250 Mr. Knight's note respecting the demonstration 
By changing A', A", A'", &c. into the coefficients of each 
of these expressions, successively, we have p particular values 
of equation (4); viz. 
L( 1 -fcT)=L( 1 -H0+ «"L 2 (i -j-a?)+«'"L 3 ( 1 +^)-J-«""L 4 (i +£-)+ 
-j- a ‘P L/( 1 
+b" |L’(i+*) + 0"'L 3 (i+®)+ j3""L‘(t +.x)+ + 
+V" {L 3 (iH-*) + /"'L‘(i+0+ + 0~ f L^i+O} 
+6""|l*(i+*)+ +r-f L^(i+*)j 
+r-'*L'(i+i) 
from which we derive the equations b"-\-a'=o, 6 "' 4 - 6 ''/ 3 "'+ 
cc'"=o, &""+6"V""+6"|G""+«""=o, &c.; next we have 
L‘( 1 +*)= La ( 1 +* ) 4 i 3 "'L 3 (i 4 *) +/ 3 ""L‘( 1 +*) 4. 
4/3""'> , L ; (i 40 
=<"' {L 3 (i 4 ® 4 ? ""L*(i40+ 40 -^( 14 -^)} 
4c""{l 4 (i40 4 +r~fv[i+x)} 
+ X L* (l+a?) 
whence we get the equations c'" -f- /3 /7/ = oy. c"' f -|- c'" -j- 
/3"" = 0, &c. 
The next particular solution is 
L I (i-^'<r)=L 3 (i +<*’)+y ,/, L 4 ( -f L* (l-j-a?) 
-J- 1 L 4 ( 1 . -\~y-r L^(i-fa?)| 
K"'NL^(i^) 
whence d"" 4- <y""=o, &c. and by proceeding in the same way 
we have as many equations as the coefficients which are to be 
determined. 
