133 
The first two equations show the solutions of the system to be functions of 
i —— ey — Va 2 es 1) 
The remaining equations then take the forms 
n M n ‘a ° n = 
df f df df dt 
Sigs =2i 445 -— vy —- } = divs =o. ........ 2 
gee hig |e ae dys “y * dos 2) 
The second of these equations has solutions 
uj = $j Vj, Vk = 0, Vx, (K=38.... n) 
With u and vy as new variables, the first and third of equations (2) become 
ig ee 2 df - df 
JESS i= el eae gee | ee ee (3) 
du, Vi? dux vk dv J 
EA se iE) ive di fee di 
Se Se Se Gye en (4) 
ahs U2 5 Uk Vs 3 dvx 
The solutions of (3) are found to be 
Vk VK? 
7 —— oo — 1 — 
Uk : Uk 
Equation (4) then reduces to 
n = 
if Ubi oes cats) 
a Coe ee are 
whose solutions may he written 
vane Dy == 4 eerl\)s 
1 
16 a— an : 
OK $1 23 
Since any functions of I’, J will be the solutions of (1), we may choose 
Ox 
a | 
= = [12k | and Dy= Hy ty =|182/, 
k 
= 
oO 
4 Xj yi l 
where | ijk |=J| xj yjl|, 
Xk Yk 1 
as solutions, and, therefore, as the 2 n — 5 invariants of the group. 
The forms of I’ and D show that the special linear growp leaves invariant all 
areas. 
26. The general linear group 
P, 4) Xq, XP—Yq, YP, XP+-yq) - 
This group furnishes a complete system of six linear partial differential equa- 
tions, the first five of which are identical with equations (1) of the preceding sec- 
tion. Hence we need only determine the functions of I’ and D which satisfy 
n “of 
df di 
i ~ Xxi—— isz—-,—0. 
= i ‘dx, 1 dyi ). 
