276 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
The three laws of formation, which are here written down at length, are in sub- 
stance the same ; for from the last the others are made by merely writing the roots 
successively in lieu of D in the factors. 
Ex. 2. Let the equation be 
du 
Referring to (2.), we have in the first instance, 
((D — 2y — -f-7?(D— 2)(£^®^^)^-(^(D — 4)D-{-4y+r)£‘‘®M=G, 
whence 
(D‘^— (^D^+r) (£^®m) ; 
roots of y*«^=0 are n and —n. 
First complementary series, 
c,x^\Ao+A,x-\-A^x‘^+..-\-A^x”^-\- 
where 
Ao=l, and ({n+my—n^)A^+p{n+m)A„_,+{q{n-^my+r)A^_^=0 
is the law of formation. 
Second complementary series, 
Co^"”(Ao-l-A,a:4-A2a?"-f .. +A,„.X''"-f ..), 
where the law is, as before, changing the sign of n. 
Particular solution, 
where 
and 
((D-l-m)" - rf)'<P„(D) -f j9(D + w.)'4/„_i(D) -f- (^(D -f m)"-f r)-4/^_2(D) =0 
is the law of the formation of the operative functions. 
If p=0, q=0, and r=l, so that the equation becomes 
the law is 
whence 
A — _ . , 
"" m{2n + m) 
w=c.j:”|i— ^( 1+w) V-f^(l+w) \2+n) (1 +w) '(2-!-w)-’(3-|-n)-‘a'®-f , 
+ “|l— ^(1— w)-V+^(l— n) *(2-w) V- (1 — r2.)-‘(2-w)“’(3 — 
Ex. 3. Let the equation be 
^ +3^ +x^ +yx"«=G. 
