LV. Note on some Definite Integrals.

By R. PENDLEBURY, St. John's College, Cambridge*.

N a recent communication to the Philosophical Magazine†,

IN Mr. Glaisher pressed the claims of the integral

for admission into the rather too scanty family of known functions, and gave a tolerably long list of integrals expressible linearly by this function. The object of this note is to point out that a fresh set of integrals can be found expressible by the squares, cubes, and higher powers of the integral (1). With Mr. Glaisher's notation,

the integral being transformed by the equations xp cos 0, y=p sin 0. The new double integral divides into two, with different limits; viz.

Putting in the first integral on the right hand tan 0=x, and in the second cot 0=x,

θα

In combination with the last equation may be used the equation (5) of Mr. Glaisher's paper quoted above, which gives

Putting, in (5), erfa=√T —Erf («), we get

2

(6)

S

If we multiply both sides of (7) by da and integrate between

the limits

, we get the curious result,

*The equation (7) can be obtained directly without much difficulty. For

It is clear now that, by the various transformations to which

the integral

S

-a2x2 dx

1+x2

can be subjected, we have a new series

of definite integrals opened out, which may perhaps be worth the trouble of arrangement and tabulation.

The fourth power of the Error-function can be easily expressed as a single integral by the method adopted to determine the value of the square.

We easily get from (7),

The last of these three terms vanishes. The others are equal, and give

If we differentiate (9) with respect to a, we obtain an expression for (Erfa) as a definite integral, viz. :

√1+x

arc tan

:

Of the three formulæ (7), (9), (10), the first may open a series of interesting integrals; the other two are perhaps too complex to be any thing but a matter of curiosity.

Cambridge, October 31, 1871.

LVI. On a Problem in the Calculus of Variations.
By ISAAC TODHUNTER, F.R.S.

To the Editors of the Philosophical Magazine and Journal.
GENTLEMEN,

IN

N the Philosophical Magazine for July Professor Challis has discussed three problems in the Calculus of Variations. He states, in connexion with the first problem, that a certain conclusion obtained by Legendre and Stegmann, and tacitly accepted by myself, is erroneous.

There is, however, no error. Professor Challis does not understand the problem in the same sense as Legendre and Stegmann. I have enunciated the problem thus:-required to connect two fixed points by a curve of given length so that the area bounded by the curve, the ordinates of the fixed points, and the axis of abscissæ shall be a maximum. It is intended that the curve should be confined between the indefinite straight lines which coincide in position with the extreme ordinates. The enunciation involves this condition; for otherwise nothing would be gained by introducing the ordinates of the fixed points. And the investigation given would show, if there were any doubt, that this is the precise meaning intended.

It is in fact this condition which constitutes the chief interest of the problem; and there can be no doubt that the solution of Legendre and Stegmann is correct. Stegmann's investigation