OF DIFFERENTIAL EQUATIONS. 551 



can but amount to the addition or subtraction of certain terms of the form Zatfc k . Let us 

 suppose it, then, established from the beginning : that is, whatever sign the term in x h may 

 have, (k<l), the terms in x k+l , .z?* +2/ , &c. have the same. If these terms be negative, in order 

 to make them all positive, we must add 2 (a k v k + a k+l x k + l + ...) to (px. This, 1, X, X s , ... X' _1 

 being the Zth roots* of unity, is 



- { (px + X l ~ k (p (Xx) + (X 8 )^ (X 3 *) + } , 



and for every negative term which occurs in the first I terms, a similar addition must be made. 



Hence, if (px be of the form a ± a v v =t a^pc 2 ± , with a cycle of I in the signs, the series 



a + a v x + a 2 <r 2 + in which all the terms are positive, is of the form 



A (px + A x (p (Xx) + A^(p(X 2 x) + + A t _ x (p (X i_1 a?). 



From this it follows, by the last case of the theorem, that divergence, if it ever begin, 

 begins immediately at or after a real and positive value of x which makes some differential 

 coefficient of the above infinite. 



One or more cases, then, of d)'"' (X c x) must become infinite for a real and positive value of 

 a; and the theorem is proved for every case in which a cycle of signs is ultimately established, 

 with the addition that if the cycle of signs go through an extent of I terms, 



cos 9 + sin . ^/ — 1 



must be an /th root of unity in the critical value which gives the least modulus. 



Lastly, suppose that no cycle of signs is ever established. Let fx be the series of positive 



terms a v + a t x+ , and let Fx be the series which has cycles of signs, with I in each cycle, 



and each cycle agreeing with the first I terms of (px. A process similar to the one already 

 used will give 



Fx = A fx + AJiXx) + A 2 f(\'x) + + A^fiX 1 -^), 



and the greater I is taken, the more nearly will Fx represent (px through the whole period of 

 convergency, and, consequently, for all values of x, unless we suppose (px discontinuous. 

 Many terms may disappear by the vanishing of their coefficients; and, in particular, the cases 

 in which A = will be very frequent. If / < "' ) A=co, we shall always have f <m) {X c x) =eo 

 when x = ^X" c , and thence F m (h\~ c ) =eo . And if f {m) a =co cannot be true for any modulus 

 of x less than h, then neither can F {m) x = 05 for any modulus less than h. Up to the limit then, 

 and at the limit, when I is infinite, and Fx = (px, the theorem is true, the directing angle being 

 possibly incommensurable with 2tt when the case gives an infinite value of c. 



It is thus proved that divergence cannot begin before the lowest critical modulus of (px; it 

 has been proved that convergence cannot last beyond that value : and these two assertions 

 contain the whole theorem. 



When the series is always divergent, we might at first suppose it indicated that (px or 

 some differential coefficient becomes infinite when x = : that is, we might suppose it proved 



* This ought to be called Simpson's Theorem. It was given by Thomas Simpson in the Philosophical Transactions, as 

 read November 16, 1758. 



Vol. IX. Part IV. 71 



