theory of the comparison of transcendents. The following examples, taken from the interesting papers of Mr Talbot, already noticed, will tend to explain the manner in which this principle is applied. The solutions are not quite the same as his.

20

da, and transform it by as

= a, a being a new variable, we have thus

and

dx

=

xa

da;

But this equation being a

quadratic must have another root, which we shall call y, and therefore, x and y being symmetrically related to a,

which, as by the theory of equations x + y = a, becomes

where xy 1. Taking the integrals between limits, we have

S'"'; (~`~**")'du + ['('+"')'du = 8{('*'*')' - ("'"';')"}

might be obtained at once from the general theorem that if Fu= 0 be an equation of the nth degree whose roots are ay

This theorem is so essential in the subject we are illustrating, that we shall give a simple demonstration of it, which is perhaps new.

Let be a root of the equation Fuu" - p1u”―1 — &c. = 0, and consider it as a function of Pn- thus we have

But p, and P- may be supposed independent of each other, therefore p, does not vary for a variation of Pn-k Hence

Y2

du

du

+

(1 − u3) §

where a1y11, 2y are the corresponding roots of Fu = 0 for two values a1a, of a.

The ambiguity of sign of the radical must always be borne in mind in considering equations similar to the last. For the fact that y for instance is a root of Fu = 0 does

Thus we must either leave

a determinate sign to the radical. the sign of the radical undetermined, or if we determine it, i.e. if we assume that it shall be always taken positively or negatively, we must look on the integrals themselves as liable to be taken with a negative sign.

(4) In these examples we have considered as a function of a new variable a. But we might have considered it as a function of two new variables, a and ẞ, or more generally of any number of variables a, ß, y, &c. It is true that we cannot conversely determine a, B, &c. in terms of x, but this circumstance is for our purpose unimportant.

In the last example, suppose we were to assume

The difference between this and the previous result is that here a1, 2, Y, Y2 are quantities to which we may assign any values we please, while in the former case when the value of a was assigned that of a became known, and hence those of y and were both determined. This restriction is wholly unnecessary. We may, if we please, suppose that the inferior limits a, y are zero: in order to this we have merely to make a = 0 and ẞ = 1, when Fu = 0 becomes u3: = (). Thus all its roots are zero, or ≈, is equal to zero if x, and y, are so. Hence the last equation may be replaced by

(5) The integral which we have been considering is a case of the following general integral

where a, ß, &c. are real constants, and R is a rational function of a. All such integrals may, by suitable transformations, be reduced to three standard or canonical forms, which are called elliptic integrals.

The reason of this designation is that an elliptic arc may be represented by an integral included in the general form above written for if s be the arc corresponding to the abscissa a in an ellipse whose semi-major axis is unity and eccentricity e, we have

where, as we see, the denominator is the square root of a rational and integral function of a of the fourth order.

Legendre was the first writer by whom elliptic integrals were treated in a systematic manner, but since his time the subject has assumed a new form in consequence of the researches of Abel, Jacobi, and others.

We shall here merely prove the fundamental property of elliptic arcs.

where as heretofore a and B are two new variables. Then

a is a root of the equation

u2 (u2 + a)2 – ẞ2 (1 − u3) (1 − e2 u2) = 0 or Fu = 0.

F'x

{Δαβαβ + βαλα},

{(a + ở) dB - Ba da} (1 - er).

If we take the sum of this for all the six roots of Fu every term will disappear except that whose coefficient is

Σ which as we know is unity. Now as Fu involves

F'x

only even powers of u, it must have three pairs of roots, the roots of each pair being equal and of opposite signs. Take

• Another assumption might also be made; v. Legendre, Théorie des Fonct. Ellipt. 111. p. 192.