30 EFFECTS OF WINDS AND OF 



These monthly normal equations were later combined by addition, 

 term by term, to form the final set of normal equations for the solution. 



The final normal equations for solution K 2 are given below as table No. 5. 

 Solution K 2 served to give the final determination of the barometric effects 

 at Milwaukee. 



TABLE No. 5 Normal Equations, Solution K 2 , Milwaukee. 

 +179445u>o- 25525>i- 39555to 2 - 24475u> 3 + 48025n - 5735m + 28175n 2 4- 3655n 3 - 17427 = 



- 2552Bwo+ 163365>i + 33585u; 2 - 46225^- 33045n + 41305m + 20245n 2 - 14825n 3 +41911=0 



- S955Bwo+ 33585u>i+220355, 2 - 31475u> 3 - 49165n - 11815m + 42695n 2 + 27745n 3 + 53880 = 



- 24475^0- 46225>i- 31475u> 2 + 151045u, 3 - 3475n - 14385m- 42455n 2 + 29045n 3 + 11035 = 

 + 48025 WO - 33045^- 49165 2 - 3475t0 3 + 134445n - 14375m- 33005n 2 - 40655n 3 - 8746 = 



- 5735u> + 41305ti- 11815, 2 - 14385, 3 - 14375n + 121495m- 4575n 2 - 655n 3 -61166 = 

 + 28175w + 20245^+ 42695u> 2 - 42455tc 3 - 33005n - 4575m + 172065n 2 - 15865n 3 - 73936 = 

 + 3655u, - 14825^4- 27745*, 2 + 29045^- 40655n - 655m- 15865n 2 + 101685n 3 + 2900 = 



These normal equations depend upon observations extending over 220 

 days, which were expressed in 186 observation equations. Some of the 

 observation equations covered two or more days each. 



The solution of these normal equations gave the following values for 

 the unknowns, expressed in the units used in the observation equations: 



B w0 =-1.78 B n0 =+1.99 



w i=-4.34 nl =+6.53 



u , 2 =-2.92 n2 =+6.34 



B wS =- .90 n3 =+2.03 



EXAMPLE OF SUBSTITUTION IN OBSERVATION EQUATIONS 



FOR BAROMETRIC EFFECTS. 







The above values by substitution in the observation equations for Septem- 

 ber 1910 as given on page 26 gave the equations which served to determine 

 the residuals v for this particular month. From such residuals the probable 

 errors were computed. These residuals are the discrepancies between the 

 theory on which the observation equations were based, on the one hand, and 

 the observed facts, on the other hand. 



In the following tabular arrangement of the substitution in observation 

 equations the heading on each column identifies the term of the observation 

 equations as shown on page 26. As a specific example, note that the value 

 given in the B w \ column for September 9, namely, +156, is the product of the 

 quantity 36 in the September 9 observation equation, which is the coeffi- 

 cient of B w i times the final value of B w i, namely, 4.34. 



The values of v, the residuals, were obtained by adding all terms in the 

 first member of the equation. If the agreement between theory and ob- 

 servation were perfect, all v's would be zero. 



Note that there were 8 days in September 1910 when the observed rise 

 (or fall) at Milwaukee, corrected for rainfall, inflow, outflow, and wind, was 

 more than 0.1 foot (corresponding to 100 in the 7 column in the following 

 substitution in observation equations), namely, September 6, 14, 18, 20, 21, 



