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Caption: Mathematical Models by Arnold Emch - Series 2 (1923)
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B y this example Peano demonstrated the falsity of s o m e of the hitherto accepted criterions for m a x i m a and m i n i m a of functions of several variables. In most cases the errors are fundamentally due to the theorem, which, in general, is not true, that in Taylor's development of a function of several variables, the ratio of the remainder following a certain term of the expansion, to this term, approaches zero as a limit w h e n the increments of the variables approach zero as a limit. Peano's important criticisms are illustrated in a very conspicuous m a n n e r by the quartic surface with the Cartesian equation z = \(y-2px)(y2-2qx). T h e model, Fig. 1, represents the form of this Peano surface, w h e n X = - 1 / w , P = 2.5yq = 0.S. Projective Description of Ruled Quartics 20. Ruled Quartic. Generated by projectively related tangent planes of two cones, Fig. 2. Fig* 2 [6]
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