1074 
Op ¢ Ou Ou Ou Ou 
——+yAu+uX,—pl — +u—+v—+w— }) vete 
Ow (5 Ox Oy En ' 
Ou Ov afte AY 
the last of which we shall replace by a different one which follows 
from the whole set of four viz.: 
Òudv Ovdu dvdw Owdv Owdu Oudw 
hp in Aa es alae ee eS eee ren, KE 
Oxdy Ody dydz Oydz Ode dx Dede 
If the motion of the liquid is the result of the friction of an 
immersed body of revolution rotating about its axis (the z-axis) and 
if, moreover, the boundary of the liquid, if it exists, is also the 
same in alle meridian planes, we may put 
USED OY SS OS EY FAO a Se ie de 
where ¢ and w are functions of the cylindrical coordinates @=V a*+-y? 
and z and of the time. . 
For small velocities we may write: 
EE Est By tts = ’ Zes ’ P=Pit P+Pt+-.. (3) 
where each successive term of the series is considered as infinitely 
small with respect to the preceding one. We therefore treat the 
motion of the liquid as the result of a composition of a series of 
conditions of motion, the velocities of which diminish very rapidly, _ 
the further we go down the series"). Consequently the equations (1) 
can now be separated into a series of sets each of which determines 
a condition of motion. Putting in the „th set: 
Ou Ou Ou 
ais opie 
Nn Yn,Z, are to be looked upon as the components of a force 
generated by the inertia of the liquid; they are completely determined 
by the preceding approximations. Also by (1') in each successive 
approximation the distribution of pressure is determined by the 
preceding approximation. | 
Kn Ee ae eat 
case the field of gravity (X= 0, Yo =0, Zj =g). A is the symbol for LAPLACE’s 
operator. 
It will be supposed, that neither w nor y are functions of the coordinates or 
of the time. In a piece of apparatus of ordinary dimensions this condition is pract- 
ically fulfilled, even for a gas, when no greater differences of pressure occur than 
are occasioned by gravity. (Comp. on this point Zempién, Ann. d Phys., 38, 81, 1912). 
1) This method of treatment of the problem was given by A. N. Wurteneap 
(Quarterly Journ. of pure and applied Mathem., 23, 78, 1889) for the purpose of 
finding a second approximation: he applied it to the case of a uniform rotation 
of a sphere. Comp. also Zemprén, Ann. d. Phys., 38,74, 1912. 
