[24] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



the velocity in the direction of w, u> the angle between the line ■&■ and the plane of xy. Then 

 p, u, and q will be functions of x, w, and t, and we shall have 



v = q cos w, w = q sin to, y = ■ar cos at, ss = sr sin w, 

 whence 



•ar s =y 2 + a:*, &j=tan -1 -. 



We have now to substitute in equations (2) and (3), and we are at liberty to put 

 co = after differentiation. We get 



d (I sin w d d 



— — = cos id — — - , = — — when w = 0, 



ay dur w du> disr 



d* d 2 



T~s " T=5 when w " °» 

 dy 2 d-ar 2 



d . d cos a> d id 



— = sin to — + , = — — when a> = 0, 



dss dw nr dw nr dw 



d 2 1 d Id 3 , 

 —-=--— + —— -j when a) = 0, 

 dz' tit din- "nr dw 



whence we obtain 



dp ld?u d?u 1 du\ du 



dp Id'u d'u 1 du\ du . _ 



IT 1 -* Tl + w=» + -T- )-/°-j7' (,6) 



dx \dx z d-sr •&&■&■/ dt 



-*& + £* i£-{A'-& ..... 07) 

 Vda?"' d?zr •ar d^ar w 2 / d# 



dw da a , . 



T +T L + 1 =0 ( 18 ) 



dx dip T& 



Eliminating p from (16) and (17), and putting for ju. its equivalent f/p, we get 



, d l d 2 d 2 1 d \ , d / d 2 d 2 Id IX _rf /d« . d<7\ 



d-ar Vda? 2 dw 2 or d'ar/ da? \d# 2 d'ar 2 w dur ■&-) dt \d-ur dx) 



( d 2 d 3 Id 1 1 d\idu dq\ , ' 



\dar d'ar -ar dw -ar u d£/ Vd'ar da?/ 



By virtue of (18), "sr {udur — qdx) is an exact differential. Let then 



■ar (ud-nr - qdx) = d\J/ (20) 



Expressing t* and q in terms of \^, we get 



du dq 1 / d 2 d 2 1 d \ 

 dw do? bt \da? 2 dw 8 <& dw) 



Substituting in (19), and operating separately on the factor -ar -1 , we obtain 



/ d? d 2 id 1 d \ 1 d? d- 1 d \ , . ,. 



[— - + . ; + _ — \|/ ft 0. . . (20) 



\dx 2 d 2, sr tit dur /u. dtl \dx~ dsr- •& d-ar; 



