This is the binomial theorem for a positive integral index.

The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

On the one hand it was declared that the kingdom of Hungary was an integral part of the Habsburg dominions and inseparable from these so long as a male or female heir of the kings Charles, Joseph and Leopold should be found to succeed to them.

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We have now to substitute these data in the universally valid circuital relations - namely, (i) line integral of magnetic force round a circuit is equal to Orr times the current through its aperture, which may be regarded as a definition of the constitution of the aether and its relation to the electrons involved in it; and (ii) line integral of the electric force belonging to any material circuit (i.e.

In tables of logarithms of numbers to base io the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.

There are no numeral auxiliaries or segregatives used in counting, as in many languages of eastern Asia, though words expressive of a collective or integral are often used after the tens, sometimes after a smaller number.

If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by JPds.

Suppose the whole space in which induction exists to be divided up into unit tubes, such that the surface integral of the induction over any cross-section of a tube is equal to unity, and along the axis of each tube let a line of induction be drawn.

To prove that this system is complete we have to consider (f, o) 2, 04') 1, (f,Q) 1, (f,Q) 2, (f,Q) 3, 0,Q) 1, (o,Q)2, and each of these can be shown either to be zero or to be a rational integral function of f, 0 Q and R.

If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.