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



VI. The series U# 2 , giving x the successive values 1, 2, 3, 4, 

 5, 6, 7, 8, 9, is 1, 4, 9, 7, 7, 9, 4, 1, 9. This series enables Tables 

 of squares to be verified within certain limits of correctness. 



VII. A similar series for TJx 3 is 1, 8, 9, 1, 8, 9, I, 8, 9. 



VIII. A similar series for Vx~ l is (excluding the values of 

 x = S, 6, and 9) 1, 5, — , 7, 2, — , 4, 8, — . This series enables 

 unitates to be assigned to such circulating decimals as \. 



The operations \J 8 x } U 9 a?, U I0 a?, V n x, U 99 a?, and U 999 # are the 

 most useful and easily performed. 



When any series of numbers, the powers of natural numbers 

 for instance, is to undergo examination by unitation, it is useful 

 to make a rectangular Table with the unitates of the powers in 

 vertical columns; each vertical column belonging to one power 

 — the square for instance. The unitates of the powers of the 

 roots are in horizontal columns, each horizontal column referring 

 to the powers of the same root — 2 for instance. 



T 



XLV. On, Riccati's Equation. By A. Cayley, F.R.S.* 



HE following is, it appears to me, the proper form in which 

 to present the solution of Riccati's equation. 

 The equation may be written 



which is integrable by algebraic and exponential functions if 

 (2i + 1) q = + 1, i being zero, or a positive integer. To effect the 



integration, writing y = - -j- , we have 



dx* ~ X U ' 



The peculiar advantage of this well-known transformation has 

 not (so far as I am aware) been explicitly stated ; it puts in evi- 

 dence the form under which the sought-for function y contains 

 the constant of integration. In fact if u = Y } u = Q, be two par- 

 ticular solutions of the equation in u, then the general solution 

 is w = CP + DQ; and denoting by P', Q' the derived functions, 

 the value of y is 



_ CP' + DQ' 

 y ~ CP-fDQ' 



* Communicated by the Author. 



