390 REPORTS ON THE STATE OP SCIENCE. — 1919. 



and so 



+M,[/(.o+/(c,')} 



+ U,|/(o,)+/(c,')}, 



where 



^0=5^^ + 3)' 

 Ci, c / = \{'p + q)±\{p — q)a^, 

 c.„ c,' = \{'p + q)±\{];i—q)a.2. 



As an example, let this rule be required to find an approximate 

 value of 



Ji a;' 

 that is, of log,.10. Here 



/(«n)= A ' 



A^^)"^ 114.9„ ' /(^l')=il_9„ ' 



~-|-T5^a2 ^^^ — -a_, 



when these are substituted in the formula, and reduction is effected, we 

 find approximately 



'=•05 17 + -0957 + -1078 

 91] X 



= •2552, 



and therefore the approximation, given by this rule, for log,,10 is 2-2968. 

 The accurate value to the same number of places of decimals is 2 '3026 ; 

 so that the error is about -25 per cent. 



III. Let 11 = 7, so that pi = 3. The quantities a,*, a.^^, a-^^ are the 

 roots of 



429^3-693^2 + 315^-35=0, 



arising out of F-j{y)=0 on the rejection of the root y=0 and the substitu- 

 tion y- = t. When this cubic equation is solved by Horner's method, its 

 roots are 



a, 2= '90081 . . . , a22 = -54987 . . . , a32=-16471 . . . , 

 and so 



ai = -9491 . . . , a2=^7415 , . . , a3 = ^4068 . . . 



