OF THE MOTIONS OF FLUIDS. 299 



Now ^x=:^r + aSs 

 dxzzads 



d/j(.zzadu 



d^fx,=:ad-u 



^vzz^^r-i-ah) sin 0-^ (r+as) cos 6 (Sfi + aSw) 



dvzzr cos 6 adu + ad* . sin 6 



d"v:=ir cos fl ad^w + ad^^.sin 6 



^l^^-ar + adu 



d|=:7id^ + aSu 



d2|i=ad% 

 Hence ^x d"x-=.(^r + a^s).{ad'S, cos 2fi — ad«M.r sin cos fi) 

 — (^d + aStt). (ad^^.r sin cos 5 — ad"ur" sin 2fl); in which 

 a^s and aSu may obviously be omitted : again, ^y^^y + 



hd^'z z= I (Sr + aS*) sm fl + (r +05) cos 6 (Sd+aSw) } . { (r 

 cos flad^M+ad^*. sin &)—2andtdv,r sin 5 (—J 8(v«).»2di2 

 +(J''sr+a5i;)< 2(^cos Q adw + ad*. s\nQ)ndt, r sin B -k-ovd^v,r* 



sin«5 J=:(3r. smd+r cose85)a< (r cos5d*w-fd«5.sinfl) 



-—^ndtdv ,r sin fi \--\^{f)m^dr^-^d'ui.al^r^ sin cos d dw.ndt 



+ r sin^fl wd^ . d* + d^y. r* sin ^0); consequently Sjcd^x + 



lyd^y^lzdHzz, 



Sr.a(d-5. cos^d — d*M.r sin cos fi-f-d^M.rsin cos fl 



+ d^*. sin 2^— 2d«— dr.m sin d) 

 + S5.a( — d^^.r sin cos d + d^w.r* sin«fl + d«M.r« cos*fl 

 + d^s . r sin cos fl — 2d^di;.r2« sin cos d) 

 — iS(v2).»2d<« 

 + I'ss^ai^dudt, rH sin cos 6 + 2dsdLrn sin ^d + d*t;.r« sin «tf); 



