36 
MR. HARGREAVE ON THE SOLUTION OF 
equation is made to assume Riccati’s form by equating n to m, the restriction on 
the values to be given to n takes effect. 
If for n be written — w, we obtain the other form of the equation, viz. 
d/D 
l)“^(c2;s 2K+i_j_ (^(y2_|_ — m{m — l))z“^) =0, 
which is subject to the same restriction, when assimilated to Riccati’s form. 
The solution of 
, / c lA 
is 
and that of 
1 
(D2-j-c2)"“'{'^~'(^sin cjr+^'cos c.t)}, x being ; 
d’^u , / c 
5 ? + (27+1) "'“=0 
is 
j7^”'*^i(D2+c2)”{a:"*(A: sin cx-\-k' cos cx } ; 
from which, general expressions for the solution of the two corresponding forms of 
Riccati’s equation may be deduced, subject to proper precautions with reference to 
the arbitrary constants. 
If we now, in a similar manner, apply the equations (6.) in conjunction with the 
original theorem (1.), we shall find, making (p(D)—T)--\-hD, that equations of the 
form 
,fi,rn, 2rl/'a?\ fbm f rrM/’x 
U-- 
are soluble ; the solution being 
M= &D)“ D'~”'(x-^x.P) j ; 
which assumes, m being integer, the form of the finite series, 
2 
X Ttt • 
And again, making 6=0 and Q,=.-^-\-^x, we have for the integral of 
D2m+2QDm+ Q2+Q'- 
(f-0' 
M=P, 
m r 1 / m \ 1 
By processes in all respects similai-, integrable forms of equations of the third and 
higher orders may be obtained. For equations of the third order, it will be found 
