560 Proceedings of the Royal Society of Edinburgh. [Sess. 
Differentiations are effected by aid of 
= V = ) 
«rV = V , = ^D 2/ | '' 
When it is desirable to emphasise the distinction between partial and total 
differentiation in some defined sense, we may in manuscript use y for 
partial differentiation, and in print we may use 0 if there is need of 
frequent indications. Below I have decided to use y. 
By (18) § 6 
dcr - Scf/o|y.cr, dp = Sc?<x|y'.p ( 3 ) 
Thus the fictorlinity S( ) | y.cr converts dp into dcr. We therefore define 
dcr/dp to mean a fictorlinity * namely S( )|y.cr. Or 
dcr l dp = S( )|y.tr, dp/ dcr = S( )|y'.p = (dcr/dp) -1 . . ( 4 ) 
Since [(26) § 7] we always denote the discriminant of cp by [cp], [dcr /dp] 
means the discriminant of dcr /dp. Or from [c p] = Vn | (<p*l)n and by (12), § 7 
[dcr / dp] ijl !) * • • • ynl^n (r i cr 2 ■ • • j 
= ( n •) -1 y»lf» ) r • • (5) 
A?vsr ) ) 
(5) sufficiently defines yf, A[ w) . More generally we may say that 
Sc^y^Vo • . • ■<Av c = (^v)f c) i , 6 v 
i ; * • ' w 
(5) at once shows that [cZo-/cZ/o] is a Jacobian, namely 
[dcr/dp] = d(y 1 , y 2 , . . . . , y n )/d(x li x 2 , . . . . , x n ) . . . (7) 
From the definition that do- /dp is the fictorlinity that converts dp into 
dcr we at once have 
dcr dp _da 
dp dr dr 
( 8 ) 
but we have not in general d p/d-r. dcr/dp — dcr/dr ; though the corresponding 
statement in discriminants, or Jacobians, is true because [<p^] = [<p][fx] = 
[\/rcp]. Taking the discriminant of (8) we at once have the well-known 
theorem 
• • • • H n) . d(x v x 2 ... . x n ) = d(y v y 2 ... . y n ) 
K X r X 2 • • • • X n) Z 2 • • • • Z n) d ( Z V Z 2 • • ■ • Z n) 
but this is clearly only a very particular property of the much more 
general meaning of (8). 
* I find (April 1908) on p. 116 of Sci. Papers that Professor Gibbs recommended identi- 
cally this notation in 1 886. 
