440 Mr. Murphy on the General Properties 



The equation of Riccati, .- + Au- = B . t"', may be put under 



a form more convenient for its solution by making u = • , 

 the equation then becoming 



^^-A.B.V'.y: 



-T^ = A.B.V" .i 



and if we make ??j + 2 = -, and t=ax, a being determined by the 



i 1 



equation a . AB = — , the equation comes under a form best 



adapted for resolution ; namely, 



d-y 1 !-■; 



ax- a- •' 



Let now the symbol 2„ denote the finite integral of the quantity 

 under it, with respect to n from w = o to w = « , then if 



S= 2 



(1 .2...w).(a+ 1 .a + 2...tf + «) ' 



^g 0*^1; — - — = -^ x" 2 ^ 



^ dar a"' ' " " (l .2...w,- l) .(a+ i . « + 2...a + w- 1) ' 



the quantity under the latter sign of integration diflfers only from 

 the former in having w-l instead of n. Consequently, between 

 the above limits the two integrals are equal; we therefore have 



