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676 LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 
of differential equations, of which the Seiche-eq uations are special cases, is well known 
to mathematicians through the important rdle it plays in the theory of the associated 
LecenpDRE-functions. But, as far as I am aware, its solutions have been investigated only 
in the special cases where n represents integers (see WHITTAKER, Modern Analysis, 
p. 235). In the theory of the C,“(w) polynomials, which are particular integrals of the 
general equations here considered, the two synectic integrals, with which we were 
particularly concerned in this investigation, are of less importance than some of the 
other hypergeometric series. But their property as synectic solutions renders them 
particularly useful in the special physical problem before us. It is not improbable that 
in other problems—for instance, in such which are based on the differential equation of 
the LrcrnpRE-functions (a =4)—the corresponding synectic integrals, the LEGENDRE 
Sine and Cosine, might also be of special importance. Since the latter, as well as their 
associated functions, may be derived from the Seiche-functions by differentiation, we 
may consider these as the typical representatives of the whole class; and this fact, 
doubtless, gives to Professor CHRysTAL’s investigation a considerable importance from 
the mathematical point of view,—an importance still more enhanced, on the one hand, 
by the introduction of the corresponding hyperbolic functions, to which he has now for 
the first time directed the attention of mathematicians, and, on the other, by the 
remarkable relations between this general class of functions and another class 
represented by simple transcendents. 
