OF FLUIDS ON THE MOTION OF PENDULUMS. 



[45] 



whence if we calculate 



U x = 2tn -1 , U 2 =- ^B tit" 2 , M 3 - T^g- J?! «' 



«1 



,= (-i) ,+, i5,_,8- | +*m-', 



. (113) 



we shall have, changing the sign of v — 1 in (112), and writing 8 for e 4 » 

 k + y/ - 1 ft' = 1 + m,8 + « 2 8 2 - w 3 8 3 + w 4 8 4 - « 5 8 5 + ... 



k= 1 + -^£"1 + ^2^3- «*+'\/i. W »"\/i tt 7+ M 8- \/2 M !)"- 

 A' = <y/£ M, + tt 2 - >\/\ U 3 + y/\ M 5 - U 6 + ^\ U-, - y/± u $-" 



If l u I.... be the common logarithms of the coefficients of HI -1 , m~ 2 ... in the last two 

 of the formula? (113), 



Z, = .1505150 

 4 = 1.6989700 

 L = 2.6453650 



l t = 2.4948500 

 h = 2.2371251 

 It = 2.4046734 



Z 7 = 2.3646348 ; 

 Z 8 = 2.7019316 ; 

 4 = 2.6017045; 



and if the logarithms of the coefficients of tit -1 , III -2 ... in w,, w 2 ... be required, it will be 

 sufficient to add .1505150 to the 1st, 3rd, 5th, &c. of the logarithms above given. 



36. It will be found that when tn is at all large, the series (113) are at first convergent, 

 and afterwards divergent, and in passing from convergent to divergent the quantities u t become 

 nearly equal for several successive terms. If after having calculated i terms of the first of the 

 series (113) we wish to complete the series by a formula involving the differences of u { , we 

 have 



u^ - u i+1 8 i+1 + u i+ .,H i+2 - ... = 8* {l - 8 (l + A) + 8 2 (l + A) 2 - ...} Ui 



= 8 ! {l +8(1 + A)}" 1 ^ 



8 r 8 / 8 \ 2 . , 



= {l A + ) A 2 - ...} «,-, 



1 + 8 ' 1 + 8 \1 + 8/ ' ' 



,' ' . 7T / . IT IT -\T^l , „ , , 7T -V-l 



and 1 + 8 = 1 + cos- + V - 1 sin - = 2 cos-e 8 , 8 (l + 8) _1 = \ sec - . e 8 , 



so that the quantities to be added to k, k', are 



to k, (- l)*£sec — {cos — - — ir.Ui - ^ sec - cos — Tr.Au { 



8 



S 



(* sec ?) 



21+1 A2 J 



COS IT. A^ttj...} 



,l , .it 7T , . 2l — 1 7T . 9,1 



to ft , (- 1)* i sec — }sm v . u, - 1 sec - sin — w . A« 4 



d 8 l 8 a 8 8 



(*-i)' 



. 2* + 1 

 sin ir. A 2 Mj...} 



(114) 



37. The following table contains the values of the functions ft and ft' calculated for 

 40 different values of HI. From m = .1 to ttl = 1.5 the calculation was performed by means 



