
Principles of Aerodynamics. 669 
to the quantity of heat transferred into it by conduction and 
to the work done by the stresses on its surface, which last 
term may be calculated in known manner (Lamb, Hydrodyn. 
p: 317). 
D 
Thus we get, using div and ;-; as symbols for 
Dt 
Ou, ov, dw Ps) Ps) 
—— ++ — and — +u— + _ +w E respectively, 
Of OY 02 ot Ox OY Oz 
e Dé , 5 
Gays == pid VP OA WAGE, os ant ox) a el ) 
where the first term on the right side represents the effect of 
adiabatic expansion, the second one the dissipation function of 
viscosity 
et a div? yf 2[(S*) + =) (s2)] 
ae 2 2 
ee ee Ge) 
the third one the effect of thermic conduction. 
This equation, which with regard to the equation of con- 
tinuity may be written in the more convenient form 
Se +kpdiv=(k—1)[®+«A7@], . . . (2) 
denoting by & the ratio of specific heats, has to be added to 
the usual equations of motion. 
If the gradients of temperature are considerable, however, 
these latter ones are to be corrected by additional expressions 
arising from the thermal variability of viscosity. In this, 
most general, case they take the rather clumsy form : 
Du bes =. OP +m [dtu Oy 4298/94 — Fain] 
PDE 02 Ze Biles 3 
Ox for , Ov] , OH [Ow , Ow 
ee a eee OS Oe Rear eK 
Three such equations, the equation of continuity, the law 
of Boyle-Charles, equation (2), constitute the fundamental 
equations of aerodynamics. 
It is useless, of course, trying to get exact solutions of this 
system, unless for very specialized conditions. Some examples 
of this kind have been given in a paper of mine, published 
in the Bullet. d. ? Acad. Cracovie, 1903, together with some 
applications of the method of successive approximations, 
