- The
**vanishing**of this invariant is the condition for equal roots. - The stars, as if knowing that no one was looking at them, began to disport themselves in the dark sky: now flaring up, now
**vanishing**, now trembling, they were busy whispering something gladsome and mysterious to one another. - In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the
**vanishing**of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n). - Bidwell 2 accordingly found upon trial that the Wiedemann twist of an iron wire vanished when the magnetizing force reached a certain high value, and was reversed when that value was exceeded; he also found that the
**vanishing**point was reached with lower values of the magnetizing force when the wire was stretched by a weight. - Each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the
**vanishing**of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.

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