there is neither maximum nor 

 minimum value unless n, p, or <r be zero. 



Restricting the consideration of the expression to those 

 curves which correspond to positive values of <r, they will, 

 for zero values of the abscissa, and whatever the values of 

 p or n, pass through one of the points defined hereunder 

 according to the conditions specified, viz: — 



Origin of curve .v = 0, ?/ = 0; cc = Q, ?y = l; x = 0, i/ = =» 



Condition p— and n - ; p 4- and n ±; P- and n + 



When the abscissa has the value unity, all the curves pass 

 through the point x—1, y = e Q , the values of y which define 

 the points for negative values of n being the reciprocals of 

 those for numerically equal but positive values of n. 



The asymptotes to the various members of the system 

 are the axes, and also the line // = 1, the y- axis and the 

 line y — \ being I he asymptotes for the curves for p negative. 

 The values of y are greater than unity when n is positive, 

 and less than unity when h is negative. 



When n is negative and p is positive, the asymptote is 

 the line y = 0. The effect of decreasing the numerical 

 value of n is to cause the points 'N' and v n' (see Fig. 4) to 

 move toward the line y=l, with which they become iden- 

 tical when n is zero. 



Fig. 4 will fully illustrate the different types of curves for 

 positive values of x ; for negative values of x the curves 

 may be either the images of the corresponding positive 

 curves themselves, or of the reciprocals of the correspond- 

 ing positive curves : the determination of their locus will 

 depend upon the conventions adopted as to the fields of 

 representation (as already indicated). 



Fig. 3, curves 15 to 21, shews the curves c~ m when n= -0 

 to - cc and a- is positive, the line y = being the asymptote; 



