Change of Form of Long Waves, 439 



On writing out (39), neglecting such terms as are of a 

 higher order than the fourth compared with 77, A, and k, 

 which latter quantities are of the same order, we obtain 



7) f * = ahkr) + {a{h—k) + abhk}rf + \—a + ab{]i — k)}rf —abif; . (41) 



and by differentiation, 



v" = iahk+{a(h — k)+abhk}r) + {— J a + |a6(/i — k^rf — Zabrj 6 . (42) 



From (40), by successive differentiations and substitutions, 

 retaining all terms up to the third and the 3J th order, we 

 deduce : — 



y=(V+2^+3fyV; 



f n = \arhk + { ar(h —k) +ab rhk + Sashk } rj 



+ {—§ar + ^abr(h—k) + ±as(li — k)}rf + ( — 2a6r — 5as)rj d ; 

 f" = \_ar (h—k) + abrlik + Sashk 



+ {—3ar + 3abr(h — k) + 8as(h — k)\r} + ( — Gabr—lbas)^]^ ; 



f*=±a 2 rhk(h—k) + {o?r(Ji-kf-%a*rlik\t) 



- l±a*r(h-k)rP + l£a?rrf', 

 / v = [a*r(h-k) 2 -%a*rhk-15a 2 r{h--k)r ] + ^aV]V 5 



where 77' is a quantity of the order §. 



Substituting these values in equation (1), where y = l + r], 

 we have, retaining terms of the third order : — 



u x =f~ \Vf - Irif + Jjjy* = g- Jar Z 2 M + AflWtt (A - £) 

 + {r— Ja?'Z 2 (A — &) —\abrl 2 lik—^ad 2 hk—\avllik 



+ ^a 2 rl i {h-k) 2 - 1 ^a 2 rl*hk} V 



+ {5 + lal 2 r-labrl 2 Qi-k) - 2asP(h -h) 



-arl(h-k)-^a 2 rV(]i-k)}rf 



+ {t + abrl 2 + %asl 2 + %arl+&a 2 rl±W (43) 



We find in the same way, including terms of the 3^ th 

 order : — 



= ^ [ _ r l + l a rl\h - k) + la&W 3 M + iasZ 3 M - ^a?rV>(]i - &) 2 



+ i 6 a?rl b hk + { - 2sZ- r — \a rl s + Ja^rZ 3 (/* - fy 



+ §aslXh- k) + \arV {h -k)+ ^rl b {h -k)\rj 



+ { -3^- 2s - aM 3 - f asZ 3 - |aW 2 -^aVjfl . . (44) 



2G2 



