of the binomial theorem . 249 
value found above for <p(i+.r) is found to be identical, and 
the truth of the solution is proved. 
We may now attempt the solution of the general problem, 
viz. Let x, x', x" , x'", .... x"-^ be independent quantities ; it 
is required to find <p( i-f-a:) from the following equation, 
:$.<?>( i+.r" - ”' K )=2=^ . <p( \~\’X; l '" m . i+x"” ,n ) — t . <p(i i-f x"“' n 
l-f x""‘ r . l~\-x""' s .l-\-x‘'- t ) 4. &c* 
Assume as usual <p (1+^) — A -j- A'# 4 A!'x 1 -{-A."'x 3 -\- ; but 
instead of attempting to find the law of the coefficients, we 
may easily convince ourselves that will have the fol- 
lowing form, viz. 
<p(i +u:)=A / | L (1 x) + a" L 2 (1 4- x) 4. a.'" L 3 (1 4- .r) 4- 
oc"" L 4 ( 1 4— x^j -f— 4” a (1 4”’^) | 
+A" { L*(i+x) +/3'"L'(i +*)+ ,3""L*(i+x) + + (4) 
/3'-?L>( i+*)} 
+ A"'{ L 3 (i +x)+y"'L'( i+x)+ +*) } 
+A'"*L*(i4-;r) 
This form evidently includes, as particular solutions, 
L(i4-^)i L 2 (i4~^) 5 L 3 (i 4-^), L*(i4-#) : 
and, by means of these particular solutions, we are enabled 
to find the coefficients a!', cx!" , a!"', &c. |G ,y , fi"", &c., f", &c. 
& c, For let 
L ( 1 +x)=x+b'^+b"'x?-{-b""x*-\- 
L 2 ( 1 4-#)= 
L 3 (i4^)= x?+d""oc*+ 
&c. &c. 
K k 2 
