230 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



example, manifestly consists of a single branch, whilst the hyperbola, with its 

 two opposed and separable branches, is as clearly multiramic; and still more 

 justly, perhaps, may this quality be predicated of the figure represented by the 

 equation y^ — a x^ = 0, which consists of two distinct right lines. These ex- 

 amples serve to distinguish a reo2 geometrical division into branches, from that 

 apparent division which belongs to the analytical function representing the 

 curve, and which owes its multiramic character not only to the nature of the 

 latter, but to the place of the origin, and the directions of the co-ordinates, as 

 will be sufficiently apparent by considering the different branches under which 

 the circle and lemniscate appear when expressed by means of linear co-ordi- 

 nates. 



Curves really monoramic, and possessed of the first degree of continuity, 

 would thus either return into themselves, or continue to infinity, or terminate 

 abruptly at one or both of their extremities; the last of which conditions ap- 

 pears to require transcendants of a higher order than those elementary func- 

 tions which are usually alluded to when we speak of the continuity of func- 

 tions, or, as might be more accurately expressed by saying— when we speak 

 of functions having a continuity of the first and second degree. 



The variations admitted into this latter class have been illustrated by the 

 examples with which we commenced this section ; and we have there also re- 

 marked that its peculiar character is, that any one point of the same branch 

 of such a curve can only have a single tangent. It is this property, therefore, 

 which includes the ordinary definition of continuity ; and as it has been found 

 of some importance in the theory of vibrating cords, and of some other parts 

 of applied mathematics, and as we also perceive it to form a well marked cha- 

 racter in the arrangements of continuity, it may not be amiss to give this pro- 

 perty an analytical, in place of a geometrical form : to effect which it will be 

 sufficient to design the general algebraic function of the first degree 



y = ax -f b, 

 as a linear function, and to say that every function F x has a continuity of the 

 second degree, Avhen each distinct branch of it has, for any given value of x, 

 a single linear osculate. 



The singular points exhibited in the annexed figures will aflford an evident 

 illustration of these remarks : the first, although determined by the usual rules 

 for points of reflection, owes its peculiarity to the mere position of the axe of 



