4 
MR. LUBBOCK’S RESEARCHES 
From the preceding development, that of r 3 S . — may be immediately in¬ 
ferred. 
r 5 = 1 + 3 e 2 ^1 + — 3 e ^1 + -jj- e 2 ^ cos a; — A e 4 cos 2 x + cos 3x + cos 4 x 
[0] [2] [8] [20] [38] 
The following approximate value of r l y will probably be found sufficient. 
rji- = {(l + (r 3 + r 4 ^ }cos2* + |n 2 -r 0 j 
[ 1 ] 
e cos x 
[ 2 ] 
+ jr 3 — j ecos (2 t — x) + |r 4 — j ecos (2 1 + x) 
[3] [4] 
+ r 5 e,c osz + r^cos (2 t — z) + r-e / cos(2< + z) + | r a-j 1 e 2 cos2x 
[5] [6] [7] [8] 
+ |r 9 - ^--il}e 2 cos(2<-2:r) + |r 10 -^ e 2 cos (2< + 2x) 
+ 
[9] [10] 
cos(x+z) + |r 12 - -^ } ee^cos (2 t-x-z) 
[11] "* [12] 
+ jr 13 -]ijee / cos(2* + x + z ) + |r H - ^}ee,cos (x - z) 
[13] [14] 
+ |r 15 -^-|e ei cos (2*-x+z) + { Tie — ■§■} ee.cos (2 t + x - z) 
[15] “ [16] 
+ r„ e, 2 cos2 z + r 18 e, 2 cos (2 t — 2z) + r 19 e, a cos (2 1 + 2z) 
[17] [18] [19] 
a ;( S 1 V-r 2 4- r? e^rl #r£ #r£ .gll+ e 2ll+ e 2r£ 
V?j "" 0 + T + 2 + - ‘T + 2 + 2 + 2 T 2 
[0] 
+ {2 r 0 r, + e 2 (r 3 + r 4 ) r 2 + e* (r 6 + r 7 ) r 5 } cos2 < + { (r 4 + r,) r, + 2r 0 rj ecos.r 
[1] [2] 
+ {fir 0 + 2r 0 r 3 } ecos (2 t — x) + {r,r 4 + 2r 0 r 4 } cos (2t + x) 
[3] [4] 
+ {r x T 1 + r,r 6 + 2r 0 rJ e,cosz + {r 5 r 4 + 2r 0 r 6 } e t cos (2 t — z) 
[5] [6] 
