550 Mb DE MORGAN, ON VARIOUS POINTS 



Let a ± a x x ± be one of the differential coefficients of <j>x, and let 



r (cos 9 + sin 9 . -y / - l) 

 be a value which makes it infinite. Consequently, one of the two series 



a ± «! cos9.r ± a s cos2#.r 2 ± and ± a^ sin 9 . r ± a 2 sin 20. r 2 ±. 



must be infinite. Hence c + a,r + a 2 r 2 + must be divergent: whence the original series 



for <bx must be divergent when x is greater than r. We cannot undertake to say this when 

 x = r : at the point of transition, convergency or divergency must be discovered by the closer 

 rules. 



So much as this is then proved, namely, that <bx cannot remain convergent after x has 

 passed any one critical modulus whatsoever. It remains to shew the converse, that <bx cannot 

 become divergent until it has passed a critical modulus : that is, <px remains convergent until 

 x reaches the least of the critical moduli. 



First, let all the coefficients be positive, or let <px = a + a x x + a 2 x* + Let 



^n+i' T " +1 = ^b+i^" 41 + an d kt tne point of transition be at x = h. Let F be a quantity 



which may be as small as we please, and take n so great that a + a x x + + a„x", at 



x = h + £, is greater than <px, if <px be real, or than the real part of <px, if <px become 

 imaginary after x = h. Let <ph be finite, for if <ph be infinite, the theorem is obvious. Con- 

 sequently, at x = h — £, P„ +1 *" +1 is positive, being the sum of a converging series of positive 

 terms : but at x = h + £, P n+l x"* 1 is either negative, or imaginary with a negative real part. 

 This change, which begins at x — h, since we may suppose £ of any finite degree of small- 

 ness, is not made by passage of P„ + i through zero. For if it were so, we should have 



dth = a + + a„h n , which is absurd if the series a + a l h+ be convergent ; and equally 



so if it be divergent, for we may begin by supposing that n is so great as to make 



a + + a n h" greater than (ph. The change then must be made, either through infinity, or 



by an absolute discontinuity. In either case <j> ( " +v h is infinite, for if it were finite (since we 

 know that (j5> (,,+1) # never becomes infinite before x = h by the previous part of the theorem) we 

 should have P„ +1 = ^> ( " +1) (\A) : 2.3...(w + 1), where \< 1, that is, P„ +1 would be finite and 

 continuous at x = h, and for some subsequent interval. 



Consequently, when all the coefficients of development are positive, it follows that before 

 divergency begins, some differential coefficient of <px becomes infinite, and for a real and posi- 

 tive value of x. This may mean when divergency begins : that is, the series may become 

 divergent at the transition value of x. And it is evident that the coefficients need not be posi- 

 tive from the outset: it is enough that they are never otherwise after a finite number of terms. 

 And hence ultimate permanence of positive sign in the differential coefficients of <px, when 

 x = 0, indicates one of two things ; either the series is never divergent, or the root which 

 gives the least critical value is real and positive. 



Next let us suppose, in a ± a y x ± a 2 a? 2 ± ..., that instead of having all the signs positive, 

 the signs are varied, but in such a manner that there is ultimately a recurring cycle of signs, 

 I in number ; as for instance, 1 = 6, the cycle being (+ + - + -+)(++- + -+) & c . If 

 this cycle be permanently established after a finite number of terms, the alterations necessary 

 to establish a cycle commencing with a will not effect the critical values of <px, since they 



