When the unknown function in the proposed equation is relative to more than two variables, the same artifices may be employed. I will give a single instance to shew the manner of proceeding, it being unnecessary to dwell at length on this case.

when the equation will be reduced to the equation of differences,

0 = F{x, 4(x, h), p(x, h+1)}.

The most general forms of 0 (x, y, z) and of the arbitrary constant, or function of x, y, z which has the same value for y= P and ≈=p as for y=Q and x=q, are then to be determined, and substitution being made the function required is found.

I will merely add one instance of the application of the method pointed out in the foregoing problems. Suppose the proposed equa

tion were

*( * ) = 2 (2, z).

We first assume 0 (x, x2) = 0, which gives 0 (x, y)=(y-x) 0 ̧ (x, y),

The r in the denominator may be included in the arbitrary function. This done, we get by substitution,

Χι

C =ƒ'(x) + (xy − 1) (y—x2) ⋅ x1 (x, y); and the simplest function which satisfies the condition is

SLOUGH, Feb. 5, 1820.

ry-a3

↓ (x, y) = 2

J. F. W. HERSCHEL.