and 
ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 639 
+ c/^j ja =0,.(6) 
f3 sm z <f) x dl3-\-ct cos^c/a — (a 3 —/3 3 ) sin <f) x cos <^ 0/^=0 .(7) 
The limits of the integrals are <f> x and 0 , or <j> x and tt/ 2, according as the origin lies 
on a maximum or minimum ordinate. In the latter case E —E x and F — F x must be 
written for E and F. 
But 
fsin 2 0, 1 fl — (1— ft 9 sin 3 <f>) . , 1 fd<f> F 
J ~A rd< P~j^] A 3 ' ~ d< P— /?Ja" 3 “/> 
and 
Hence, 
and (7), 
substituting from these expressions, and eliminating d(^ l between (6) 
da{/3~ F—a 2 E — cd a x cot ^} + a 3 c//3(E—F—A ] tan^> 1 )=0 .... (8) 
Again, differentiating the expression for Y given in equation (5), we get in the 
general case, when c/a, d{3, and (A/>, are independent, 
c/Y/7r=c/a-j (2a 3 d-2a^8+^/S 3 )E — 
+ ^(3ot s —/3 3 ) A j sin 2‘j'! j \ t/>/. 
l/SVsin**,, l/SV 3 -^) • „ ■ nJ 
sm * sin 4,1 - 
+^{(« i +h/8)E-^/5F-“|A 1 sm 2^ 
+ 
'2 a, ,, , 2 WsinV l/3 3 fsin 2 0 
i a-+«^+3^)-j~^+3 -J^T# 
, 1/3 (a 2 —/3 2 ) . 9J . 
-f-- 7 -sin- 0 ! sm 2(/> 1 
6 aAj 
l«/3 2 
3 A, 
.1/0 02\fO A o . S^ 33 ^! 
+ 6 a ( a ''—^ )( 2 A l cos 2 <£i-^- 
