86 PROCEEDINGS OP THE [Oct., 1875. 



mark 16, 32, and on —1, mark 19, 35.5. Then an Archimedean Spu-al con- 

 structed to pass accurately through points 7, 23, 39 on ray +1, will pass accu- 

 rately through several of the others, and very near to each of the rest. It has 

 already been remarked that element B=ll which stands in the table as a posi- 

 tive triad in series A, might be transferred as a negative pentad to series B, 

 without marring the existing continuity of the whole series or sequence of 

 numbers ; so Gl=9.3 and Mg=24 positive dyads in A and B might be placed, 

 with the same numerical fitness, as negative hexads in series B and C. But in 

 the diagram just described there will be no necessity for removal, the same end 

 will be attained by continuing to read in the negative direction, past ±4 on- 

 wards to +3 now regarded as — 5, and to +2, regarded as —6. If the diagram 

 of preceding section with rectangular coordinates were wrapped around a cylin- 

 der whose circumference equals interval in diagram from +4 to— 4, the result 

 just mentioned would be exhibited with the same accuracy, a helix being sub- 

 stituted for the spiral. 



Taking the series of numbers from Li=7 to Ba=137 it is evident that no 

 straight line or linear equation, adjusted to give accurately certain atomic num- 

 bers, can give more than rude approximations to many of the others. Curve 

 lines or equations of a higher degree than the last, may do better, as 



27 1 23 1 1 1 



«,=7-t--a=+g-a^. or ^=8+16^+2^^-4200^-500000^' 



where w signifies the atomic weight required, and x the number of steps or the 

 distance from Li=7, counting every Une and unfilled gap as one step including 

 those of line whose index «=0 in first column, for an equation ought to fiU 

 every gap. By using more complex fractions for the coefficients, better results 

 would be obtained in several cases, if it were worthwhile to compute them, but 

 it is not difficult to see that curves which would give closer approximations than 

 the above, must have two points of inflection, or contrary curvature, one 

 between the ordinates or numbers 40 and 50, the other somewhere between 

 87.5 and 108, and if V, Cr, Cb, and Mo be retained in the places now assigned 

 them, it may become worth our while to construct equations for such curves, 

 especially if any of the gaps now existing were filled by discovery of new ele- 

 ments. We might use a transcendental equation such as 



2.0531 +.0067aj=com. log. of («/j+105), 



but the remark just made shews that it can give tolerable approximations only 



in certain parts of the series. 



But all these processes are arithmetic or geometric exercises, and though it be 

 curious or remarkable that such manipulations are possible with numbers ex- 

 pressing chemical relations, we must beware of laying undue stress upon them. 

 None of the equations or constructions give results exactly coinciding with the 

 atomic weights throughout any one series of numbers, and in not a few cases 

 the values of the atomic weights themselves, cannot be regarded as yet defin- 



