4G0 Vtl^consin Academy of Sciences, Avis, and Lcitcrs, 



Since the coefficients of tan ^/2 c, which depend on the quantity 

 /", do not vary ^eatlv, the quantity 0.37*, which is scarcely 10^ 

 different from the extreme values, may be written as a mean 

 value/ so that we have: 



There is no nood to continue this tahle, which shows aj\proxi- 

 mately the proluible error correspond in/^ to any particular value 

 of Cy since for any value c < 00^ another equation 



cos* Ic = cos* l{a-\-b)-\-sin a sin 6 cos* ^G 



is employed, the sohition of which is the same and which has this 

 same probable error for values of c from c = 00^ to c = 180°. 



The 3)rn])al)ility W of an error witliin certain limits, or of an 

 error within ± 1 second, is computed from the equation [17]. 



A ^/>, and this difforence is soupht in the column of the table marked c-ot. Then 

 from the column marked cos the corresponding logarithm is taken, which subtracted 

 from log A gives the required log C. But if IS^ A the diflFerence log B — log A is to be 

 souplit in the cot column and the corresponding logarithm from the cos column is to 

 b3 subtracted from log B. That difference is alwaj'S taken which is the logarithm of a 

 quantity greater than 1, and the corresponding logarithm from the cos column sub- 

 tract*id from the greater of the given logarithms. In the computation of the probable 

 error the rale is so used that for ju^i.o° the change is made from cotangent to sine. 



•0.38 in the original. E. F. C. 



