the use of Integral forms 173 
quantity, w is a velocity of translation, and the full specification 
of the transformation is 
2=0, USN, Zane uy), v= v(t we), va ee 
X’=9(x-), Y= (y+ >), Pats 
W—y(o+s), Y= (b-“) € =E val 
The value of X in (4) is V(V + w)/(V — w), and 
p= ai =e 7) (2 + Vt’)? — ( ee 7) C= ve} 
but when the conclusion that only a linear transformation 1s 
possible, is once reached, there is no point in using the function @. 
The form (1b) may be derived from a linear one containing 
vector and scalar potentials, viz. 
Q, (m)= [Fae + Gay patie = Vand ee (7). 
Its invariance under the transformation (6) yields 
HG GH’ = y(H-" Y) ap! = =y(y-"7)- (8). 
The transformation of the derivative of (1a) leads to further 
results. This form is 
Jodeay dz — pu,dtdydz — pv,dtdzdxu — p (w+ w,) dtdady 
if the relative velocities, where density is p, are u,v,w,; which 
on transformation becomes 
| p (" - wr) dax'dy'dz' — pu,dt'dy'dz' — pv, dt'dz'da' — ypw,dt'da'dy’. 
The invariance demands 
/ 1 J / 7 y / / 
p =p mares > Pp Ur = ptr, Pr =Plr, pW, = yew, .-.(9 a), 
and therefore 
1 Lr a yw, 
7 an ra yz = ye 
