ON HANSEN’S LUNAR THEORY. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 112—125.] 
The following paper was written in order to exhibit, in as clear a form as may 
be, the investigation of the remarkable equations for the motion of the moon established 
in Hansens “Fundamenta Nova Investigationis Orbitse verse quam Luna perlustrat,” 
&c., Gothse, 1838. I have availed myself for this purpose of the remarks in Jacobi’s 
two letters in answer to a letter of Hansen’s, Crelle, t. xlii. [1851], p. 12; it may 
be convenient to remark that the quantity there represented by A, and which does 
not occur in Hansen’s own investigation, is in this paper represented by ©. 
The position of the moon referred to the earth as centre is determined by 
r, the radius vector, 
L, the longitude, 
A, the latitude. 
Suppose, moreover, that the attractive force at distance unity, = k(M + E), is represented 
by n 2 a 3 , then the principal function will be F= — , and the disturbing function R 
may be represented by w 2 a 3 il; the expression for the half of the vis viva is 
and the equations of motion are therefore 
where il is considered as a function of r, L, A.