11 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS
d . d cos co d Id, A
-7- = sin co -7= — 1 --- 7- ğ = — r~ when co = U, a# cfe -or a&> OT aw
d2 1 d 1 d2 , n
— = — ~ — — - -j— when co =.Q, <Lz ix dm tx du>
whence we obtain
dp __ ./<Fu d*u 1 du\ du ~ + +~
(18).
Eliminating p from (16) and (17), and putting for /* its equivalent /jfp, we get
.
-or dx W d^
JL-l(^-
dt \cZt3r dx
or i -, Q -] -, - <) r — ~T • aw
By virtue of (18), -or (iid-cr — qdx) is an exact differential. Let then
-BT (^t!J — ^3?) = dty...................(20).
Expressing u and q in terms of ^, we get
cfe ^ 1 / d2 d2 1 cZ \
Substituting in (19); and operating separately on the factor -cr"1, we obtain
_L_ _a. _ f-ft C9n'^
.dğ2 "*" dw2 Ğr rfiB- /*' At) \dtf + d^ -aM Y~ ' '^ >' Since the operations represented by the two expressions within parentheses are evidently convertible, the integral of this equation is
........................ (21),
* If we denote for shortness the operation
JL JL. "'2