390 Mr. Airy on Laplace’s Investigation, &c. 
in which, in conformity with the preceding remarks, the principal 
parts of this theory ought to be taken. 
First, it must be shewn, that the expression for V’ is of the 
form 
U (0) Uw U (2) 
— +-—~— + - 3 
; > + &e. 
“id Lb Tr 
Secondly, the equation 
dV 27h 
Paes =F +1V must be proved. 
Thirdly, by the application of this equation, the equation 
as. £4. oc ey FL 
Aan’ .y = Tan == ae a 
+ &e.. 
must be demonstrated. 
Fourthly, the form of ¥, supposed a rational function of x, 
J/1i—n. cos, and ./1—x”°.sin », must be found. 
Fifthly, it must be shewn that y, if a rational function of 
the same quantities, can be resolved into such a series, and only 
into one. 
Lastly, by substituting this in the equation above, the values 
of U, U, U®, &c. are to be found. 
The theory, after this poimt, presents no. difficulties which 
it is necessary to consider here. 
G. B. AIRY. 
Trinity CoLLece, 
April 29, 1826. 
