350 Prof. A. Cayley on Riccati's Equation. 



and it is easy to see that we may herein change the sign of x q } 

 thereby obtaining another finite particular solution, 



\ q(q-l) q{q-l)2q(2q-l) J 



Reverting to the equation in z, we have next a particular so- 

 lution of the form 



^=A^ + B^ +1 + C^ +1 + I)<£ 3?+I + &c., 



giving between the coefficients the relation 



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



(Zq + l)-B + (2q + l)2qC = 0, 



&q + l)C + {Sq+l)3qD=0, 



(7 ? +l)D + (4^ + l)4gE = 0, 



&c. 



If A=l, we have 



A= 1, 



B = 



(g+i)q 



c=+ te+^+i) 



(q + l)q(2q + l)2q 



D= (g + l)(3g + l)(5g + l) 



(q + l)q(2q+l)2q(3q+l)3q 

 &c, 



where, as in the former case, the series is considered to terminate 

 as soon as there is an evanescent factor in the numerator, with- 

 out any regard to the subsequent coefficients which contain in 

 the denominators the same evanescent factor. 



Hence if we have (2i+l)q= — 1, the series terminates, and 

 we have for u the finite particular solution, 



from which, changing the sign of x q t we deduce the other finite 

 particular solution, 



