156 Proceedings of the Royal Society of Edinburgh. [Sess. 
and the similar transformations of the variables Pj , Pj , P3 , P4 , P5 , Pj are 
Pf - Pi = - (-n, + QF, + riJ>, + C1P3 + - C4P5 } . 
P 3 '-P 3 = e{ f,Pi-(fi+yP 3 + ^,P 3 -^ 4 P 4 +? 4 PJ, 
P 3'-?3 = ^{ + +l 4 P 5 -’l 4 ? 6 }, 
p;-P4 =€{ + 
1 * 5 ' - P 5 = «{ - + ^1^3 - ’Jsl’4 - (% + ^4)I’6 - ■'JlPJ > 
p; = P, = e{ - e,7, - + e,)7,} . 
« 
The variables X^", X2", X3", X/' are cogredient with X^ , Xg , Xg , X^ . 
It is, however, unnecessary to take their infinitesimal transformations into 
consideration ; for, if 
^(X,,X2,Xg,X,) 
is a point-covariant, so also is 
(f>{X^ + XX{, X2 + AX2", X3 + AX3", 
X4+AX;); 
and therefore 
* ^‘a'aX + ^■'aX + vi) V(X„ i 
X3, X4) 
is also a covariant for all powers of p ; that is, we can introduce the 
variables X" by means of the polar operator 
r 
operating on a covariantive function of Xj , Xg , Xg , X^ . 
Similarly, it is unnecessary to consider the infinitesimal variations of 
JJ2", Ug'', U4" ; they can be introduced by means of the polar operator 
?’=i ' 
operating on a covariantive function of , Ug , U3 , U4 . 
Lastly, it is unnecessary to consider the infinitesimal variations of 
P/', . . ., Pg"; they likewise can be introduced by means of the polar 
operator 
dp = 
S=1 ^ 
operating on a covariantive function of P^ , . . Pg. 
Variations of the Coefficients and their Derivatives. 
12. Next, we require the corresponding infinitesimal variations of the 
coefficients a, . . n oi the quadratic differential form. We have 
(a', b', c', d', ^ dx^^ dx^^ dx^Y = (a, h, c, d, . . . \ dx ^ , d.x ^ , dx ^ , dx^^ ; 
