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III. 
OlSr THE DETERMINATION OF THE NUMERICAL EACTORS 
IN THE EXPANSION OE LAPLACE'S COEEEICIENTS. 
Br ERANCIS A. TARLETON, LL.D., E.T.C.D. 
[Read November 30, 1887.] 
If the expression (1 - 2Xh + ^^)'^, 
where A. = i^ix' + a/i - />t^ v^l - /x'^ cos - <^'), 
be expanded in a series of ascending powers of the quantities by 
which the powers of h are multiplied are termed Laplace's coefficients, 
and are usually indicated by the symbols Zi, Zg, &c., Z,-. 
If ju- be substituted for A in the expression given above, the multi- 
pliers of the powers of h in the corresponding expansion are called 
Legendre's coefficients, and may be indicated by the symbols Pi, 
P2, &c., Pi. 
It is proved in Williamson's Differential Calculus that 
n n 
Li = %anU^u'^I)''PiD^'P^(^o^n{<i> - ^'), 
where u = fx^ - I, u' = /jl"^ - 1, 
P/ is the same function of which Pi is of /x, and is a function 
of i and n, independent of />i, fx', <f), and <^'. 
The value of a,^ has been usually arrived at by means of a laborious 
trigonometrical expansion. The object of the present investigation is 
to show that it can readily be obtained by the use of the fundamental 
theorems of the Laplacian analysis. 
By using the well-known theorem 
