14 Sir G. Greenhill on 



Then, with a = c cos a, b — c sin a, 



1 ^1 . 1 cfoo 



c doe = ^ C0S a ~" 7?4 Sm "' c S = V * Sm a + 7?4 C0S *' ^ 11; 



1 <^h . 1 dv± 



c dx ~ Vl C ° S a ~ 7?2 Sm *' ~/= *7i sin a + *?2 cos a, (12) 



C7?1 = S cosa + Jsina, C772 = „g sina+ ^ cosa? (13) 



jcn^dx = 773 cos a + ?7 4 sin a, ^crj 2 dx = — 773 sin a + 774 cos a. . 



1 ^1 o • o 1 &V1 A - A 



7 2 dx*= VlG0 s2a - 7 72Sin2a, ^ ^£ = 77 1 cos4«-77 2 sm4a, . 



1 d 2 V2 • n o 1 ^2 • , 



~c*~da? = fl 7i sm2a + fl 7sCOS 2a, ^^~ = ^ sm 4a + 77 2 cos 4a, , 



c*~dx 2 = ^ C ° S 2a ""^ 4 sln 2a > ? d^T = Vt cos 4a -774 sm 4a, . 



1 d 2 7)A • . n 1 d^i)t . , 



~ 2 ~T^ =Z ^3 sin 2a -t ?7 4 cos 2a, -j -T-^f = 773 sin 4a + 77 4 cos 4a. . 



Just under the ice, where ?/ = 0, 



^ = 1]%, kJp + A t *p = UB 1 ^! + UB 2 ^- 2 ) . . . ri9' 



dy dx dy dy dx dx v ' 



- 4- 1 = £ a * cos ( — bx-rJu) sinu— e™ sin ( — bx + h,) cos a 

 c dy \ 1/ 



= e** sin (6#— A x + a) 



= (ch a# + sh aa?) [sin bx cos (Ai — a) —cos 6# sin (Ai— a')] 



= (^2 + ^4) cos (Ai-a)-(^7i + %) sin (7*i— a), .... (20) 



'JP2 _ ^-aj: cos (frx + / t2 J gm a _ e -ax gm (£# + 7^) C OS a 



£ a?/ 



= e~ ax sin ( — &#— /t 2 + a) 



= (ch a#— sh ax) [ — sin 7># cos (li 2 — a) — cos Zw sin (h 2 — «)] 



= (^2 — ^4) cos (7i 2 — a) -(771- 773) sin Qi 2 — a) (21) 



