58 Prof. S. U. Pickering on 



proportion to the actual experimental errors : similar errors, 

 amounting to 1,000 to 3,000 cal., could be mentioned in several 

 of the other cases. 



These last observations give, in fact, as I believe, the clue 

 to an explanation of these supposed constants. By taking a 

 sufficiently high fraction and allowing a sufficiently wide 

 margin for " error/' it is obviously possible to represent any 

 two numbers whatever as multiples of the same constant. 

 The whole question, therefore, resolves itself into one of 

 probabilities. 



Thus in No. 1 the difference between the two quantities is 

 68,810 cal., and the error which the first of them shows as 

 compared with the second is (111,590 — (3 x 36,880) = ) 1,150 

 cal. ; while the error which the second shows as compared 

 with the first is (5 x 37,197-184,400 = ) 1,585 cal., or a total 

 error of 2,735 cal. is made allowable. Now it is obvious that 

 if the error allowed had been as great as the difference 

 between the two values (184,400 and 111,590), every quantity 

 between these two numbers would have been regarded as 

 thus related. As it is, however, only 2,735 out of 68,810, or 

 1 out of 25, numbers between these limits will show the rela- 

 tion ; or, to put it in a more general form, of the numbers 

 between 184,400 and zero, 1 out of 25 will be, within the 

 given range of error, some submultiple by 5 of 184,400. 

 Now, on examining the table (vol. ii. p. 401) which contains 

 the data relating to the iodine compounds, we find 28 quan- 

 tities less than 184,400, i. e. rather more than sufficient to 

 give, through purely mathematical chances, the one case 

 discovered by Thomsen. 



With No. 2 there should be 74 numbers less than 72,970 

 cal. to give one coincidence within the limits assigned; there 

 are, as a matter of fact, 49 such numbers amongst the nitrogen 

 compounds (p. 406). With No. 3 there should be 28 num- 

 bers less than 300,080 amongst the phosphorus compounds 

 to give the one coincidence, whereas we find there are really 

 24 such numbers. 



With No. 4 it is certainly not legitimate to consider a 

 quantity such as 192,920 to be a simple multiple of 103,240, 

 when it can only be done by introducing the large fraction 

 of -fg and a margin of 1,007 cal. as error. Taking, there- 

 fore, the numbers for S,0 2 and S,0 3 only, we find that there 

 should be 74 numbers less than 103,240 to give the coinci- 

 dence, whereas there are 35 such quantities amongst the data 

 for sulphur. 



Thus in these four cases there are two in which the data 

 are nearly exactly as numerous as they should be if the coin- 



