52 
MR. HARGREAVE ON THE SOLUTION OF 
VII. Miscellaneous Forms. 
In the course of the preceding investigations, I was led to attempt the solution of 
some forms of equations by means of successive operations not consisting exclusively 
of D combined with constants, but involving also functions of x. The only important 
result which I obtained is the following, being a slight generalization of the method 
originally employed by me in elfecting the solution of the equation of Laplace’s 
coefficients. 
The equation 
is solved as follows. 
Let 05 and (3 be the roots of z^-\-bz-\-c'^=0, and assume 
— au=u^. 
7/ 
Then Dmj — jSw. — 1 ) — 
COS oc 
COS X /y'wo I 7 1^ I o \ 2 cos^ . sin^ ^ / COS^ 5? 
“1+*®“!+'' “i) „ - („+!) (D»i-f3«i) - (D-“)(„-(STr) 
or — 2tanx.(DMi— /3 mi) — aX— 2tanj7.X=X^suppose. 
X 
Assume Dw^ — 0 . 11 ^ — 2 tan x.u^ — Wg, 
then by a similar process we obtain 
D 2 w^ + &DM 2 + c 2 M 2 — 4 tanj:.(D^< 2 —/ 3 Ma)—^^^^^^^^^M 2 =X',-aX,- 4 tan.r.X^=X, ^suppose. 
Similarly, if we assume 
we obtain 
0 ^ 2 — aM 2 — 4 tan x.u^-u-^^ 
D 2 m 3 +Z»Dm 34 -c 2 m 3-6 tan x.{J}u^ - (3u^) -aX ^ -6 tan 
suppose. 
Proceed in like manner until we arrive at the assumption 
05?/^^ — 2 w tan x.u^=u„+^. 
Then D2 m„^,4-^)Dm„+, + c2m„^, — 2(w+l) tan 
Let Dm„+i— ^m„+, = Q, 
then DQ— 05 Q — 2 (n+l) tan a’.Q=X(„+,,, 
and Q=g®''(cos j:')”^^”+‘y£'“^(cos 
M„^,=a/yg(“-^y(cos .z’)-"^"+y£-“ycos xy^^^^X^„^,,dxdx, 
W„=e“'(cos J:’)“^y’£~“''(coS xyu„+idx ; 
