OF WHICH IS MOVING ROTATIONALLY AND PART 1RROTATIONALLY. 391 
To find the third, substituting for y' its value b /\J r ~— 2 
dx! 
d x 
2 Vf 
therefore 
a 2 —b 2 . 
x= 
a% 
\/ 1 mcft—fx 
'2 
therefore one value of y satisfying the partial differential equation, which may be 
called x 
Also 
Whence 
Thus 
, f « 3 —& 3 , , 
x=f--^f x y 
(/)($+£ 
x' 2 y' 2 
' x' 2 y' 2 \dy\r _ y' 
~a 2 +V 2 )'dx'~~~2f 
y z ,y' 2 \d^ _ x' 
a 2 b 2 )dy' 2f 
, dV 
u — ~j—e 
dx 
+£-«= tf$+£-*+b)+% 
a 3 b 2 ) dx' 
2f) 1 dx' 
i) f |= -4&+£-*+£+& 
dy' 
b 3 ' a 3 
2 f) ' dy' 
If, therefore, e be put =2/(tu — ^ 
M -£ ; v+ e '/')+ e (? + ^“ i )S 
a 3 5 3 
/ d , , , fx' 2 y’ 2 \(Alr 
” +^> +V+F _1 )*7 
Thus the proper value to take for the y of Clebsch’s forms is y'-J-et/;. Omitting, 
as unnecessary, terms containing t only 
