Weber at the same time deduced the mathematical laws of induction from his elementary law of electrical action, and with his improved instruments arrived at accurate verifications of the law of induction which by this time had been developed mathematically by Neumann and himself.
With this conception of the infinite as absolutely unconditioned should be compared what may be described roughly as lesser infinities which can be philosophically conceived and mathematically demonstrated.
Stefan's law of radiation according to the fourth power of the temperature is too difficult to pursue, but if we are content with cognate results we can follow them out mathematically in a hypothetical law of the first power.
This equation, which is mathematically deducible from the kinetic theory of gases, expresses the behaviour of gases, the phenomena of the critical state, and the behaviour of liquids; solids are not accounted for.
Oliver Heaviside showed mathematically that uniformly-distributed inductance in a telephone line would diminish both attenuation and distortion, and that if the inductance were great enough and the insulation resistance not too high the circuit would be distortionless, while currents of all frequencies would be equally attenuated.
The combination of ductility, which lessens the tendency to break when overstrained or distorted, with a very high limit of elasticity, gives it great value for shafting, the merit of which is measured by its endurance of the repeated stresses to which its rotation exposes it whenever its alignment is not mathematically straight.
This is mathematically expressed by the statement that dE is an exact differential of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.
In treating mathematically the propagation of other kinds of waves, it is necessary to analyse them into their simple-harmonic components, which may be treated as being propagated independently.
For Kutuzov this was mathematically clear, as it is that if when playing draughts I have one man less and go on exchanging, I shall certainly lose, and therefore should not exchange.
For long periods he was mathematically unproductive, but then sudden inspiration would come upon him and his ideas and theories poured forth far more quickly than he could record them.