1876.] the Product of two given Functions. 267 



angular motion to the rotating disk equal to its own linear motion, the 

 other gives a linear motion equal to its own to the centre of the rolling 

 globe. 



The machine thus described is immediately applicable to calculate the 

 values H x , H 2 , H 3 , &c. of the harmonic constituents of a function ^(o?) 

 in the splendid generalization of Fourier's simple harmonic analysis, which 

 he initiated himself in his solutions for the conduction of heat in the 

 sphere and the cylinder, and which was worked out so ably and beauti- 

 fully by Poisson*, and by Sturm and Liouville in their memorable papers 

 on this subject published in the first volume of Liouville's ' Journal des 

 Mathematiques.' Thus if 



be the expression for an arbitrary function \j/cc, in terms of the generalized 

 harmonic functions ^(V), <j> 2 (^), 3 (#)> ^ c -> these functions being such 

 that 



we have 



H= JoW a? )*K a O c fc 



2 hmr^' 





 &C. 



In the physical applications of this theory the integrals which consti- 

 tute the denominators of the formulae for H x , H 2 , &c. are always to be 

 evaluated in finite terms by an extension of Fourier's formula for 

 J x xufclx of his problem of the cylinder f made by Sturm in equation (10), 

 § iv. of his "Memoire sur une Classe d'Equations a differences partielles" 

 in Liouville's Journal, vol. i. (1836). The integrals in the numerators 

 are calculated with great ease by aid of the machine worked in the manner 

 described above. 



The great practical use of this machine will be to perform the simple 

 harmonic Fourier-analysis for tidal, meteorological, and perhaps even 

 astronomical observations. It is the case in which 



0(a?)= (noe) ; 



rv ' cos v ' ' 



* His general demonstration of the reality of the roots of transcendental equations 

 essential to this analysis (an exceedingly important step in advance from Fourier's 

 position), which he first gave in the 'Bulletin de la Societe Philomathique ' for 1828, is 

 reproduced in his ' Theorie Mathematique de la Chaleur,' § 90. 



t Fourier's < Theorie Analytique de la Chaleur/ § 319, page 391 (Paris, 1822). 



x2 



