360 



putting therefore 



we find in first approximation (for »V <^ :r) 



I — cos •') 



(8) 



^I' = t\ rri i /l•i^/ sm ,> 



2 log 



1 -4- C05 /y 



• (8') 



1 4- cos i'J 



Hence, as long as (p is not too near n, eqnation (2) in connection 

 with (7) gives 



j; z=z R^ sit I (f- — ^ ^'/«"o* *'^* <p 



1 — COS (f 



(I — cos fp){\ 4- 2 cos 7') 



^k'R^' sim^ 



(I -|- cos rp) 



— ^P/^j^ sm f f log 



1 + ^'OS (f 



(1 — 3 COS (fi -\- 6 cos' Y' -\- 8 cos* <ƒ) 

 1 -)- cos ^/A*) 



(9) 



Fig. 2. 

 .c becomes minimum etc. 



§ 5. In fig. 2 OAB represents 

 the mei'idian section of the capillary 

 surface for y(:>0, OA' B' gives the 

 section for k<^0; both have been 

 drawn for a positive R„ (for R^ nega- 

 tive the diagram must be turned 

 upside down about the .<?-axis) ; the 

 dotted curve between the two is the 

 circle with radius R^ (corresponding 

 to ^1= 0). In both cases .r goes tiirough 

 a maximum (in A and A' respectively), 

 but, whereas in the first case the 

 curvature keeps the same sign all the 

 way, so that y passes a maximum 

 (in B), X a minimum etc. (§ 7), the 

 curve in the second case has a point 

 of inflection (in B'), beyond which 



The maximum value of .r is obtained by putting <( = - in (9), 



the result being: 



xa{xa') ^K- \ kli,' + it ^'^.' (6 log 2-1) , 



(10) 



') This degree of approximation (the 4t'') is one higher than what is obtained 

 by simply using the relation (4). 



