The preceding example is from Gauss (Theoria Motus,) where he has proved that the Method of Minimum Squares gives the most probable values of the unknown quantities. For a more detailed account of this method, the student may consult Galloway's "Treatise on Probability" (1839.) We shall add only one example by way of exercise.

Ex. Suppose that by observation the four following equations have been formed, viz.:

it is required to find the most probable values of x and y, by the method of least squares.

Ans. x-
=—6′′.14, and y =-7" 86.

CHAPTER IX,

TO CHANGE THE INDEPENDENT VARIABLE.

If we reduce an equation between x and y to the form

y = f(x)

a is called the independent variable and y the dependent variable.

Let it be required to change the differential co-efficients found on the supposition that y = ƒ (x) into others where x = 7 (y); that is, where y is the independent variable.

Let h and k be the contemporaneous increments of x and y.

Substituting the value of k found in (1) in this last equation

It is required to change a differential expression found on the supposition hat y is a function of x, into another in which both x and y are considered functions of a third variable t.

Let the contemporaneous increments of x, y, t, be h, k, l.

Substitute for h in this last equation its value from 1st series.

Compare this with value of k (3) and equating similar powers of l

And we wish to change some differential expression, found on the supposition that y = f (x), into others where x = (y), t = y in the equations (A) and (B), whence

ON THE APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF CURVES.

Then the angles 6, a, are in the order of their magnitude. The form of equation to tangent will be

where X and Y are the variable co-ordinates of the straight line, and x and y those of the point of contact.

Which three series must be in the order of magnitude, whatever be the value .. their first terms must be in the order of magnitude.

of h,

3o. Making Y = 0 in the equations to the tangent and normal we find the values of AT and AM, the abscissas of the points in which the tangent and normal cut the axis of a's, hence we find

x and y are the co-ordinates of any point in the curve; if x y be the co-ordinates of the point of contact in the present instance; we have of course,

The equation to the ellipse referred to its centre as origin is