58 
Mr. Knight on the Expansion 
"...(r — Mi+i) "...(r— i) l"...r\q—i "...(r — i) 
x c x c x [ c J , (where c is either the 
last factor, with respect to the number of strokes, or the last 
but one, accordingly as q is equal to or greater than one ) ; if 
"...(r — i) 
we make this term vary, with respect to c , we shall 
have the combination marked (a) over again. 
Let us next consider when it will be necessary to make c 
"...n l"...m\p 
vary. In the term /(c)xPx| c I (/3), if we make c 
"...(«+ 1 ) , l"...m\p 
vary, there arises the combination f (c) x c x P x [ c I ; 
but the same coefficient (as B) that contains (/3), will also 
r 
"...(«+ i ) , "...(m — i) I "...wz \ p—~ i 
contain f (c) xcxPxc x \ c / which when we 
"...(?n— 0 0 , 
make it vary with respect to c gives also f (c) x c x 
l"...?n\p i) 
P x 1 c ) . Now c was here the last quantity or the 
last but one. We may then affirm, in general, that, if we 
make every term in B vary with respect Jo the last quantity, 
n—i 
and the last but one also, when this immediately precedes the 
last, not in place only, but in the number of its strokes, we 
shall get all the terms we ought to have, any further varia- 
tion only giving the same over again. From equation (g) we 
have then the following 
Rule . 
To find B take thefiuxion of B zvith respect to the last of the 
n n— i 
i a 
quantities c, c, c, &c. in each term, and the last but one also , if it 
immediately precede the last in the number of strokes . 
