158 G. Siueal: 



suitable, value of A to be used in the formula. Tlie same method was 

 followed at first in reducinf; the present series, and considerable time 

 was spent in ofroupin^; and examining the values of A before it was 

 discovered that the method was incorrect in principle, though yield- 

 ing approximately correct results so long as the number of observa- 

 vations was not too small. 



By the application of the method of least squares to the more 



general formula 



.V --'///- AB(7-/'), 



in which tliere are two constants, rj and A, to be determined, a single 

 value of each is obtained to represent the whole set of observations. 

 The corresponding formula with these numerical co-efficients has the 

 property that it gives the value of .c with the least probable error 

 from observations of the values of the other quantities concerned.. 

 Now this is evidently the result which is required ; in practice the 

 psychrometer is used alone, and we are required to determine from its 

 readings the actual value r of the vapour-pressure in the air. We 

 therefore seek a fornmla of the recognised type with such numerical 

 co-efficients that the value of r will in the long run of similar trials 

 be given with the smallest possible margin of error. If the simple 

 formula 



X = f - AB{/ - O, 



be used, nr, in other words, if the co-efficient 7/ be assumed to be unity, 

 this end will be attained by a direct application of least squares to the 

 equation as it stands, and not to the severally determined values of 

 the constant A. Taking the arithmetic mean of the values of A is an 

 application of least squares which makes the errors of A a minimum, 

 instead of those of r. The correct value of A which is appropriate to 

 the whole set of observations is therefore given by the equation 



where 3 is put for convenience in place of B (f - /'). Since the indivi- 

 dual value of A is given for each obsei-vation by 



A = (/- .r)/s, 



the correct result is the same as would be obtained by weighting the 

 individual values proportionally tn z^, that is practically to (f-t)'^. 

 Since t ■ f \s freiiuently small, and is in the denominator, this makes 

 it seem probable that the correct value for the present purpose would 

 also be a better value than the simple mean, if the object were to 

 determine A with the least margin of error. (This latter might be 

 the case, for example, if the formula were assumed to be absolute 

 and not merely an a[>i)roximatioii of varying accuracy : then the value 

 of A might be considered as an aid to investigation of the projjerties 

 of air or water-va|)Our. It need haidly bo said that sucli a procedure 

 would be absurd.) 



