50 STATISTICAL METHODS. 



surface, but may be regarded as situated from those axes at 

 the respective distances h and k. These values may be 

 found from the formulae 



_ e _ tlf 

 TT /() 



a, b, c, and d being known, h and k are found from Table IV. 

 Then 



and 



-= - 



V27T V27T 



of which the values may be looked up in Table III, or, better, 

 their product may be calculated by logarithms as follows: 



Find also log hk, h 2 , and k 2 . To find r solve the following 

 equation to as many terms as may be necessary: 



(/i 4 - 67i 2 + 3) (A; 4 - GF + 3)r 5 



+ -_Lhk(h*- lOh 2 + "15) (k 4 - lOfc 2 + 15)r 6 + etc. 



This gives us a numerical equation of the nth degree which 

 can be solved by ordinary algebraic methods, using Sturm's 

 functions and Horner's method. Or it can be solved by 

 successive approximations as follows: The first approxima- 

 tion is made by neglecting all powers of r above the second 

 and solving the quadratic (remembering, that if ax 2 + bx + c = 0, 



