MR. A. CAYLEY ON A DIFFERENTIAL EQUATION. 113 



where the right-hand side is 

 and the equation may be written 



that is^ 



which^ since each side of the equation is=[n6—iy+'^-^, 

 is obviously true. 



The foregoing differential equation is developable in the 

 form 



na \du/ ^ ' 



but to find the coefficients u^, a^, . . . Un-x, I start jfrom 

 this form, and proceed to substitute in the equation the 

 value of ifj which on the left-hand side I use in the 

 original form, and on the right-hand side in the form 

 obtained by writing ^+ i in the place of 6, viz. 



The equation to be satisfied is 



+ «„_, [(»- 1)^+ 1]»-) =i ^g^+ff[ i{n- 1) e+n}"-', 

 or, what is the same thing, 



SEE. III. VOL. II. I 



