As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.
Thus the most prominent subdivision of the older school, the Taittiriyas, in their Samhita, have treated the main portion of the ceremonial in this promiscuous fashion, and to add to the confusion they have, by way of supplement, put forth a so-called Taittiriya-brahmana, which, so far from being a real Brahmana, merely deals with some additional rites in the same confused mixture of sacrificial formulae and dogmatic explanations.
It may reasonably be supposed, not only that they constructed the external framework of many chapters, and also made some additions of their own - a necessary process in order to weld their motley collection of fragments into a new and coherent book - but also that they fabricated anew many formulae and imitative passages on the model of the materials at their disposal.
As corollaries to the general formulae he adds the formulae relating to the representation of a number as a sum of five squares and also of seven squares.
These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.
This use of formulae for dealing with numbers, which express magnitudes in terms of units, constitutes the broad difference between mensuration and ordinary geometry, which knows nothing of units.
He proposed to apply the same principles to the calculation of the disturbances produced in the rotation of the planets by external action on their equatorial protuberances, but was anticipated by Poisson, who gave formulae for the variation of the elements of rotation strictly corresponding with those found by Lagrange for the variation of the elements of revolution.
The n formulae of this type represent a normal mode of free vibration; the individual particles revolve as a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor.
This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.
In the development of the atomic theory and the deduction of the atomic weights of elements and the formulae of compounds, Dalton's arbitrary rules failed to find complete acceptance.