392 H. S. Uhler — deviation Produced by Prisms. 



[See prefatory remarks.] Apparently, the only other fact 

 which flows directly from (3) alone is this : if r) l is kept con- 

 stant while the azimuths are allowed to vary, JD and E will 

 decrease or increase together, so that if either passes through a 

 stationary value so also will the other pass through the same 

 kind of critical value simultaneously. 



In addition to the relations given above, the following equa- 

 tions will be found useful : 



sin rj 1 = rasin 77/ (4) 



sin y 1 = v sin y/ (5) 



siny 2 '= vsiny 2 (6) 



where v = n cos 77/ sec rj 1 = VW 2 4- (n*— 1) tan 2 ??,. 



Since, as will be shown later, the general deviation may be 

 expressed as an explicit function either of 7/ and 77/ or of 

 7 X and rj 1 (c or n and /3 being supposed given), two methods, 

 at least, for exhibiting all possible variations of D suggest 

 themselves. One plan would be to construct a surface outside 

 of the sphere of figure 1 which would be the locus of the 

 extremity of the extension either of the radius OS or of OP, 

 the length of the extended segment to be taken proportional 

 to the value of the deviation at the points S or F respectively. 

 This process would be analogous to the scheme often used for 

 showing the distribution of static electricity over the surface 

 of an insulated conductor. For various reasons the model 

 with extended radii would be inconvenient for publication. 

 The other method is to construct a surface with rectangular 

 coordinates, considering D as dependent variable, (axis of 

 2), with 7/ and 77/ or with 7, and ^ as independent variables, 

 (axes of x and y). In the following pages the properties of the 

 surfaces referred to rectangular frames will be discussed, and 

 diagrams of certain plane sections of these surfaces will be 

 given. 



Instead of deriving an explicit function of D in terms of 7/ 

 and 77/ it will be found preferable to replace the coordinate 7/ 

 by X, where \ = 7/ — -|/3. Furthermore, all square-root oper- 

 ators will be understood as meaning only the positive root. 

 From (2) 



cos J£=cos/3sin y l sin y,/ + sin /3 sin y 1 cosy/ — sin (3 cos y, sin y 2 ' 

 -f cos (3 cos y 1 cos y 2 '. 



From (5) 



siny^vsin (X + ^/3) ; 

 from (1) and (6) 



siny 2 '=vsin (X—i/3) 



