30 
THE VOYAGE OF H.M.S. CHALLENGER. 
III. I, ■ >f ih 77 p. ro'iitages of sulphuric acid reported in the general table is based 
u[ u • 1 t-t two w -11 agreeing determinations, the mean of which was adopted as the 
in : ■! result. In each case the mean deviation of the individual result from the 
m m ina\ In taken as a guess at the probable error, and the mean of the 77 mean 
: v • us >hould be a fair approximation to the probable error of the individual per- 
. • i : ■ i_. - ..f sulphuric acid reported. I have calculated this general mean, and found it 
»±0'00026 per .r = O l 158. Adding on O’OOOOG for the influence of the uncertainty- 
in the chlorine determination, we have for the presumable analytical error the value Ax 
±0 00032. which is not much below either of the calculated r’s, and is nearly double the 
did- pm. between the., of the “shallow” and the x 0 of the deep-sea waters. Hence, 
although 1 find in my list of values for x— oj 0 not a few which I could not well admit 
t-. fill u thin tie probable limits of analytical errors, there is no chance of tracinga relation 
bet w. . n this ./• and depth. Rut supposing the irregularities to be of the nature of acci- 
- Mai « Hoi ', they should be amenable to the law of frequency of error, and it should be 
possible to calculate the prob ability of the actual difference of 0*00017 between the x 0 for 
t ; p o' • nel the ay for the shallow waters, being brought about by mere accident and 
i; ■' h\ tin .'xistciice of a law prescribing to the one x 0 a higher value. Or to formulate 
t • qu- sti. hi in more definite terms. Supposing out of our 77 values for x we take 34 cases 
"t ■ / • /. ; we take their mean a^, and compare it with the mean x i3 of the rest. What 
- ih' probability that the difference x 34 —x 43 amounts to 0‘0001 7 ? The treatises on the 
m-tlcl "I tin h ast squares, as far as my knowledge goes, do not give a formula that 
d<l . a d ilc on< to calculate the precise numerical value of this probability. As a make- 
1 :t I . dop' d for this and the subsequent analogous cases the following mode of 
’ "i 11 According to the law of “frequency of error,” the probability of an error 
. r. at- f than four times the probable error is 0‘007, 5r corresponding to ’0007. 
II m v -ay, the unknown errors of our two means jc 0 arc sure to fall short, in the 
case of the 34 «1< op-sen waters, of 5 times their r 0 , which is 0’0004 ; in the case of the 43 
■ 1 , of 5 x their /•„, i.e., of 0'00025. Supposing them to be of opposite signs, 
I ■ (in ' dental difference x 34 — x 43 may amount to 0’000G5, which is considerably 
more than the actual difference 0 00017. 
The Quantities of Magnesia. 
(Taken in ternm of 100 parts of total Salts.) 
*0 
r 
h> 
For all the 77 cases analysed, 
62145 
not calculated. 
For the 34 deop-ooa waters, 
6-222 
0-019 
0 0032 
For the 43 shallow waters, 
\ sloe of for de<-p-#ea minus that for 
6-209 
0022 
0-0033 
•hallow w stem, 
+ 0013 
