﻿Paper on a new Harmonic Analyser, 129 



where H(V) is the integral of Q(at) and may be tabulated as 

 Q(x) is tabulated. Now in many cases the part between the 

 square brackets is zero, but this is of little consequence in 

 comparison with the fact that the part j*H(#) . dy may be 

 evaluated by a machine somewhat like Prof. Henrici's first 

 or 1889 instrument. I have worked with this 1889 instru- 

 ment, and I am not disposed to think it so inaccurate as 

 its inventor thinks it. Its defects are really defects of 

 mechanical construction ; for example, the amplitude of the 

 simple harmonic motion of its table is very much too small. 



I have already put my machine in hand and hoped to 

 exhibit it here to-day in working order, but unfortunately the 

 Easter holidays have prevented it being finished in time. It 

 is arranged to develop an arbitrary function in Bessels of 

 the zeroth order, or rather Fourier cylindric functions. 

 Thus it is required to determine the constants A lt A 2t &c, in 



f(r)=A 1 J (fJL 1 r)+A 2 J {n 2 r)+ &c., . . . . (1) 



where fi i9 fi 2 -> & c * are the successive roots of some such equation 



as J (/*a)=0, (2) 



or fiaJ x (fia)— \J (fia) = 0, (3) 



where X has a given value. 

 It is well known that 



A.=Mf%/W . J.&v) • dr, 



'•-J. 



where M = 2/a 2 [Ji(/v*)] 2 , if fa, fa, &c. are the roots of (2), 

 and 



M=2/*V( x9 + ^V 9 )[ J o(^«)] 2 , if Pi> th, &c - are the 

 roots of (3) . 



la every case the practical difficulty consists in finding the 

 integral. I exhibit to the Society an easy example of such 

 an analysis worked out numerically (I suppose that such a 

 thing has never been done before) by two of my students, 

 Mr. H. F. Hunt and Mr. W. Fennell. 



It will be seen that the work is rather tedious. It was 

 made more tedious by their having found it necessary to 

 calculate numbers and tabulate them in a handy form, 

 interpolating between the numbers given in Lommel by 

 the use of his formula. Before this work was finished we 

 discovered Dr. MeissePs elaborate tables, from which the 

 remainder of our handy four-figure tables is merely copied. 

 These handy tables of J (#) and Ji(#) are at the service of 

 the Society; they would occupy just four pages of the 



Phil. Mag. S. 5. Vol. 38. No. 230. July 1894. K 



