Hydrometers of Total Immersion. 507 



equation ~- — 0, which gives 



^ B _l9 C Vi , B ' & «\ /i B C 9 \/B' a C' \ n 



U +2 a t A 1+ ^ t+ ^ 72 )-( 1+ a t+ a t2 Xa' +2 a^ t ) =0 ' 



or 



B B' , , /C C'\ /C B' C B\ . _ 



a-a^ + Ha-a') t+ (aa'-aTa) t ^ - 



In the actual experiments A, A' are nearly 162, B, B', 

 0, C are of order -004 at most. Thus -r -j-, — p- -r- is very 



B R' P f ' 

 much smaller than either T --r;or t-~"tt- One root of 



A A' A A' 



the equation is verv large ; the other is the value looked for. 

 Thus 



B _B' 



A A 1 



2 



. n » q y approximately. 



U"a) 



We requh-e to estimate the influence on r of given small 

 errors in the values of A, B, C, A', B', C. We find 



Bt A ^t A\A A'/ Bt AVAA' A' A 



bb~ /c cy sc- /c_cy ba~ 9 /C_cy 



^U'~aJ '\A' A J ^VA' A/ 



I _1/B_B'\ I/5?_C' B\ 



Bt _ 'A Bt _ A'VA A7 Bt _ A f \A A' A' AJ 



BB'~ /C 0\? BC' _ 9 /C CV' BA'~ 9 /C c\« • 



2 va'~aj j U'~av va'~a; 



Errors in A, A' are thus of little consequence compared 

 with equal errors in B, C, B', C/. 



The value found by Mr. Warrington for his hydrometer 

 from all his experiments is Y = Y + at + bt' i , where 



V = 161-93021 --00001365c?; a= + -003864; 

 b= +-000,0012. 



The probable errors, calculated from all the observations, 

 are for 



V ±-000,129, a +-000,017, 6 +-000,000,45. 



