1907-8.] On the Theory of the Leaking Microbarograph. 445 
If x be small, and we may neglect or x 5 , then equation (17) may be 
replaced by 
! ■ 
which gives, to the degree of approximation supposed, 
ii =x — X 2 + (j + 2)x 3 - (£ + 
U. ^ = x - 3-1415V + 6*93480 x 3 - 13-54519 x * . . , (19). 
In like manner, under the same assumption, (18) leads to 
£2 = x - 27r x 2 + { ( 2?r2 + ^)x 3 - x% + ]U 2 3 } - | (-|- 7r3 + \ * )x 4 - 27r x% [ ; 
k = x - 27 r x 2 + 27 r 2 x 3 - y ^ 3 ~ 27 r )x 4 ; 
ie. £ 2 = x - 6*283 19 x 2 + 19 '7392 l x 3 - 37*91 397 x 4 . . . (20). 
Numerical Example. 
X= '!• 
This gives in degree measures 
x = 5° 43' 47". 
Hence 
/(£) - sin ^ - •0733646e' 1000336 £i = 0 
is the equation for . 
The equation (19) gives for a first approximation 
£ = -1 - -0314159 + -0069348 - *0013545 = -07416. 
Calculating more closely, we get the following table — 
: ?1 I 
sin fj 
^•100034^ 
Mi) 
A i 
•073 
•072935 
•074187 
- -001252 
991 
•074 ! 
•073932 
•074193 
- -000261 
987 
•075 i 
•074929 
•074203 
1 
+ -000726 
whence 
£ 1 = -074264. 
Hence, for the time of the maximum we get 
nt + *1 = 7r — 07 426 ; 
$ = *■/»- -17426/» (21). 
Since fhjn = tan x = *10003, 1/^ = 300*03; and the acceleration of the 
maximum is *l7426/^ = 52 sec - In this case T = 27 r/?i = 1885 sec - =31 - 43 min - > 
