8 
MR. LUBBOCK’S RESEARCHES 
+ 9*5691515 ee t cos (2 t + x + 2 ) -f 9*8697180 e e t cos (x — 2 ) 
[13] [14] 
— 0*0466780 ee ; cos (2 t — x + z) 4- 0*4139940 e e, cos (2 t + x — z) 
[15] [16] 
— 0*0479097 e, 3 cos 2 2 - 0*7991728 e, 2 cos (2 f — 2 2 ) 
[17] [18] 
— 9*5709386 y- cos 2 2 — 9*5761195 y°- cos (2 t — 2 y) 
[62] [63] 
where the logarithms of the coefficients are written instead of the coefficients 
themselves. 
„ m. a-f 34 20 0 , , 38 ,38 /0 , 20 /r) . 
R - i — <! — — - _ — cos 2t + ~ e cos x + — e cos (2 t — x) - — e cos (2 t + 2 ) 
A ~ «/ 1 137 27 77 17 ' 27 v ' 
[0] [1] [2] [3] [4] 
— || e, cos 2 — ~ e, cos (2i — z)+~e t cos (2 / + z) + 12 e 2 cos2x 
[5] 
[ 6 ] 
[7] 
[ 8 ] 
— e 2 cos (2 t — 2 x) — e 2 cos (2 f + 2 x) + e e t cos (x + 2 ) 
15 ->/ 2 / 
[9] 
[ 10 ] 
[II] 
+ ^ e e, cos (2 < — x — 2 ) + 1? e e, cos (2 < + x + 2 ) +^ee ( cos (x + 2 ) 
[ 12 ] 
27 
[13] 
27 
[14] 
— e e, cos (2 t — x + 2 ) — || e e ( cos (2 < + ^ — 2 ) — e, 2 cos 2 2 
[15] [16] [17] 
— ^ e, 2 cos (2 t — 2 2 ) — 1| y 2 cos 2 >/ — || y 2 cos {2t — 2y) nearly. 
[18] [62] [63] 
I make use of these approximate coefficients in the following* development 
solely in order that it may occupy less space. 
s R*= r h^ 
68 
137 
r °' + + % { T ' + x '} “ ff e "' r * “ if e: { ? ' 3 ' + } + & e °' { + 
* See Phil. Trans. 1831, p. 275 
1 
t r 5— = r 0 ' + cos 2 t + er 2 ' cosx -f- er.J cos (2 1 — x) &c. 
[0] [1] [2] 7 [3] 
S X = sin 2 1 + e X 3 sin (2 < — x) + &c. 
[I] [3] 
