OF DIFFERENTIAL EQUATIONS. 549 



the given limits neither the function itself, nor any one of the first n differential coefficients 

 becomes infinite or indeterminate. 



There has been some controversy on M. Cauchy's demonstration in Liouville's journal. 

 With this I have nothing do, as my own demonstration has no affinity whatever with that 

 in dispute. That M. Cauchy has not fully stated the conditions, is evident from instances 

 alone. But further, it is plain from the last paragraph in this page, that if <p'x = CO enters 

 the conditions, <p "x = co must do the same : and so on. 



Let x = r (cos 9 + sin 9 . %/ - l), and let r be called the modulus of CO, and let 9 be called 

 the directing angle, or simply the angle of x. Let the modulus* be always taken as positive; 

 the simple negative quantity requiring the angle tt, or one of equivalent direction. It would be 

 little more than legitimate extension if we were to call r the value of x, and cos 9 + sin 9.^/ — 1 

 its sign : but in that little would lie the seed of much confusion. 



Let any value of x, real or imaginary, which makes (px, or any one of its differential 

 coefficients, infinite, be called a critical value of (px itself. Thus x — ± \/ — 1 are the only 

 critical values of (l + a?*)*. The theorem which I now proceed to prove is as follows : — The 

 development of (px, in ascending powers of x, remains convergent from x = up to x = the 

 least of the critical moduli ; after which it is always divergent. 



I avoid all discussion of the different kinds of convergency, and of the effect produced 

 upon it by variations of sign in the terms, by the following explanation. The given series being 



« ± a x x ± a-iX 2 ± , the series a + a^x + a-^af + may be always convergent when 



x<h, and divergent when x > h. The value of h is the object of the inquiry, without any 

 consideration of whether there be convergency or divergency when x = h. When a„ +1 : a n 

 finally obtains a permanent approach to a fixed limit, the limit is A -1 , and after x = h 

 the terms finally increase without limit. But when the value of a n+1 : a„ never ceases to 

 undulate, it may be possible that from x = h up to some other definite value, the undulations 

 may prevent a n x" from finally increasing without limit, and may perhaps even allow it to 

 diminish without limit, the series being still divergent. I omit all such inquiry, because it is 

 not essential here ; merely remarking that undulating convergency and divergency, in series 

 with all their terms positive, have received so little consideration, that it would not be safe to 

 hazard any assertion upon how they might or might not take place in a series of the form 

 o + a t x + , in passing from one value of x to another. 



In the series a + a^x + a 2 x 2 + in which all the letters are positive, the limit of con- 

 vergency is the same for the series and all its differential coefficients. First, a divergent series 



gives a divergent differential coefficient, for a, + 2a 2 x + is of the same character as 



a x x + Za^pt? + which certainly diverges with a + a^ + Next, if a + a x x + 



be convergent, that is, if x < h, it is convergent when we write x + m for x, x + m being still 



< h: hence a + a x x + + (a { + 2a s x + ) m + can have no divergent coefficient, 



and all the differential coefficients of a + a x x + are convergent. 



* We often want to signify the value of a real quantity, 

 independently of its sign : that is, to speak of the quantity, if 

 it be positive, and of the quantity with its sign changed, if it 



be negative. What we ought to mention, then, is the modulus 

 of the quantity. 



