88 Mr. Nixon on the Measurement by Trigonometry of the 



lations to the use of logarithms, treating them in the notation 

 as natural numbers, F (or the distance required) will be equal 

 to the sum of the base and sine of the angle opposite F, minus 

 sine of the angle subtended at a by the base. — Here we may 

 observe that tne value of the angle at e affects the calculation 

 merely as it serves to determine that of the unobserved an- 

 gle a. 



Admitting that one only, as well as both of the angles de 9 

 may have been inaccurately observed, the subjoined list will 

 include every possible arrangement of the angular errors to^- 

 gether with the consequent alteration of the correct value of 

 the unobserved angle a : 



I. Make the error of d -f 1', and e + 1'; whence a = — 2'; 

 II. +1 -1; 



III. +1 -1; 



IV. +i-i o. 

 As the log. sines increase from 0° to S0°, and diminish from 



90° to 180°, it will be necessary to divide our investigations 

 into two classes. 



Class I. — The angles at a and d being both acute. Call a! 

 the difference of the log. sine of a and the log. sine of a + 1' 

 (or of a— 1'); and d 1 the corresponding difference of d and 

 d + V (or of d and d — 1'). Then as the correct length of F 

 has been shown to be equal to A 4- sin d — sin a, its value 

 with an arrangement of the angular errors as given in No. I. 



will be come 



=■ A -f sin d + S — sin a — 2 a'<; or simply 

 = F + d' + 2 a' ; whence x or the lineal error required will 

 be = d 1 + 2 a'. For the whole class of the angular errors the 

 values of x will be as follows : 



I. .r...= d! + 2a' 



II d' + a' 



III a' 



IV d\ 



Hence the probable error of calculation will be equal to the 



mean of the above four quantities, or to a 1 -\ — — . 



Class II. — When a is obtuse. Selecting No. I. of the ar- 

 rangement of the possible errors of observation, the calculated 

 length of F becomes 



B + sine? -f dP— sin a + 2a n ; whence x will be equal to 

 d'oo 2 a', and its value for the complete arrangement of errors, 

 I. . . x = d'cv>2a' 



II d'ooa' 



III a' 



IV d': 



Consequently 



