165 



the quantities a, b, c, being given in the preceding table. The 

 values of c, and consequently those of x and y, are probably in 

 every instance affected by errors of observation ; probably they 

 are also affected by peculiarities of rocks and hills. To obtain 

 from the whole series the most general values of x and y, or 

 in other words the mean rates of increase of dip toward the 

 North and towards the West, we must combine them into a 

 general result. The most approved mode of doing so is called 

 the method of least squares, because it reduces the sum of the 

 squares of the errors (or differences from the mean value) of all 

 the observations to a minimum. This is accomplished by 

 giving to each equation ax + b y zz c, first a value propor- 

 tioned to a, and next a value proportioned to b ; whence arise 

 two general equations of the form, 



a^x:t.abi/ = ac 

 aba;±b^y = bc, 



yielding by reduction the values of x and y, the direction of the 

 isoclinal line compared to the meridian, and the ratio or aug- 

 mention of dip (r) along the normal to it. Without encumbering 

 these pages with the calculations, which are somewhat trouble- 

 some, it may be sufficient, to give an example of the form and 

 state the result. 



a ^ ab. b^. ac. be. 



Newcastle 16 208 2704 100 1300 



Lincoln 1849 227 2809 1419 1749 



Sheffield 1936 968 



The sums of the several columns are, 



a ». 31587 a b. 4537 a e. 8269 



b\ 21261 be. 11961 



whence we obtain as the most probable mean values, in the 

 whole series, 



y = -521 toward the North. 

 X = '184 toward the West, 

 r = -553 toward the N. N. West. 

 u = TO'-Sl 



