Oct., 1875.] ELLIOTT SOCIETY, 83 



already given in the second paragraph of this paper ; for example, the numbers 

 inline —1, for this sequence, would be, 



19, 19+1x16.5=35.5, 19+2x16.5 + 1x28=80, 2x19+2x16.5+2x28 = 127; 

 for line —2 they would be 



16, 16 + 1x16 = 32, 16+4x16=80, 16+7x16=128; 

 for line —3, they would be, 



14, 14+17=31, 14+17+44=75, 14+17+2x44=119, 



and so for others ; for the artiad lines, the numbers used by Dumas, were of 

 course only half those of the Table, and the last number just given, 119, agreed 

 better with the atomic weight of Sb used by Damas, than with that at present 

 received. It is worthwhile to remark, in connection with this part of our sub- 

 ject, that the numbers in line +2, are very approximately represented by the 

 multiples of 8 upon the following numbers, 1, 3, 5, 8, 11, 14, 17 and 26, for the 

 series. A, B, C, D, E, F, G and K respectively ; hence it evidently follows, that 

 this line or group will furnish no less than live of the " triads" of Dumas, 

 namely 1, 3, 5 ; 5, 8, 11 ; 8, 11, 14 ; 11, 14, 17 ; 8, 17, 26. The numbei-s in 

 line or group — 2 are represented even more closely by the multiples of 8 upon 

 2, 4, 6^, 10, 12, 16, 23, 25, but they furnish only one "triad" 4, 10, 16. .No 

 number of the last series of multiples coincides with any one of the former, but 

 the first three of the last series are arithmetical means between each pair of 

 the first four of the former. It is also evident that the average diilerence 

 between series B and each adjacent series is about 16=8x2, between each pair 

 of series from C to G is about 24=8x3, and between G and K is 72=8X9; 

 this last series is marked K to indicate this large interval from G, and the in- 

 termediate column is marked H, I, to indicate uncertainty as to the number of 

 intervening or missing series, and the probability that there would be at least 

 two, with common difference of 24 ; the only two numbers at present in col- 

 umn H, I, indicate a difference 1 5 with respect to series G, nearly equal to that 

 existing between series A, B, C. Another mode of expressing or obtaining the 

 numbers in line +2, is the following ; it is allied to that of Dumas just given 

 and may also be applied to the other Jines in the Table is this : 



for the series ABCDEFGK 



write the numbers 10 32 55 78 102 126 198 



their differences are 16 16 23 23 24 24 72 



then to each of these numbers is to be added a certain constant quantity for 

 each line of the Table; for lines — 4, — 3, —2, add — 4, —2 and respectively, 



