1 48 Royal Society. 



classify these equations into sets having peculiar and distinguishing 

 properties in regard to this subject. 



The first set includes those equations whose solutions can always 

 be found in convergent series of ascending powers of the independ- 

 ent variable ; and if in such case the equation be solved in series of 

 descending powers (which can be done by this process), those series 

 are certainly always divergent. 



The distiiiguishing marks of this class of equations are, — that the 

 factor of the highest differential coefficient contains one term only; 

 and that (the terms being arranged in an ascending order) when 

 this term is x^, the factor of the next differential coefficient must 

 not contain a term lower than xP-'^, the next riot lower than x^~^y 

 and so on to the end. 



The second set includes those equations whose solutions can always 

 be found in convergent series of descending powers of the independ- 

 ent variable ; and if in such case the equation be solved in series of 

 ascending powers, they are always divergent. 



The distinguishing marks of this class of equations are, — that the 

 factor of the highest differential coefficient contains one term only ; 

 and that when this term is x'^, the next factor must stop dXxp-^, the 

 next at x''~'^, and so on to the end. 



The third set includes equations whose solutions can be found in 

 series oi ascending powers which for some values of the independent 

 variable are convergent, and for other values divergent ; and whose 

 solutions can also be found in series of descending powers which are 

 divergent for all values for which the other series are convergent, and 

 convergent for all values for which the other series are divergent. 



The distinguishing marks of this class of equations are, — that the 

 factor of the highest differential coefficient contains two terms only, 

 and that with reference to the first of such terms the equation is 

 under the restriction mentioned with regard to the first set, and that 

 with reference to the second of such terms it is under the restriction 

 mentioned with regard to the second set. 



The fourth set includes equations whose solutions are or may be 

 divergent for some values of x, both in the ascending and descend- 

 ing series. In some cases, the ascending series is necessarily diver- 

 gent, and the descending series convergent or divergent according to 

 the value of x ; in other cases, the descending series is necessarily 

 divergent, and the ascending series convergent or divergent accord- 

 ing to value ; and in the remaining cases, both series are convergent 

 or divergent according to value, but not so as to be necessarily com- 

 plementary to each other in this respect. 



The distinguishing marks of this class are, — that the first factor 

 may contain more than two terms ; and that either the restriction of 

 the first set is transgressed with reference to the highest term, or the 

 restriction of the second set is transgressed with reference to the 

 lowest term. In this set the divergency arising from value is of a 

 finite character; and, as the series approach without limit to ordi- 

 nary recurring series, there is a probability that the passage from 

 convergency to divergency is not attended with danger. 



