892 Mr. T. Smith on the Accuracy of the 



be of the same sign for the whole range of values of m 

 for which the instrument could be used. This conflicts 

 with the fundamental principle on which errors should be 

 treated, which requires them to lie between limits on either 

 side of the approximate result and to reach these limits 

 at each end of the range as well as at the greatest possible 

 number of intermediate points. Any satisfactory discussion 

 must be based on the range of distances for which the 

 instrument is required to give good results. The limits 

 will be determined for near points by the distances it is 

 more convenient to measure with a tape, and for far points 

 by the length of the staff. Let p and q be the limiting' 

 magnifications, and P and Q the values of M resulting 

 from the substitution of p and q for m. As m varies con- 

 tinuously from p to q, M will continuously increase or 

 decrease from P to Q. New if a and b represent any 

 real quantities which at the moment are left arbitrary, 



2(A-M)(a+5)(oJP + &Q) 



= {a + b) 2 {A 2 -M 2 )-{( : a + h)A-a~P-bQ\ 2 



+ {(a + 5)M-aP-5Q} 2 . 



Also 



{ (a + &)M-dP-6QP = |^ {(6 2 -a 2 )M 2 + a 2 P 2 -6 2 Q 2 } 



+ 2j a4^P+ftQ) (M _ p)(M , Q)> 



Thus if, as in all the applications here considered, ctP + 6Q 

 has the same sign as (a-f-6)(P + Q), that is as (a + 6)/e', 

 the last term is negative and 



0<{(a + ^)M-«P-6Q} 2 ^p-^{(6 2 -a 2 )M 2 + a 2 P 2 -6 2 Q 2 t 



or 



{(a + 6)M-aP-£Q} 2 



= -2(l + ^A:A:'J^{(6 2 -a 2 )m + a 2 p-^} > 



where 



-1<:<9< i. 



This leads on substitution in (5) to 



d= _B-- + 0(l3-l\ .... (8) 

 m \ mj 



