EQUATION AND ITS TRANSFORMATIONS. 
813 
38. Taking the differential equation in the form (1), which has been adopted as the 
standard form in this memoir, it may be observed that, although the integrals 
and 
u=xi [fJx) ( 6 'i ?*+c % e **) 
u = x i+1 
1 d\ V + c 2 e 
x clx 
(39) 
have thus been given more than once by different mathematicians, the slightly 
modified form 
u=x i+1 
seems scarcely to have been noticed.* It was this form which led me to the solution 
in §11. as follows ; if x 2 is written for £ after the performance of the differentiations, 
then 
x i+1 
n £ 
\x dx t 
i+1 
/ d\ i+l 
e aJ: =2 i+1 x i+1 (^—J e aVi 
d_ 
= 2 l+l . i\ . x i+l . coefficient of h t+l in e /ld Ke aV ^ 
— 2 l+l . i\ . x l+l . coefficient of h l+l in e aV( f+/<) 
= 2 Z+1 . i\ . x i+1 . coefficient of h l+1 in e a ' / ^ +h) 
= 2 l+1 . i\ . coefficient of h l+l in ef t ^ xa+x& \ 
In my paper “ On a Differential Equation allied to Riccati’s ” ( £ Quarterly Journal,’ 
vol. xii., 1872, p. 136), I deduced by this method from the form (39), which is the 
same as (19) of art. 29, that the solution of (1) was 
uz=x %+l . coefficient of h 1 in 
_j_ a */(x*+h) 
vV + D ’ 
but I did not then remark the far more simple form 
coefficient of h l+l in c 1 e a ’ J(x ' 1+xh> -\-c%e a ^ t+xh \ 
39. It is interesting to connect Mr. Gaskin’s definite-integral solution (38) of 
art. 37 with that given in art. 26. The latter is 
’~x p+1 i -4 
J n (x~ 
cos ag 
o 2 +r 
3\®+l 
d£. 
(40), 
* The integral is however in effect expressed in this form in Eabnshaw’s ‘ Partial Differential 
Equations’ (1871), p. 92. 
