AS COMMONLY INVESTIGATED. 239 



1 2 



where not only f z, f z, &c., vanish for z = x, but they are composed of factors 

 f.z.fz, f^^z.f.z.fz, &c., each of which vanishes when z is equal to x. 

 Viewed analytically, it will thus follow that 



1 a n 



f Z, f Z, f Z, &C. . . . Q 



are functions arranged in the order of their magnitude, although we could not 

 thence conclude that, in all cases, these functions form a converging series 

 when z was taken infinitely near to a. 



n 



To examine this question, and to show under what circumstances f z be- 



n 



comes of the form s, (z — x,) it will, perhaps, be most convenient to substitute 

 h for z — X, or to examine when F (x + h) expands under the form 



Fx + Fx. |h + Fx. |h.. .. Q. 1(h). (S) - 



And with this purpose we must commence with the expansion 



F(x + h) = Fx + Tt 

 1 

 where R is a function of x and h that is merely limited to vanish when h = 0. 



Now, we observe that if F x possesses a continuity of the first order, or if it 

 possesses such a continuity between the limits x = a — ne and x = a + n f , 

 we can apply to it the idea attached to a rate of increase, and can ascertain 

 when this increase is fastest or slowest; to illustrate which further, we remark 

 that, if the function was expressed geometrically, the portion in question would 

 evidently be a continuous curve, osculating with some straight line, and in- 

 creasing faster or slower v/ith the position of the latter ; but, as the appeal to 

 geometry is justly considered foreign to the subject, and does not apply to ima- 

 ginary functions, where we must consider separately the increase of the real 

 and of the imaginary part, we must use, in place of this illustration, the nume- 

 rical expression of the function by means of tables. 



Let its form be ' 



p+QV-i ; 



then if, when we substitute for z the successive values a ± f, a ± 2 f, &c., P 

 and Q are found to approach to some definite rate of increase as e is taken 

 smaller and smaller, or, at least, if they so far approach to a definite rate of 

 increase that we can assert the alterations to be faster or slower in one part of 

 the portion than in another part; or, again, that such rate is the same through- 

 out the whole portion, in either of these cases the portion has a continuity that 

 is finite and of the first degree. 



