22 4 Mr. Knight o« the construction 
If, for brevity, we put L, L°, L', L", &c. for L(.r — 1), L.x, 
L(x+l, L( jc-J- 2 ) , &c. the last equations give 
L° =L -f-L . * 
U S=L° -j-L . es-J-Lo/ (c ) 
L" =L' +(L'— L°)+L.«' 4 L. a " 
L'" =L/' + ( L" — L') + (L" — * 1 / 4 L° ) +L . *"+L . 
L""= L^L'"— L") + (L'" - 2L" 4 L') -{- (!/"■ — 3L" + 3U— L c ) 
+L . *' /, 4"L . a"" 
L" t*+ , )=L""» + (L“-* — 2L"-("-° + 
L'...(«-«))+ + (L" '-“-mL'- < ’ ! - ,) + 
— ( ” -3) + )+L . «"- "+L . «"•••<*+■> 
These equations are subject to a law arising from that which 
we noticed in (6), viz. that the m th term (provided it is not the 
last) in the value of L"—« is equal to L"-(«— , ) } minus the sum 
of the first m — 1 terms in the expression L •■•(«— By t erm 
I here mean the whole expression included between two 
brackets. 
If we form generally from the last term of equa- 
tion ( a ) we have 
f (« + >)« (n+i)n(n— \)(n— 2 ) 
T // „ T ) (x+n)x(*+n— 2 ) 12 x(j: + »— 4> 1>2 3 4 X &c. I 
«+7 (» + ■>(«-■) f 
L (x+n— 1 ) 1 X(*+« — 3> I,2 ‘ 3 X &c. J 
If any one should not be satisfied that the form given to 
L "••■(»+*) is general, he has only to form L'' (« + 2 )from it, and 
he will find the same form in that case. Now L" -(» + 2 ) is 
formed from L" (»+0 by changing x into -r-J-i (or L"- r 
into L"-( r + I ) in all the terms but the last L . a"—C»+ I ), and 
