356 MR. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



so that under this limitation + 2>ty z = A/ty- Now lf ^* is defined as related to p jt in 

 the same way as /3* is related to /,, that is by equation (6) then operating on both 

 sides of (25) with 2)', we have 2/i/r, = Afi'pji. Substituting in (26) 



Thus it appears that the (p)'s and (b)'s, as defined by (22) and by (G), are the 

 body and boundary values of the integrals Pj . . . P B . 



Equation (23) is therefore equivalent to the statement that an arbitrary function 

 /may be expanded in the series 



/=A 1 P 1 + A S P,+ ...A II P 1 , . -. ...... (28) 



as to its body points, and as / and the (P)'s must all satisfy the same boundary 

 condition the same expansion holds good on the boundary also. Analogous to this in 

 infinitesimals, and for a special form of <>', are the expansions of arbitrary functions 

 in series of sines, Bessel functions, spherical harmonics, and other integrals of 



(V 2 + /,:')</>= 0. 



Equations (22) mean that to determine the coefficient of any principal mode of 

 vibration in the expansion of an arbitrary function we must multiply the arbitrary 

 function by this mode of vibration and by the unifying factor, and add up the products 

 at the body points only. This is analogous to the well-known Fourier method of 

 determining coefficients. 



Squaring equation (2.3), multiplying by i t and summing over the body points and 

 using (15) and (17), we find 



I = n l = n -^ 



2 *ip\i = 1, 2 iipjipu. = 0. . . . 



L(29) and (30). 



which mean the same as SIP/ = 1, SIP*P; = . 



The above proof holds good even if any of X^ ... X n 2 are equal to one another. 

 This completes the properties of the (P)'s which we require.* It remains to consider 

 some approximations to X^.-.X/. 

 Since by (17) and (18) 



V = VA 1 !i + X/A/+...X n 2 X 2 / 81 x 



T A 



* The transformation of (^)'s to (A)'s resembles an orthogonal transformation of co-ordinates in that 

 there are two other relations similar to (29) and (30). These are found by squaring (22) and using (17) 



3 = n j = n 



and (15). They are i k 2 p jt ? = 1 and 2 p 3l p^ = 0. The first means that the sum of the squares of the 



j=i j=i 



values of the n harmonics at the &th body point is l/i fc . The second means that if we take each principal 

 mode of vibration, form the product of its values at two fixed body points, and then add up these products 

 for all the modes, their sum is zero. April, 1910. 



