VARIATION

(Lat. varius, cAawgw^.) In Music, a difference in performing the same air or melody, either by subdivision of its notes into several others of le duration, or by adding graces; but, neverthele , in such a mann<!r that the original melody is not lost in the decorations or alterations which it is thus made to undergo. VARIATION OF CURVATURE, in Analytical Geometry, is the change which takes place in the curvature in pa ing from one point of a curre to another. The circle is the only curve in whicli the curvature is uniform at every point. The curvature of a curve line the least time po ible, . In such questions the relations between the variables is not given, as in the ordinary cases of maxima and minima; the object proposed is to find that relation, or to find the equation which must subsist between the variables in order that the condition of maximum or minimum may be fulfilled. The first problem of this kind which was solved appears to have been that of the solid of least resistance. In the first edition of the Principia, published in 1687, Newton gave the equation of the curve by the rotation of which about its axis the solid is formed which, when moved through a fluid in the direction of the axis, is le resisted than any other body of the same specific gravity and bulk; but without demonstration or indication of the views by which he had been guided. About ten years later the famous question of the brachyslocrone, or curve of quickest descent, was agitated between the two brothers James and John Bernoulli, in which Leibnitz and some of the other most illustrious mathematicians of the day took a part. The more general problem of isoperimeters was solved by James Bernoulli in 1701; and it was in the analysis which he gave on this occasion that the principle on which the solution of similar questions depends was first distinctly unfolded. Euler treated the whole subject in his peculiarly luminous manner in a treatise published in 1744, under the title Method us Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudenles. And, lastly, the method was reduced to its utmost sim  [s. 1297]