238 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



But, notwitlistanding all these various changes through which R may pass, 

 we may always write 



Fz = Fa + q.fz 



f a = ; 

 since, whilst we assert nothing more concerning f z than is contained in the 

 second of these equations, we merely express in algebraic terms the truism 

 that F a = F a. It will be observed, however, that if we assume f z to have 

 the form ^ (z — a,) and much more if we assume it as expressed by (z — a)™, 

 we depart as much from the limited condition represented, as if we assumed 



iz to be log -, or sin (z — a,) or any other function that has the property of 

 a 



vanishing when z = a. 



This function f z is the prime factor of Fz — Fa that vanishes when z = a; 

 but its form, far from being a matter of arbitrary assumption, must be deter-, 

 mined by a strict analysis, which will terminate in proving that for all func- 

 tions which have a certain degree of continuity, and with the exception of cer- 

 tain values of a, f z may be assumed as any function of z — a that has the same 

 evanescent magnitude as z — a itself 



A similar analysis with regard to Q would prove 



Q = F(a) + Q.f..z; 

 and continuing this decomposition, and writing x in place of a, we arrive at 

 the theorem 



Fz = Fx + F.x.fz + Fx.f.z-{-&c Q.f.z 



of the co-ordinates, we derive y = b, a real existence, from the first, and y = b-l-av' — 1 

 from the second; which last value of y is as completely imaginary as y = a \/ — 1. 



This distinction, and its loss when the geometrical illustration is departed from, will be seen in 

 the annexed figure, where, as there are no points of the curve ex- 

 isting between the conjugate point A and the vertex of the branch B, 

 the squares, or any other functions of such non-existent co-ordinates, 

 must be wholly imaginary. Representing these co-ordinates, how- 

 ever, by y = a -v/ — 1, their squares become real; and thus it is evi- 

 dent that in applying algebra to geometry we do, usually, superadd to imaginary quantities, or to 

 such quantities when used in certain ways, the condition that no functional operation is to be re- 

 garded as removing the absurdity. It is this property for which a sign is wanted, in order to com- 

 plete the theory of correlation, and thus to remove all difficulty which has arisen from regarding 

 >'^ — 1 as a sign of perpendicularity. 



