354 



ME. L. F. RICHAEDSON: APPROXIMATE ARITHMETICAL SOLUTION 



integral, where the unifying factor I is equal to minus the ratio of the volume of the 

 elementary co-ordinate block to the product dq l dq 2 dq 3 .* 



<& denotes a summation for every difference which involves a body-number directly 

 (that is not by way of boundary equations (6)). 



In the following table the co-ordinate difference is supposed constant for each 

 co-ordinate separately : 



2)' with sign to make 

 CM plus 



Unifying 

 factor I. 



2 



r^Vr z 2 



^ 



, 1 



Unity 



Unit y *Ua + * 





+ 





Unity 



4 4\ a 



Note that the averager p. occurs in 2)' but not in V. Having now set out the 

 conditions of existence of V in a form in which they can easily be applied to test any 

 operator )' given with unifying factor and boundary conditions, let us pass on to 

 deduce the properties of the principal modes of vibration from the existence o/V. 



Let 



2T = ^ 2 +^ 2 2 +...^ 2 ........ (15), 



then 



t> { .......... (16), 



and by (14) T is essentially positive. 



Now V and T being real quadratic forms in the same variables, one of which, T, is 



* Simply putting for 3 in the infinitesimal V does not give the finite-difference V except in special 

 cases. April, 1910. 



