114 Dr. J. H. Vincent on a Numerical 



with this value of q could be found 1st when no gap was left, 

 2nd with the gap. 



If the second arrangement gave a smaller mean difference 

 than the first, then this would justify the prediction of an 

 element in the place under discussion. 



It may well be, however, that the result of such an investi- 

 gation would be different according as one regarded percentage 

 differences or the differences themselves as of more importance. 

 If the latter, then the mean error is less when no gap is left 

 between molybdenum and ruthenium. 



Thus if we regard any atomic weight w N as giving us a 

 value of q, we have a series of equations like 



or log Wx = glogl$. 



Using the atomic weights of all the elements up to and 

 including molybdenum, we have 



log 3*93 = 2 log 3, 

 log6'97 = ry log 5, 

 log 9*0 =q log 6, 



log 95*3 = ^ log 44. 



The value of q given by 



_ 2 log w 

 ^"STo^N 



will be such that the percentage errors will be least. This 

 equation may be written 



2 log IV 

 q ~ log 144 -log 8' 



from which q may be found readily by using a table of the 

 logarithms of factorials. The value obtained is 



2 = 1-209. 



Similarly q may be computed, when no gap is left between 

 molybdenum and ruthenium, from the atomic weights of the 



