42 
Mr . Herschel on the 
•Vr ' 
developed in a series of powers of (i-{-a)( i+A'). &c. con» 
tinued both ways to infinity, (which is evidently possible) 
and letK(i+A) i . (i-j-A')h & c. be any term of the deve- 
lopement. The corresponding term in the above coefficient 
will be K( i-J-a)''0*. ("i+a)*. o y . &c. -that is, K. i*. i y . &c. 
or K. **+:v+ &c - g ut * s p] a j n the performance of the 
Sec. 
same operations on | Iog ‘ — | n o x + y ~ r ^ would have led 
to the same result : and we may therefore conclude that the 
numerator of (<?) is rightly represented by this latter ex- 
pression, whose value we have already determined (equations 
8, and 9). The coefficient therefore of t*. t' y . &c. in the 
developement of f TK }”> is 
n 
B 
x-\-y + Sec. 
A ^ 
rtf ;W. 
x, y, See. 1 xxi y x &c. 
5 • • • 
• • (*9-) 
and’ the same reasoning may be applied to any function of 
J ' J " See. whatever. 
£• £ • £' 
Analogous theorems to those we have deduced respecting 
functions of one variable may easily be deduced from the 
value of A „ given in (27). Thus, since 
x,y> Sec. 
f{e n ‘ 
n't' 
>.£ &C. J =/ { (/)”,(/ f, &C- } 
we ought to have 
“ F /{(i + A)",(l+A')' ,, ) &c. }o*.o' y . &c. 
ihisldo oyy j 
f 1 1 -f- A, i-}-A / , & c. | 0 x . 0 y . &c. ...... . (30) 
which, by assigning particular values to n, n', Sic. affords an 
infinite number of theorems analogous to ( 19) and 20). 
Similar theorems respecting the product of two or more 
functions of /, &c. may be derived. For instance, if 
x y 
n .n . 
Sic. 
