Orr — stability or InsiaUUty of Motions of a Viscous Liquid. 137 



One of the former is 



a* a® a' - ' a^^ 



^3 = a2 + « ^ q + 2/i;) -^^ + (1 + 8^') -f^ + (1 + 20/.- ■+ mr 



+ (1 + 40/^ + 88F-) -^ + (1 + 70/j + 364Z;^- + 120/^0 ^-^ 

 5 15 I " !_' 



+ (l + 112^'+1120F+1296/i;^)^^+(l + 168Z;+2856F- + 7568A;3+1680^'0 



[7|_8 ^ ^[_8|_9 



+ (1 + 240^ + 6384/^2 + 31760^^ + 24096/^*) 



a 



L?L15 



+ (...+ 182736^"^ + 30240A;^) , .■...^ + (...+ 542400^;^ 



10 [11 ' LH L— 



a 



26 



+ (. . . 4- 2 . 6 . 10 . 14 . 18 . 22k') _^g + . . . (90) 



The boundary-conditions -it = 0, -y = evidently give 



xPodxPJda - ^jd^da = 0, (91) 



where, in a, r is equated to a. Denoting this value of a by ui, this equation, 

 on division by oi*, becomes 



"• 9« 2 K., 4 14.„ 6 „ 8 -.10 



+ 7^ + f4ti + pfr. + (42 + m 4V« + (132 + 40^-) 



2 L? L?L1 L2L^ L!L5 L5LZ 



+ (429 + 280/r) -^ + (1430 + 1680/v^) -y|^ + (4862 + 9240A;'+ 336^•^) -^^^ 



„ 18 20 



+ (16796 + 48048F + 6048/j^) ^r^j— + (. . . + 55684^'*) ' 

 [The terms on the left which are independent of k are those of 



2a,-' 



a{/i(2a)}V?a.] 



The lowest value of - k'^ai' is therefore approximately 

 1 W ^ 14a^ 42ai_^ 15%»i!! 429c.^ , 1430a/^ 



^ ^ H" "^ Hl? ^ [M "* Si ^ T-1- ^ TIL? " "IXlF ^ " " " 



( 4af 40ai^ 2800^ 1680ai^ 9240ai^° 48048ai" I 



^ (L£5 "" Ll^ "" r^ll ' ITf "" \3M ' "MIT "" " ' V 



(93) 



R. I. A. PEOC, VOL. XXVII., SECT. A. [19] 



