of orthogonals at differential intervals is small 

 compared to the angle of refraction. (Please note 

 that convergence or divergence is not considered 

 non existent). 



5. That a line drawn through any point midway 

 between two contours and making equal angles 

 with the adjacent contours is closely the direction 

 of a level line at that point (provided the require- 

 ment implied in assumption No. 1 is accomplished). 



B. Derivation 



. A ^1 (ci— C2) • ^ i? Ac . 



.. Aa = ^ — -j- — sm a] or Aa = ^ — sm a 



J Ci-\-C2 J Care 



i, (c.) 



Figure 12. 



considering Aq:<13° for two-place 



accuracy, or Aa<6° for three-place 



accuracy, tan Aa=sin Aa=Aa= 



R2 — Ri 

 BB' 



AA'=m'=BB'=A (a differential distance). 

 Assumption No. 4. 



and: Aa=^^ 

 d 



let c' be the velocity from Ato B (effective) 



let c" be the velocity from A' to B' (effective) 



let t be the time required for wave front to move 



from AA' to BB' 



R^=c"t; Ri = c't; then Aa=^-^^~^ 



^" = c' 



C1—C2 J, ■ 



a sm a 



then Aa= 



J 



' J 



^ sin a; but f=-r 



R Ai . 



or Aa=-y J — sm a 



R 



but for the general case -y=sec a 



AZ 



therefore, Aa=y — tan a 



(1) 



(2) 



andc'=^ 



C. Utilization 



These two equations are thus seen to be inde- 

 pendent of the scale of the chart, and they have 

 been made independent of the wave period by the 

 reduction of this factor to the dimensionless ratio 

 AX 



Equation (1) is applicable to all cases where 

 A a is less than some predetermined limit depend- 

 ing upon the accuracy desired. In general, good 

 results are obtained when Aa is less than 13°. 

 In practice Aa rarely approaches this limit. 



Equation (2) is more readily applied under 

 ordinary conditions than is equation (1). The 

 limitation of equation (2) stems from the fact that 

 as a approaches 90°, tan a becomes infinite. 

 Physically this situation may occur but is instan- 

 taneously altered by refraction so that a becomes 

 less than 90°. However, the application of equa- 

 tion (2) normally necessitates crossing an entire 

 con torn- interval at each step. The value of 

 Aa changes very rapidly in the region when a 

 approaches 90°. Thus the instantaneous refrac- 

 tion over a small distance results in a great change 

 in the rate of refraction, and the interval must be 

 crossed in a series of shorter steps. It is there- 

 fore desirable to employ equation (1) whenever 

 a. exceeds about 80°. Equation (1) readily lends 

 itself to crossing a contour interval in partial steps 

 by the judicious selection of R (the distance of 

 wave advance). 



The protractor has thus been constructed with 

 graphs for equation (2) and a special table for use 

 when a exceeds 80°, which adapts equation (1) to 

 the graph of equation (2). 



The use of the graph suffices for the great 

 majority of diagrams and, for all ordinary cases, 

 there is never any necessity to refer to the table. 

 The table therefore has been made very simple. 

 The graph requires the measurement of one factor 

 only (a) and thus is much faster to use than the 

 table, which requires the measurement of three 

 factors (a, R and J). 



19 



