362 On the Expansion of the Functions/ (x),/ (.r, y), S)C. [Ma y 



Hence we obtain /* {x -{■ A x, 1/ -\- A i/, z + A z) = ti -\- A 

 + B -f C -f D -f '&:c. 



In like manner, we may expand the functionjT (x, ?/, z, v), for 

 the several combinations of the variables abstracting the numeral 

 coefficients are as follow : 



{x + 1/ + z -i- v) = X -\- 1/ -r z -^ V 



{X + }/ + Z^ V)' =^ X- +^ X1/ + XZ + l/'^ + XV -\-i/z -i- z'^ +t/v 



<jr + ^ + 2 + v)^ = ^'^ + 07 j/« + X ;s^ + 1^ v« + y5 + y .r^ + 



y z' + 1/v' + z^ + &c. to 20'terms. " 

 i^x -^ y -\- z + vY == X* + X 1/^ + X z^ + X v^ -{■ y^ + y ^' -f 



y z^ + yv^ + «^ + &c. to 35 terms. 

 &c 



By prefixing the symbols A, ^, dr, f/', &c. before the variables, 

 x,y, X, r, and n, as in the preceding problems, we readily obtain 

 the expansion of^' (.r, y, z, v), and so on for a greater number of 

 variables. 



In the foregoing expansions, it will be observed that the index 

 of the symbol d placed before u is equal to the sum of the indices 

 in the several combinations of the variables, the symbols d and 

 A are simply placed before the variables and their powers, and 

 the numeral coefficients in the denominators are also governed 

 by the indices of the variables. Thus th.e differential coefficient 



of x° y^ z^ will be found to be ,, ^ ,. ^ ,, ,. ^ " ~/ /\ -- ; 



and the differential coefficient of a.'" ?/" z% or of ti, will be 



Tx — a ^v~7i — z 77; — s ^ ":; 7-, a general form for 



three variables, which may be extended at pleasure. — (See 

 Woodhouse's ** Analytical Calculations," p. 86. 



The differential of a function being the second term of its 

 developement, or that part of the expansion of a function which 

 contains the first powers only of the increments d x, d y, d z, 

 &c. It will be perceived that the differential ofy'(j) consists of 

 one term of / (.r, y) of two terms, of / (.r, y, z) of three terms, 

 &c. Therefore by changing A x into d x; A y into d y, A z into 

 dz, &c. we have 



df{x) = dn = i^^d X, df{x,y) = du:=^i^^dx +^rfy, 

 df{oo, y,z) = dn = i^^dx + ^ rf y + ^ J ;s, &c. 

 — ^ X, — J y, &c. are called partial differentials, and 

 :t— , -7-, J-, Sec. are called differential coefficients. 



• jr ay a z 



