266 
MR. LUBBOCK’S RESEARCHES 
Development 
of R. 
-4 --^y* e *cos( 2 x + 2 y) + A — e-cos ( 2 t- 2 x - 2 y) 
3 ay ay 
[78] 
[79] 
+ ^ ^ r® e2 cos (2 < + 2 * — 2 y) + ^ Y I e e, cos (x + z — 2 y) 
[81] 
[83] 
9 a- a , , n , , 21 a 2 . fn , _ , 
— yg ^ y-e^cos (x + z + 2 ?/) + — — j y 2 ee,cos (2 t — x — z — 2 y) 
[84] 
[85] 
3 a- 9 , n . , , n . ,27 a 5 „ , „ . 
T7T ~ 5 7 e e i cos ( 2 * + x + 2 — 2 y) + T 7 - -1 T ee i cos (x — z — 2 2 /) 
j 0 ay 1 o ay 
[87] 
[89] 
9 a® 3 a- 
- rg — y 2 ee ( cos (x -z + 2y) - — — y°- e e, cos (2 * - x + a - 2y) 
l O ay l o a t - 
[90] [91] 
+ r? % Y~ e e i cos (2 t + x — z — 2 y) 
1 0 ay 
[93] 
Y*e t a - cos (2z-2y) ^y*e*cos (2* + 2y) 
[95] 
[96] 
~ ^ Y °~ e * C0S ( 2< - 22 ~ 2 2/) + ^ 4 ~3 y* e ,*C08 (2 t+ 2 z — 2 y) 
1/ 
8 l 
1 + 3 e® + 3 e 
[97] 
11 1 “//S * 1 S „S 
® — y 2 > — cos t + e cos (2 — #) 
' 4 ' J a, 4 1 c ~ 4 
[ 101 ] 
16 a* 
3 a 3 , . 9 a 3 , . 3 a 3 , . 
+ Ta J! ecos (* + *) — o' TT5 e / cos ~ 2 ) ~ ~a — e - cos (* + *) 
1 0 o tty o (Lf 
[103] 
[104] 
[99] 
— x . 
[ 102 ] 
t + 
[105] 
— — — e 2 cos (< — 2 x) + — — - e~ cos (t + 2 x) + — — e e i cos (t — x — z) 
64 a ( 4 
[106] 
[107] 
[108] 
3 a 3 ,, , . . , 15 a 3 , . 9 a 3 ,, , . 
+ jg e e, cos (t + x + z) + -j-g —4 e e, cos (t — x + 2 ) + — — e e, cos (t + x — z) 
[109] 
16 ay 
[ 110 ] 
16 a. 
[ 111 ] 
159 a 3 „ . . 33 a 3 o . . 0 , 9 a 3 „ ,, 0 . 
- 04 V <! '' C0S - 2 - Oi < Cr C “ (< + 2 Z) _ To V r cos (t — 2y) 
[112] [113] [114] 
sin® -i-cos (i + 2y) - { 1 - 6e» - 6 e y - |- y 2 } ^ cos3 < 
[115] ' ' [116] 
• • Cl^ 
* For the coefficients of the terms multiplied by — see p. 39. 
Cl. 
