EQUATION AND ITS TRANSFORMATIONS. 
823 
§ VIII. 
Writings specially connected with the contents of the memoir. 
[When only a portion of a paper relates to the subject of the memoir, the page- 
numbers refer only to this portion.] 
Writings referred to in §§ I., II., III. 
(i.) 1868. Cayley, “On Eiccati’s Equation.” ‘Philosophical Magazine/ Fourth 
series, vol. xxxvi., pp. 348-351. 
The equation is written in the form 
and the expressions P 3 , Q 3 , IP, S 3 of art. 17 are obtained by assuming series of the 
forms in question and equating coefficients. Two of the series terminate when q is 
the reciprocal of an uneven integer. 
(ii.) 1869. - “Note on the Integration of Certain Differential Equations by 
Series.” ‘Messenger of Mathematics/ First series, vol. v., pp. 77-82. 
It is shown that if we have a solution 
A (x a -\-p : x a+lJ r 
of a differential equation, and that if one of the factors in a numerator, say a,., vanishes, 
then we may stop at the preceding term, the finite series so obtained being a particular 
integral ; but that if we continue the series, notwithstanding the evanescent factor, 
and if at length a factor in a denominator, say h,(s>r), vanishes, then the series 
recommences with the term involving x a+s , and we have another particular integral 
U\ h +1 A+A+3 / 
,0 
in which A' - may be replaced by a new arbitrary constant B. 
(iii.) 1872. Glaisher. “On the Delations between the Particular Integrals in 
Cayley’s Solution of Eiccati’s Equation.” ‘ Philosophical Magazine/ Fourth series, 
vol. xliii., pp. 433-438. 
The relations between U 3 , V 3 , P 3 , Q 3 , E 3 , S 3 given in art. 17 are obtained. These 
afford an example of the principle explained in (ii.). See the introduction, p. 763. 
