Prof. A. Cayleyow Riccati's Equation. 351 



Hence, in the equation 



dx^ V °° > 

 where q(2i+l) = + 1, we have (writing D = l) 



CP'+Q' 



y= 



CP + Q 



where C is the constant of integration, P, Q are finite series as 

 above, and P', Q' are the derived functions of P and Q. Writing 

 successively 2=0, 2 = 1, 2 = 2, &c, we may tabulate the solutions 



dy 1 J 



-JL + y 2 =#- 4 , V = xe *, Q=xe*, 



^ + y*= #-#, P= (1 _ sJ )e*^ , Q= (1 + 3^)e~ 3 ^, 



^ +2,2=a?-f, P = ^(l + 3o?-i)e- 3 ^, Q=a?(l-3a?-£)e 3 *~* 



&c. 



+ 2/2 =tr -|. P=^l-5a*+^**W, Q=(l + 5a?*+^a?*V*^ 



It is hardly necessary to make the final step of calculating P' 

 and Q! and substituting in y ; but, as an example, take the above 



equation -j- + y' 2 =x-% : we have 



y= 1 



C(l— 3a?£)e3*i+ (l + 3^)e~ 3 ^ 



which is readily identified with the solution, p. 98 of Boole's 

 'Differential Equations' (Cambridge, 1859). 



Cambridge, September 29, 1868. 



