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Published online by Cambridge University Press: 20 January 2009
In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.
page 205 note 1 The theory is given explicitly for a single pair of variables q, p (or Q, P). It is clear, however, that it may be generalised at once to include any number of pairs.
page 205 note 2 Professor Whittaker takes both signs positive; but it is more convenient for the present method to take one positive and one negative.
page 209 note 1 CfWhittaker, and Watson, , Modern Analysis (1927), 353 (Ex. 11)Google Scholar
page 216 note 1 This is a particular case of a general result. CfWhittaker, and Watson, , Modern Analysis (1927), p. 385, Ex. 50. It should be noted thatGoogle Scholar
page 219 note 1 Zs. f. Physik, 40 (1927), 809.CrossRefGoogle Scholar
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