838 



z yk::=z2 sin ^ cp ^) u y/k — log tan \ (p \- 2 cos ^ (p^). . (3) 



Substituting this value of z in the first correction-term of (2) we 

 obtain as a second approximation 



4 



i/:^' = 2«n'iy + — -— (I -ro*»èr/,) . ... (4) 

 61 y k 



or, as long as sin \ tp does not become infinitely small {(p <^1 n)*) 



X [/kz=2 sin ^ (p -\ (2 tan ^(p + sin rp) *) . . . (4') 



3^ [/k 



u \/k = loa tan \(p Y'^ cos ^(p^ (| log tan \ <p — \sec*\ip^ cos >p-\-^) (5) 



ol yk 



.T 3 -T 



Putting fp successively equal to -, -t and -— the coordinates of 

 A, B and D (fig. 1) are found as follows: 



. . 1 . 



1) In Older that this expression may stand as a first approximation, ysm^raust 



be very small as compared to kz or, since sin (p is 1 at the utmost, klz must be 

 a large number (klz» I); it follows, introducing the expression tor sand remem- 

 bering that also sm 9 < or = 1, that ll^'^k»!. Iiv^k must therelore be a large 

 number, say 100; since for water k is about 13. / has to be at least 15 cms in 

 order that the approximation may be apphcable. Vox mercury (A; = 30) it would 

 be 10 cms. 



2) From this relation it follows, that 9 may be considered as infinitely small, 

 while u itself is still small with respect to l\ i.e. ^ actually becomes very small 

 in the marginal part (provided lv^k»\). For instance for <p = '/loo (0°.6 about) 

 n^ k becomes approximately — 4. i.e. still a moderate number ; in view of this fact, 

 however, the practical limit of applicability of the approximation was possibly 

 estimated still rather low at ?^ k = 50. 



'■^) Unless (p itself is infinitely small, for in that case the correction term in (4) 

 is still much smaller than the principal term. 



*) Except for a small reduction this formula was already given by PoissoN (loc. 

 cit.). With the degree of approximation in question it is of no importance that the 

 quantity / has not the same meaning with PoissON as here, the difference being 

 small compared to I itself; indeed we might just as well have represented by I 

 any other distance which differs from I by an infinitely small amount, such as the 

 abscissa of ^4 or of D (fig. 1 ^ 



Compare also Feeguson, Ioc. cit., equation IX, where, however, the expression 

 for z^^k is incorrect owing to an error of sign. 



It may be here remarked in passing, that the manner in which Ferguson 

 integrates equation (2) comes to the same as introducing a new variable c=l<p-\-z; 

 for the rest the equation is also solved by successive approximations. The trans- 

 formation in question is unnecessarily cumbrous. 



