Prof. A. Cay ley on Riccatr's Equation, 349 



showing the form under which the constant of integration C-hD 

 is contained in y. To complete the solution, assume 



we find 



8+ 2 **- , S+(*- i )*'- s *= 0: 



considering first the particular integral of the form 



we find that the equation will be satisfied if 



(?-l)A + q te-l)B = 0, 



(o2-l)C + 32(3?-l)D = 0, 

 (7g-l)D-f4?(4 5 -l)E = 0, 

 &c. 



If A=l, this is 



A = 1, 



B = - 





r= | (g"l).(3g-l) 

 "*" q(q-l)2q(2q-l) 



D = 



( g -l)(3y-l)(5g-l) 

 ? (g-l)2g(2g-i)3^3g-l) 



where it is to be noticed that the series may be considered to stop 

 so soon as there is in the numerator a factor =0. For instance, 

 if 5q — 1=0, then if the particular integral had been assumed to 

 be z = A. + l$x q + Cx 2q , the only conditions to be satisfied by the 

 coefficients are the first and second equations giving the foregoing 

 values of A, B, C. It is immaterial that the analytical expres- 

 sions of F and the subsequent coefficients contain in the denomi- 

 nators the evanescent factor 5q — 1 ; the coefficients after C do 

 not ever come into consideration. 



Thus if (2i + \)q= + 1, the series terminates, and we have for 

 u the finite particular solution 



V q(q-X) ?(?-i)2<z(2?-i) ) 



