10 



Art. 6. — E. Kaibara : 



V35) 



= 1 + 8h, + Mh,- + 367i,^ + 56/;,' + , 



= 2;i + 13/ir + 42V+ ), 



= 8(1 + 5V+10V+ \ 



= i(6-ll/?, + 90/v-121//,' + 238V----), 



= 1(1 + 4Ä, + 517^-10/^^^ + 211 V- ), 



= i(l + 16/«, + 29/i;-+170/;,^'-31/ii'+ ).} 



Here, tlie values of ^-1, B, Q D, can be determined for a given 



value of /h either by calculating â-n, «?3, i^, à-^", ^3", or preferably 

 by direct evaluation from the series in (35). Then the correspond- 

 ing value of ^1 is found from (33) by the method of successive 

 approximation. 



For convenience of subsequent interpolations, values were 

 first assigned to h at the intervals previously chosen, and then the 

 corresponding values of A, were accurately determined from (30). 

 A further advantage of this procedure is that equation (17) is still 

 available, for, from the value of z furnished by this equation, we can 

 calculate that of z, by (32), which serves as a first approximation. 

 If h be not ver}^ near to unity, this approximate value of z, is 

 fairly close to the true one, so that by applying (33) once or twice, 

 a final approximation of sufficient accuracy is arrived at. When k 

 is close to unity, the extreme smallness of h, facilitates the work 

 with (33), and thus the trouble arising from the roughness of the 

 first approximation is compensated. 



8. For the calculation of o's, e's, /i, etc., the process is the 

 sam.e as before. Let 



^, = 2V^,(1 + V+ )' 



d^(2T^v) = 2v'//i(cosh^i + /;,^cosh 3,.-, + \ 



"dj^lr.v) = 1 + 2// ,- cosh 1z, + 2/? ,^ cosh 4.:, + ■ • ■ 



then the angles a and ß introduced in § 5 are now given by 



(36) 



tan (45° -r/) = -L£-, 



taii(45° + y9) = 





(37) 



