638 Proceedings of the Royal Society of Edinburgh. [Sess. 
units of the sixth decimal place, and under d the greatest departure from it 
of the mean of any individual of the (s 2:l + %) series. 
There are twenty-four tables of each class, referring in all to 185 
different solutions. The total number of series of observations of specific 
gravity made with the two hydrometers is 1203. Each of these series 
consisted of nine separate and independent observations, so that the total 
number of individual observations is 10,827. Dividing the total number of 
series of observations made (1203) by the number of different solutions 
experimented on (185), we obtain 6 - 50 as the mean number of series of 
observations made per solution. 
The sum of the probable errors (2> 0 ) is 532 - 4, whence the mean probable 
error of the mean specific gravity per solution is ±3 - 04 in the sixth 
decimal place. Amongst the 185 separate values of r 0 , 171 are not above 
5 ; and of the remainder, 13 lie between 5 and 10, and only 1 value is 
above 10 in the sixth decimal place. We see, then, that the mean specific 
gravities in these twenty-four tables have a probable error which hardly 
exceeds ±3 in the sixth place. Consequently the exactness of units in the 
fifth place is abundantly guaranteed. 
In the column under d we have the maximum departure from the mean (S) 
of any of the serial values which form the basis of the mean. The mean of 
the 185 values of d is 15‘4. Of these, 33 are above 20 and 3 above 30. 
Returning to the values of r 0 , the probable error of the arithmetical 
mean S per solution, we have seen that it is ±3*04 for a mean of 6 - 50 
series per solution. Amongst observations, all having equal weight, the 
probable error of the arithmetical mean varies inversely as the square root 
of the number of observations used in arriving at it; consequently the 
probable error of one of the 6’50 serial means is 3’04 J6’5 0 — 7'75. As all 
the observations were made uniformly and with the same care and 
attention, they are to be taken as having equal weight ; and we have in the 
following table the probable error r 0 of the mean specific gravity of a 
solution derived from any number s of serial means. 
s = 1 2 3 4 5 6 7 8 9 
±r 0 = 7-75 5*48 4*47 3'88 346 3*16 2'92 2-74 2-55 
This table is of use for showing the probable exactness of mean specific 
gravities depending on other numbers of serial means, whether made with 
one or more than one hydrometer. The specific gravities of the solutions 
of the Ennead (MR0 3 ) consisting of the salts of {K, Rb, Cs ; C10 3 Br0 3 , 10 3 } 
and those of the nitrates of the same metals, at 19'5° C., were all made with 
hydrometer No. 17 alone. This was done because there was not time to 
make the double determinations in the case of all the solutions, and the 
