﻿194 ON THE USE OF EQUIVALENT FACTORS 



and the probable value of the sum of tlie terms ae^ -\- a e'o -{- is 



« e„ + a' e'o + - ^ V(l ± g) V« a + «' a' + ^ I" ^^ ^ I ^^ ■^'a ' 



which is the probable error of the equation (22), (a). 

 Again, since 



A, = auJl-{-^-A, £ |3 = J ,. f 1 + ^ V 

 \ atvj \ wj 



and 



P, = „ a-l- „' a' + , Q, = /3 J + 0' J' + 



P', = „& + „' 5' + , 'Q. = j3 a + 0' a' + 



or, substituting the probable values 



we have 





INIoreover, 



The sum of all the terms ^^ « a — ^ /3 a^ = T^ p^ _ ^ 'Q, j = 0, 

 Therefore the probable sum of the terms 



will be 



(/3_^«)e„+0'-'-§i«')e'o + 



■t 1 -t 1 



which is the probable error of the equation (22), {h,). 



The other probable errors in (23) are readily supplied by analogy. 



If we neglect - g., of which the probable value is less than — , the probable errors of 



(22), (a), (6,) become 



Probable error of equation (22) (a) = ^>. Vj- , Probable error of equation (22) (J^) = ^ \~ . 



We shall now proceed to explain a third form of solution, (III.). 



