POLARISATION OP LIGHT. 



refraction, of an opposite character, placed 

 at right angles to the other two, and 

 having the same intensity as either axis 

 singly. 



If there are three axes, two of which, 

 either both positive or both negative, are 

 of equal intensity and in the same plane, 

 while the third is at right angles to the 

 other, then the resultant of these three 

 axes will be a single axis coincident with 

 the latter axis. Thus, \njig. 33, if the 

 two equal axes are A, C, and the third 

 axis O, then since A = C 

 If their characters are + A+ C O o, we 

 shall have 



The single axis at O o, which we shall 

 call x 



x= -(Oo+A). 

 If their characters are A C O o, then 



^^-(Oo-A), if Oo>A 



a?=+(A-Oo), if Oo<A. 

 If their characters are + A + C + O o, then 



#= + (Oo-A), if Oo>A 



x= -(A Oo). 



If their characters are A C + O o, 

 then 



x=+(O +oA). 



If all the axes are equal, and have the 

 same signs ; that is, if A= C = O o, then 



# = 0. 



That is, the three equal axes destroy one 

 another, when they are all of the same 

 character. 



In the preceding combinations of axes 

 we have supposed two of them to have 

 the same intensity and the same character, 

 so that the resultant is a single axis, or 

 system of rings, in the direction of the 

 strongest ; but when the axes are three 

 in number, and the resultant is a double 

 system of rings, we must combine them 

 in a different manner. 



Let ABC, for example,^-. 34, be the 



the general law given in the preceding 

 chapter, we shall have the resulting tint: 

 a sin. " 



But in order to combine this tint, arising 

 from the united action of A and C, we 

 must know the direction of it. When we 

 consider that ^ is the double of the real 

 angle of the planes in which the forces 

 from A and C act, we shall find that 

 the direction of the new plane in which 

 A and C are united forms an angle with 

 the real direction of C, or the lesser force, 

 whose complement is 



or it forms with the real direction of A, 

 or the greater force, an angle, whose com- 

 plement is 



Hence it follows that, since the direction 

 of the resultant, in relation to C E, is 

 known, its direction in relation to B E, 

 or the force with which it is to be com- 

 bined, is also known ; and, using accented 

 letters to express the same parts of the 

 new parallelogram of forces, we shall have 

 a' sin, y 



three axes the resultant of which is re- 

 quired; then, if we combine A and C by 



In order to illustrate this in a simple 

 case, where the truth of the result will be 

 immediately recognised, we shall take 

 the case of three equal axes, where the 

 resultant of all the three is or zero. Let 

 E, Jig. 34, be the point of the sphere where 

 we require to know the tint produced by 

 the three equal axes A B C, and let 



AE = 70 AG= 66 44' 



B E = 60 CG=23 16 



EG = 30 CE=37 17 



E F 20 Sin. 2 A E = . 883 104= a 



Sin. 2 BE -.75000 = b = {fol 54' 



Sin. 2 CE = . 36694 = c I = 37 12 

 u =40' 41 a+c= 1.25004 

 * =77 52 a-c= 0.51616 



Hence, if we combine A and C, we shall 

 have 



T = .7500, 



which will be + or positive, because -^ is 

 greater than 180'. 



j, ? 

 Now, we have -^-+ -~ = 49 19', 



which gives 4C 21' for the direction of 

 the new plane in which the two forces, 

 emanating from A and C, produce the 



