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The derivation o f the corresponding formulae o f dynamics from the principle o f virtual displacements, in combination with D ’Alembert’s Principle, is effected somewhat as follows: Consider a system o f particles whose masses are m^. . and whose co-ordinates are Z il ^2? J2j Za etc. Let the components o f the forces acting upon the several particles be X^, Tj, Xg, Tg, Z2) etc. The effective forces, measured by the mass-accelerations o f each particle, are given by m , mi and similarly for the other particles.

Will be the motions o f the respective bodies in view o f their mutual reactions. In the latter part o f his book D ’Alembert makes numerous applications o f his Principle; and he also succeeded in relating to it the theory o f the motion o f fluids, in his Traite de Vequilibre et du mouvement des jtuides (Paris, 1744). He subscribed to the opinion, current in his time, that the principles o f mechanics are capable o f being proved; but the alleged proofs which he adduces merely amount to saying that the proposition in question is true because there is no adequate ground for maintaining the contrary.

Euler was the first distinctly to conceive the notion o f a mathematical MATHEMATICS 53 function, with which the earlier chapters o f the Introductio deal. This conception has been well described as perhaps the most fundamental of all the creations o f modern mathematics. — Introductio, I, i, 4). , where a, b, c stand for cpnstants. Later he considers functions of more than one independent variable. , addition, subtraction, multipHcation, division, raising to powers, and extraction o f roots. Logarithmic or trigonometrical functions of the variable, and those which involve it as an exponent, are classed as transcendental.