MR. A. C. DIXON ON SIMULTANEOUS 
1 5(1 
Here the differentials represent simultaneous infinitesimal increments, those of the 
independent variables being arbitrary. The equation may also be interpreted by 
supposing aq, aq . . . x„ to depend in any manner on a single parameter p, when the 
equation 
du _ ” du dx r 
dp ~~ r=i a>v dp 
holds whatever functions of the parameter we suppose aq . . . x n to be. 
To get the idea of a double differential we must suppose two sets of simultaneous 
infinitesimal increments ; denote them by d, 8 . The bidifferential of x, y is then 
dx . Sy — Sx . dy* This vanishes if x, y are not functionally independent, just as dx 
vanishes if x is a constant. The analogy is very clearly shown if we say that dx 
vanishes when some function <f>( x ) vanishes, dx dy vanishes when some function 
(J)(x, y) vanishes. 
If u, v are functions of n independent variables aq, aq . . . x n , we have 
or 
du = S ^ dx r , §u = 2 Sx r 
r =\OX r r =iCX, 
n 0^ n 0 y 
dv — 2 — dx r , Sv = 8.x,., and hence 
r=l O0: r r= 1 CV'V 
du . Sv — Su . dv = N i y 5 ^ (dx,.. Sx s — da q. S.x r ), 
r=l s=l®f OX s 
du dv = X ^ U ’ V \ dx r dx s 
C\^X r} X s j 
the summation being taken over all pairs of different suffixes r, s. 
expression for du dv is formed by multiplying together 
Hence the 
8 u 
with the conventions 
S v- dx r and 2 x- dx r 
,. = i ox,. ,._i ox r 
dx dy = — dy dx, 
dxdx — 0 . 
We shall often use the notation d(x, y) for dx dy. 
§ 8 . For the purpose of double integration of such an expression as 2 X w d(x,.,x s ), 
• • • • • ^*i ^ 
m which the coefficients X are functions of aq . . . aq, it is natural to suppose aq . . . x„ 
expressed throughout the range of the integration in terms of two parameters, say 
p, q. The integral thus becomes 
* The clot is used here and throughout the paragraph to distinguish multiplication in the ordinary 
algebraic sense from multiplication according to the Cfrassmann conventions stated at the end of the 
paragraph. 
